Hopf monoids in the category of species
Abstract.
A Hopf monoid (in Joyal’s category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a self-contained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We study several examples of this nature. We emphasize the central role played in the theory by the Tits algebra of set compositions. Its product is tightly knit with the Hopf monoid axioms, and its elements constitute universal operations on connected Hopf monoids. We study analogues of the classical Eulerian and Dynkin idempotents and discuss the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems for Hopf monoids.
Key words and phrases:
Species; Hopf monoid; Lie monoid; antipode; Hadamard product; partition; composition; linear order; simple graph; generalized permutahedron; Tits product; group-like; primitive; functional calculus; characteristic operation; Hopf power; Eulerian idempotent; Dynkin quasi-idempotent; Garsia-Reutenauer idempotent; Poincaré-Birkhoff-Witt; Cartier-Milnor-Moore2010 Mathematics Subject Classification:
Primary 16T30, 18D35, 20B30; Secondary 18D10, 20F55Introduction
The theory of species and its applications to enumerative combinatorics was developed by Joyal in [34] and further advanced in [12]. The basic theory of Hopf monoids in species was laid out in [6, Part II]. The account we provide here includes most of the topics discussed there and some more recent results from [1, 3, 4, 7], as well as a good number of new results and constructions.
The category of species carries a symmetric monoidal structure, the Cauchy product. We study the resulting algebraic structures, such as monoids, Hopf monoids, and Lie monoids. They are analogous to algebras, Hopf algebras, and Lie algebras (and their graded counterparts).
Species will be denoted by letters such as and , monoids by and , comonoids by and , bimonoids (and Hopf monoids) by and , and Lie monoids by .
A species consists of a family of vector spaces indexed by finite sets and equivariant with respect to bijections . A monoid is a species together with a family of product maps
indexed by pairs of disjoint sets whose union is . They are subject to the usual axioms (associativity and unitality). Iterations of these maps yield higher product maps
indexed by decompositions of (sequences of pairwise disjoint sets whose union is ). When decompositions and are related by refinement, the higher product maps and are related by associativity.
The notion of a comonoid in species is defined dually. The higher coproduct maps are again indexed by set decompositions. Decompositions of finite sets go on to play a defining role in the theory.
Given two decompositions and of the same set , there is a natural way to refine using in order to obtain another decomposition of . We refer to as the Tits product of and . This turns the set of all decompositions of into a monoid (in the usual sense of the term). We refer to the algebra of this monoid as the Tits algebra of decompositions of .
A bimonoid is a species which is at the same time a monoid and a comonoid, with the structures linked by the usual compatibility axiom. The Tits product appears implicitly in this axiom. This becomes evident when the compatibility among the higher product and coproduct maps is spelled out. The axiom then involves two arbitrary decompositions and of , and their Tits products and .
For any bimonoid , the set of all endomorphisms of species of is an algebra under convolution. If the identity of is invertible, then is a Hopf monoid and the inverse its antipode. There is a formula for the antipode in terms of the higher (co)product maps (in which decompositions necessarily enter).
Complete definitions of a bimonoid and Hopf monoid are given in Section 2. The discussion on higher (co)products is deferred to Section 10, with the groundwork on set decompositions and the Tits product laid out in Section 1.
The category of species supports other operations in addition to the Cauchy product and one such is the Hadamard product, discussed in Section 3. The corresponding internal Hom and duality functors are also studied. The Hadamard product preserves Hopf monoids. This is one aspect in which the theory of Hopf monoids differs from that of graded Hopf algebras.
We pay particular attention to set-theoretic Hopf monoids (in which the role of vector spaces is played by sets) and connected Hopf monoids (in which , the base field). These are discussed in Sections 4 and 5. The latter includes a discussion of canonical antipode formulas (Takeuchi’s and Milnor and Moore’s) and the formulation of the antipode problem, a question of a general nature and concrete combinatorial interest. The primitive part of a connected comonoid is also introduced at this point.
The free monoid on any species is studied in Section 6. We concentrate mainly on the case when is positive: . In this situation, the monoid is connected and carries a canonical Hopf monoid structure. Corresponding discussions for commutative monoids and Lie monoids are given in Sections 7 and 8. The free commutative monoid on a species is denoted and the universal enveloping monoid of a Lie monoid by . They are Hopf monoids. Theorem 20 proves that the Hadamard product of two connected Hopf monoids is free, provided one of the factors is free. In the case when both factors are free, Theorem 19 provides a basis for the Hadamard product in terms of the bases of the factors. Similar results hold in the commutative case.
Many examples of Hopf monoids are given in Section 9. Most of them are connected and at least part of their structure is set-theoretic. They arise from various combinatorial structures ranging from linear orders to generalized permutahedra, which then get organized into Hopf monoids and related to one another by means of morphisms of such. Solutions to the antipode problem are provided in most cases. Among these, we highlight those in Theorems 36 and 44.
Two additional important examples are studied in Section 11. The bimonoid has a linear basis indexed by set decompositions . It is not connected and in fact not a Hopf monoid. Its companion has a linear basis indexed by set compositions (decompositions into nonempty sets) and is a connected Hopf monoid. They are both free. There is a surjective morphism of Hopf monoids which removes the empty blocks from a decomposition. Each space and is an algebra under the Tits product, and the interplay between this and the Hopf monoid structure is central to the theory. This important point was made in [5, 6, 54], and is discussed extensively here.
To a species one may associate a vector space of series, studied in Section 12. A series of is a collection of elements , equivariant with respect to bijections . If is a monoid, then is an algebra, and formal functional calculus can be used to substitute any series of with into a formal power series to obtain another series of . In particular, one can define and , the exponential and logarithm of , and these two operations are inverses of each other in their domains of definition. If is a connected bimonoid, they restrict to inverse bijections
between primitive series and group-like series of . The former constitute a Lie algebra and the latter a group.
The characteristic operations are discussed in Section 13. They are analogous to the familiar Hopf powers from the theory of Hopf algebras. The role of the exponent is played by a decomposition , with the Tits monoid replacing the monoid of natural numbers. For any decomposition of , we refer to the composite
as the characteristic operation of on , where is a bimonoid. Extending by linearity, each element of induces a linear endomorphism of , and thus a series of induces a species endomorphism of . If is connected, then the characteristic operations factor through , and a series of induces a species endomorphism of . Among the most important results in this section we mention Theorems 82 and 90. The former states that, when the bimonoid is cocommutative, the characteristic operations endow each space with a left -module structure, and similarly for if is connected. Under the same hypotheses, the latter theorem states that the operation of a primitive element maps to . Moreover, if the coefficient of in is , then is an idempotent element of the Tits algebra , and its operation is a projection onto . The basis element of is indexed by the composition with one block. The characteristic operations of some important series of on connected bimonoids are given in Table 1.
Section 14 deals with the analogues of certain classical idempotents. They are elements (or series) of . For each integer , the Eulerian idempotent is a certain series of , with being primitive. The latter is the logarithm of the group-like series defined by . Since operates as the identity, operates as its logarithm. It follows that the logarithm of the identity projects any cocommutative connected bimonoid onto its primitive part. This result is obtained both in Corollary 80 and in Corollary 101. Another important primitive series of is , the Dynkin quasi-idempotent. It operates (on products of primitive elements) as the left bracketing operator. The analogue of the classical Dynkin-Specht-Wever theorem is given in Corollary 114. This section also discusses the Garsia-Reutenauer idempotents, reviewed below.
| Series of | Characteristic operation on | |
| Corollary 101 | ||
| Proposition 103 | ||
| Formula (308) | ||
| (antipode) | Formula (313) | |
| Corollary 110 |
The Poincaré-Birkhoff-Witt (PBW) and Cartier-Milnor-Moore (CMM) theorems for Hopf monoids in species are discussed in Section 15. It is a nontrivial fact that any Lie monoid embeds in . This is a part of the PBW theorem; the full result is given in Theorem 119. In a bimonoid , the commutator preserves the primitive part . The CMM theorem (Theorem 120) states that the functors
form an adjoint equivalence.
The structure of the Tits algebra has been studied in detail in the literature. It follows from works of Bidigare [13] and Brown [17] that admits a complete family of orthogonal idempotents indexed by partitions of . In fact there is a canonical choice for such a family, which we call the Garsia-Reutenauer idempotents (Theorem 102). The action of these idempotents provides a canonical decomposition of each space (Theorem 117). The Garsia-Reutenauer idempotent indexed by the partition with one block is the first Eulerian idempotent, hence its action is a projection onto the primitive part. This leads to the important result that for any cocommutative connected Hopf monoid , there is an isomorphism of comonoids between and (Theorem 118). This is closely related to (and in fact used to deduce) the PBW and CMM theorems. PBW states that as comonoids, and CMM includes the statement that as bimonoids.
Section 16 collects a number of results on the dimension sequence of a finite-dimensional connected Hopf monoid . The structure imposes conditions on this sequence in the form of various polynomial inequalities its entries must satisfy. For example, Theorem 122 states that the Boolean transform of the sequence must be nonnegative, and Theorem 124 that if the Hopf monoid is set-theoretic (and nontrivial), then the same is true for the binomial transform. The latter result follows from Theorem 15, the analogue of Lagrange’s theorem for Hopf monoids.
Among the topics discussed in [6, Part II] but left out here, we mention the notion of cohomology and deformations for set-theoretic comonoids, the construction of the cofree comonoid on a positive species, the construction of the free Hopf monoid on a positive comonoid, the notion of species with balanced operators, and the geometric perspective that adopts the braid hyperplane arrangement as a central object. The latter is particularly important in regard to generalizations of the theory which we are currently undertaking.
The notion of Hopf monoid parallels the more familiar one of graded Hopf algebra and the analogy manifests itself throughout our discussion. Much of the theory we develop has a counterpart for graded Hopf algebras, with Hopf monoids being richer due to the underlying species structure. To illustrate this point, consider the characteristic operations. For connected Hopf monoids, they are indexed by elements of the Tits algebra, while for connected graded Hopf algebras they are indexed by elements of Solomon’s descent algebra. The symmetric group acts on the former, and the latter is the invariant subalgebra. In some instances, the theory for graded Hopf algebras is not widely available in the literature. We develop the theory for Hopf monoids only, but hope that this paper is useful to readers interested in Hopf algebras as well. Some constructions are specific to the setting of Hopf monoids and others acquire special form. There are ways to build graded Hopf algebras from Hopf monoids. These are studied in [6, Part III], but not in this paper.
Acknowledgements
We warmly thank the editors for their interest in our work and for their support during the preparation of the manuscript. We are equally grateful to the referees for a prompt and detailed report with useful comments and suggestions.
1. Preliminaries on compositions and partitions
Decompositions of sets play a prominent role in the theory of Hopf monoids. Important properties and a good amount of related notation are discussed here. In spite of this, it is possible and probably advisable to proceed with the later sections first and refer to the present one as needed.
1.1. Set compositions
Let be a finite set. A composition of is a finite sequence of disjoint nonempty subsets of such that
The subsets are the blocks of the composition. We write to indicate that is a composition of .
There is only one composition of the empty set (with no blocks).
Let denote the set of compositions of .
1.2. Operations
Let be a composition of .
The opposite of is the composition
Given a subset of , let be the subsequence of consisting of those indices for which . The restriction of to is the composition of defined by
Let be the complement of in . We say that is -admissible if for each , either
Thus is -admissible if and only if is, and in this case, and are complementary subsequences of .
We write
| (1) |
to indicate that is an ordered pair of complementary subsets of , as above. Either subset or may be empty.
Given and compositions of and of , their concatenation
is a composition of . A quasi-shuffle of and is a composition of such that and . In particular, each block of is either a block of , a block of , or a union of a block of and a block of .
1.3. The Schubert cocycle and distance
Let and . The Schubert cocycle is defined by
| (2) |
Alternatively,
Let be another composition of . The distance between and is
| (3) |
In the special case when and consist of the same blocks (possibly listed in different orders), the previous formula simplifies to
| (4) |
where the sum is over those pairs such that and appears after in . In particular,
| (5) |
Note also that
where .
1.4. Linear orders
When the blocks are singletons, a composition of amounts to a linear order on . We write for the linear order on for which .
Linear orders are closed under opposition, restriction, and concatenation. The opposite of is . If and , their concatenation is
The restriction of a linear order on to is the list consisting of the elements of written in the order in which they appear in . We say that is a shuffle of and if and .
1.5. Refinement
The set is a partial order under refinement: we say that refines and write if each block of is obtained by merging a number of contiguous blocks of . The composition is the unique minimum element, and linear orders are the maximal elements.
Let . There is an order-preserving bijection
| (8) |
The inverse is given by concatenation:
Set compositions of are in bijection with flags of subsets of via
Refinement of compositions corresponds to inclusion of flags. In this manner is a lower set of the Boolean poset , and hence a meet-semilattice.
1.6. The Tits product
Let and be two compositions of . Consider the pairwise intersections
for , . A schematic picture is shown below.
| (9) |
The Tits product is the composition obtained by listing the nonempty intersections in lexicographic order of the indices :
| (10) |
where it is understood that empty intersections are removed.
The Tits product is associative and unital; it turns the set into an ordinary monoid, that we call the Tits monoid. The unit is . If , it is not commutative. In fact,
The following properties hold, for all compositions and .
-
•
.
-
•
.
-
•
If , then .
-
•
If is a linear order, then and is a linear order.
-
•
and .
-
•
.
The last property makes the monoid a left regular band. Additional properties are given in [6, Proposition 10.1].
Remark.
The product of set compositions may be seen as a special instance of an operation that is defined in more general settings. It was introduced by Tits in the context of Coxeter groups and buildings [74, Section 2.30]. Bland considered it in the context of oriented matroids [16, Section 5, page 62]. It appears in the book by Orlik and Terao on hyperplane arrangements [48, Definition 2.21 and Proposition 2.22]. A good self-contained account is given by Brown [17, Appendices A and B]. The case of set compositions arises from the symmetric group (the Coxeter group of type A), or from the braid hyperplane arrangement.
1.7. Length and factorial
Let be a composition of . The length of is its number of blocks:
| (11) |
and the factorial is
| (12) |
The latter counts the number of ways of endowing each block of with a linear order.
Given a set composition refining , let
| (13) |
where is the number of blocks of that refine the -th block of . Let also
| (14) |
In particular, and if is a linear order, then .
1.8. Set partitions
A partition of is an unordered collection of disjoint nonempty subsets of such that
The subsets of which belong to are the blocks of . We write to indicate that is a partition of .
There is only one partition of the empty set (with no blocks).
Let denote the set of partitions of .
Given and partitions of and of , their union is the partition of whose blocks belong to either or .
Restriction, quasi-shuffles and admissible subsets are defined for set partitions as for set compositions (Section 1.2), disregarding the order among the blocks.
Refinement is defined for set partitions as well: we set if each block of is obtained by merging a number of blocks of . The partition is the unique minimum element and the partition of into singletons is the unique maximum element. We denote it by . The poset is a lattice.
Lengths and factorials for set partitions are defined as for set compositions:
Here is the number of blocks of that refine the block of . In particular, and .
The cyclic factorial of is
| (15) |
It counts the number of ways of endowing each block of with a cyclic order.
The Möbius function of satisfies
| (16) |
for , with as above. In particular,
| (17) |
1.9. Support
The support of a composition of is the partition of obtained by forgetting the order among the blocks: if , then
The support preserves lengths and factorials:
If refines , then refines . Thus, the map is order-preserving. Meets are not preserved; for example, but . The support turns the Tits product into the join:
We have
| (18) |
Both conditions express the fact that each block of is contained in a block of .
1.10. The braid arrangement
Compositions of are in bijection with faces of the braid hyperplane arrangement in . Partitions of are in bijection with flats. Refinement of compositions corresponds to inclusion of faces, meet to intersection, linear orders to chambers (top-dimensional faces), and to the central face. When and are nonempty, the statistic counts the number of hyperplanes that separate the face from . The factorial is the number of chambers that contain the face . The Tits product is, in a sense, the face containing that is closest to .
This geometric perspective is expanded in [6, Chapter 10]. It is also the departing point of far-reaching generalizations of the notions studied in this paper. We intend to make this the subject of future work.
1.11. Decompositions
A decomposition of a finite set is a finite sequence of disjoint subsets of whose union is . In this situation, we write
A composition is thus a decomposition in which each subset is nonempty.
Let denote the set of decompositions of . If , there are decompositions into subsets. Therefore, the set is countably infinite.
With some care, most of the preceding considerations on compositions extend to decompositions. For , decompositions of may be concatenated with decompositions of . Let be a decomposition of . The restriction to is the decomposition
(the empty intersections are kept). The set is a monoid under a product defined as in (10) (where now all intersections are kept). The product is associative and unital. We call it the Tits product on decompositions.
Consider the special case in which is empty. There is one decomposition
of the empty set for each nonnegative integer , with corresponding to the unique decomposition with no subsets. We denote it by . Under concatenation,
and under the Tits product
Thus,
| (19) |
with concatenation corresponding to addition of nonnegative integers and the Tits product to multiplication.
Given a decomposition , let denote the composition of obtained by removing those subsets which are empty. This operation preserves concatenations, restrictions, and Tits products:
| (20) |
The notion of distance makes sense for decompositions. Since empty subsets do not contribute to (3), we have
| (21) |
The notion of refinement for decompositions deserves special attention. Let be a decomposition of a (possibly empty) finite set . We say that another decomposition of refines , and write , if there exists a sequence , with a decomposition of , , such that (concatenation). In this situation, we also say that is a splitting of the pair .
If , then must be the empty sequence, and hence . Thus, the only decomposition that refines is itself.
If and are compositions with , then has a unique splitting, in view of (8). In general, however, the factors cannot be determined from the pair , and thus the splitting is not unique. For example, if
we may choose either
to witness that . Note also that (for we may choose ) and in fact
for any and .
The preceding also shows that refinement is not an antisymmetric relation. It is reflexive and transitive, though, and thus a preorder on each set . We have, for ,
Concatenation defines an order-preserving map
| (22) |
According to the preceding discussion, the map is surjective but not injective. A sequence in the fiber of is a splitting of .
2. Species and Hopf monoids
Joyal’s category of species [34] provides the context for our work. The Cauchy product furnishes it with a braided monoidal structure. We are interested in the resulting algebraic structures, particularly that of a Hopf monoid. This section presents the basic definitions and describes these structures in concrete terms.
2.1. Species
Let denote the category whose objects are finite sets and whose morphisms are bijections. Let be a field and let denote the category whose objects are vector spaces over and whose morphisms are linear maps.
A (vector) species is a functor
Given a species , its value on a finite set is denoted by . Its value on a bijection is denoted
We write for the space . The symmetric group acts on by
for , .
A morphism between species is a natural transformation of functors. Let be a morphism of species. It consists of a collection of linear maps
one for each finite set , such that the diagram
| (23) |
commutes, for each bijection . Let denote the category of species.
A species is said to be finite-dimensional if each vector space is finite-dimensional. We do not impose this condition, although most examples of species we consider are finite-dimensional.
A species is positive if . The positive part of a species is the (positive) species defined by
Given a vector space , let denote the species defined by
| (24) |
2.2. The Cauchy product
Given species and , their Cauchy product is the species defined on a finite set by
| (25) |
The direct sum is over all decompositions of , or equivalently over all subsets of . On a bijection , the map is defined to be the direct sum of the maps
over all decompositions of , where denotes the restriction of to .
The Cauchy product turns into a monoidal category. The unit object is the species as in (24).
Let be a fixed scalar, possibly zero. Consider the natural transformation
which on a finite set is the direct sum of the maps
| (26) |
over all decompositions of . The notation stands for the cardinality of the set .
If is nonzero, then is a (strong) braiding for the monoidal category . In this case, the inverse braiding is , and is a symmetry if and only if . The natural transformation is a lax braiding for .
We consider monoids and comonoids in the monoidal category , and bimonoids and Hopf monoids in the braided monoidal category . We refer to the latter as -bimonoids and -Hopf monoids. When , we speak simply of bimonoids and Hopf monoids. We expand on these notions in the following sections.
2.3. Monoids
The structure of a monoid consists of morphisms of species
subject to the familiar associative and unit axioms. In view of (25), the morphism consists of a collection of linear maps
one for each finite set and each decomposition of . The unit reduces to a linear map
The collection must satisfy the following naturality condition. For each bijection , the diagram
| (27) |
commutes.
The associative axiom states that for each decomposition , the diagram
| (33) |
commutes.
The unit axiom states that for each finite set , the diagrams
| (44) |
commute.
We refer to the maps as the product maps of the monoid . The following is a consequence of associativity. For any decomposition with , there is a unique map
| (45) |
obtained by iterating the product maps in any meaningful way. We refer to (45) as the higher product maps of . We extend the notation to the cases and . In the first case, the only subset in the decomposition is itself, and we let the map (45) be the identity of . In the second, is necessarily and the decomposition is ; we let be the unit map . Thus, the collection of higher product maps contains among others the product maps as well as the unit map .
The Cauchy product of two monoids and is again a monoid. The product
| (46) |
is the sum of the following maps
over all and .
If is a monoid, then is an algebra with product and unit .
2.4. Comonoids
Dually, the structure of a comonoid consists of linear maps
subject to the coassociative and counit axioms, plus naturality. Given a decomposition , there is a unique map
| (47) |
obtained by iterating the coproduct maps . For this map is defined to be the identity of , and for to be the counit map .
Comonoids are closed under the Cauchy product. If is a comonoid, then is a coalgebra.
2.5. Commutative monoids
A monoid is -commutative if the diagram
| (48) |
commutes for all decompositions . When , we speak simply of commutative monoids. When , -commutative monoids are rare: it follows from (48) that if , then .
If is -commutative, then its product is a morphism of monoids.
A comonoid is -cocommutative if the dual to diagram (48) commutes.
2.6. Modules, ideals, and quotients
Let be a monoid. A left -module is a species with a structure map
which is associative and unital. When made explicit in terms of the components
The free left -module on a species is with structure map
Submonoids, (left, right, and two-sided) ideals, and quotients of can be defined similarly. Every monoid has a largest commutative quotient . Depending on the context, we refer to either or the canonical quotient map as the abelianization of .
The dual notions exist for comonoids. In particular, every comonoid has a largest cocommutative subcomonoid, its coabelianization.
2.7. The convolution algebra
The set of morphisms of species is a vector space over under
for and .
Assume now that is a monoid and is a comonoid. The space is then an algebra over under the convolution product:
| (49) |
The unit is the morphism defined by . Explicitly,
| (50) |
If is a morphism of monoids and is a morphism of comonoids, then
| (51) |
is a morphism of algebras.
Let be a monoid and a comonoid, for . If are morphisms of species, then
| (52) |
The convolution product on the left is in ; those on the right in , . Here we employ the case of the Cauchy product of (co)monoids (46).
2.8. Bimonoids and Hopf monoids
A -bimonoid is at the same time a monoid and a comonoid with the two structures linked by axioms (54)–(66) below. They express the requirement that
are morphisms of comonoids, or equivalently that
are morphisms of monoids.
Let be two decompositions of a finite set and consider the resulting pairwise intersections:
as illustrated below.
| (53) |
The compatibility axiom for -bimonoids states that diagrams (54)–(66) commute.
| (54) |
| (65) |
| (66) |
If is a -bimonoid, then is an ordinary bialgebra with structure maps , , and .
A -Hopf monoid is a -bimonoid along with a morphism of species (the antipode) which is the inverse of the identity map in the convolution algebra of Section 2.7. This requires the existence of an antipode
for the bialgebra , and of a linear map
for each nonempty finite set such that
| (67) |
Proposition 1.
Let be a -bimonoid.
-
(i)
Suppose is a -Hopf monoid with antipode . Then is a Hopf algebra with antipode .
-
(ii)
Suppose is a Hopf algebra and let denote its antipode. Then is a -Hopf monoid with antipode given by
and
(68) for each nonempty finite set .
The sum is over all compositions of . A proof of Proposition 1 is given in [6, Proposition 8.10], when . The same argument yields the general case.
Thus, a -Hopf monoid is equivalent to a -bimonoid for which is a Hopf algebra.
Proposition 2.
Let be a -Hopf monoid with antipode and . Then the diagram
| (69) |
commutes.
Thus, the antipode reverses products (the reversal involves the braiding ). Similarly, it reverses coproducts. These are general results for Hopf monoids in braided monoidal categories (see for instance [6, Proposition 1.22]) and hence they apply to -Hopf monoids. Other such results [6, Section 1.2] yield the following.
-
•
If is a -(co)commutative -Hopf monoid, then its antipode is an involution: .
-
•
Let and be -Hopf monoids. A morphism of -bimonoids necessarily commutes with the antipodes, and is thus a morphism of -Hopf monoids.
-
•
Let be a -Hopf monoid and a monoid. If is a morphism of monoids, then it is invertible under convolution and its inverse is . If is commutative, then the set of morphisms of monoids from to is a group under convolution.
-
•
The dual statement for comonoid morphisms holds. The inverse of is .
A result of Schauenburg [64, Corollary 5] implies the following, confirming the earlier observation that -(co)commutativity is rare when .
-
•
If is a -(co)commutative -Hopf monoid and , then .
2.9. Lie monoids
We consider Lie monoids in the symmetric monoidal category . A Lie monoid structure on a species consists of a morphism of species
subject to antisymmetry and the Jacobi identity. The morphism consists of a collection of linear maps
one for each finite set and each decomposition , satisfying the naturality condition (27) (with replacing ). We refer to as the Lie bracket of and .
The antisymmetry relation states that
| (70) |
for any decomposition , , and .
The Jacobi identity states that
| (71) |
for any decomposition , , , and .
Every monoid gives rise to a Lie monoid with Lie bracket defined by
| (72) |
We refer to (72) as the commutator bracket. This does not require the monoid to possess a unit. If is commutative, then the commutator bracket is zero.
2.10. Algebras as monoids
Let be a vector space over and the species defined in (24). If is an ordinary algebra with unit and product , then is a monoid with , and all other . Moreover, as algebras.
In the same manner, if is a coalgebra, bialgebra, Hopf algebra, or Lie algebra, then is a comonoid, bimonoid, Hopf monoid, or Lie monoid.
There is another way to associate a species to a vector space . Define by
| (73) |
for every finite set . On a bijection , is the identity of . If is an algebra over , then is a monoid with and for all . If is a coalgebra, bialgebra, Hopf algebra, or Lie algebra, then is a comonoid, bimonoid, Hopf monoid, or Lie monoid. When is a Hopf algebra with antipode , the antipode of is simply . This follows from either (67) or (68).
3. The Hadamard product
Multiplying species term by term yields the Hadamard product. The possibility of building Hopf monoids by means of this operation is an important feature of the category of species. We study this and related constructions in this section.
3.1. Species under Hadamard product
The Hadamard product of two species and is the species defined on a finite set by
and on bijections similarly. This operation turns into a symmetric monoidal category. The unit object is the exponential species defined by
| (74) |
for all . The symmetry is simply
(Comparing with (73), we see that . In particular, is a Hopf monoid. Its structure is further studied in Section 9.1.)
The Hadamard product of two monoids and is again a monoid. The product of is defined by
where the first map on the bottom simply switches the middle two tensor factors. If and are commutative, then so is . Similar statements hold for comonoids.
Proposition 3.
Let be arbitrary scalars. If is a -bimonoid and is a -bimonoid, then is a -bimonoid.
In particular, if and are bimonoids (), then so is .
A proof of Proposition 3 is given in [6, Corollary 9.6]. The analogous statement for Hopf monoids holds as well, in view of item (ii) in Proposition 1 and the fact that is the tensor product of the Hopf algebras and . However, there is no simple formula expressing the antipode of in terms of those of and .
3.2. Internal Hom
Given species and , let be the species defined by
The latter is the space of all linear maps from the space to the space . If is a bijection and , then
is defined as the composition
| (75) |
Note that there is a canonical isomorphism
For species , and , there is a natural isomorphism
| (76) |
This says that the functor is the internal Hom in the symmetric monoidal category of species under Hadamard product. There is a canonical morphism
| (77) |
it is an isomorphism if and are finite-dimensional.
We let
Let be a comonoid and a monoid. Then is a monoid as follows. Given and , their product
is the composition
| (78) |
The unit map sends to the composition
Associativity and unitality for follow from (co)associativity and (co)unitality of (and ).
There is a connection between (78) and the convolution product (49); this is explained in Section 12.11.
Given , let be the composition
| (79) |
Assume now that is finite-dimensional. We then have a canonical isomorphism
Let
be the preimage of under this isomorphism. Also, if , let be the preimage of the composition
under the canonical isomorphism
With these definitions, is a comonoid.
Now let be a -bimonoid and a -bimonoid. Suppose is finite-dimensional. Combining the preceding constructions yields a -bimonoid structure on . In particular, for any -bimonoid , we obtain a -bimonoid structure on .
The analogous statements for Hopf monoids hold.
3.3. Duality
The dual of a species is the species defined by
Equivalently,
Since is a Hopf monoid, the considerations of Section 3.2 apply. Thus, the dual of a comonoid is a monoid with product
where the first arrow is canonical. The dual of a finite-dimensional monoid (-bimonoid, -Hopf monoid) is a comonoid (-bimonoid, -Hopf monoid).
A -bimonoid (or -Hopf monoid) is called self-dual if as -bimonoids. In general, such an isomorphism is not unique.
There are canonical morphisms of species
which are isomorphisms if either or is finite-dimensional. These are special cases of (77). These maps preserve the (co)monoid structures discussed in Sections 3.1 and 3.2 (when present). In particular, if is a -bimonoid and a -bimonoid, the map
is a morphism of -bimonoids. If is finite-dimensional, the isomorphism
implies that the -bimonoid is self-dual.
4. Set species and set-theoretic Hopf monoids
Many naturally-occurring Hopf monoids have a canonical basis indexed by combinatorial objects. Let be such a Hopf monoid and suppose that the product and coproduct maps of preserve the canonical basis. In this situation, the structure of can be described in a purely set-theoretic manner, leading to the notion of a set-theoretic Hopf monoid. We discuss these objects and explain how they relate to Hopf monoids via linearization.
4.1. Set species
Let denote the category whose objects are arbitrary sets and whose morphisms are arbitrary functions. A set species is a functor
where the category is as in Section 2.1. A morphism between set species is a natural transformation. Let denote the category of set species.
A set species is finite if the set is finite for each finite set . It is positive if .
The Cauchy product of set species and is the set species defined by
| (80) |
where denotes the cartesian product of sets. The Cauchy product turns into a monoidal category. The unit object is the set species given by
The set is a singleton.
The natural transformation obtained by interchanging the factors in the cartesian product is a symmetry.
4.2. Set-theoretic monoids and comonoids
Monoids in can be described in terms similar to those in Section 2.3: the structure involves a collection of maps
one for each decomposition , and a map , subject to axioms analogous to (33) and (44). We refer to these objects as set-theoretic monoids.
Given and , let
| (81) |
denote the image of under . Also, let denote the image of under . The axioms for a set-theoretic monoid then acquire the familiar form
| (82) |
for all decompositions and , , , and
| (83) |
for all . In particular, is an ordinary monoid.
A set-theoretic monoid is commutative if
for all , , and .
The situation for comonoids is different. The existence of the counit forces a comonoid in to be concentrated on the empty set. Indeed, a morphism entails maps , and hence we must have for all nonempty .
There is, nevertheless, a meaningful notion of set-theoretic comonoid. It consists, by definition, of a set species together with a collection of maps
one for each , subject to the coassociative and counit axioms. The former states that for each decomposition , the diagram
| (89) |
commutes. The latter states that for each finite set , the diagrams
| (100) |
commute, where denotes the unique map to the singleton .
A set-theoretic comonoid is cocommutative if for each , the diagram
commutes.
Given , let
| (101) |
denote the image of under . Thus, while . We think intuitively of as restricting the structure from to , and of as contracting or moding out from (so that the result is a structure on ).
The axioms for a set-theoretic comonoid may then be reformulated as follows. Coassociativity states that
| (102) |
for any decomposition and . Counitality states that
| (103) |
for any .
In particular, it follows from (103) that for we have .
A set-theoretic comonoid is cocommutative if and only if
for every and .
Proposition 4.
There is an equivalence between the category of cocommutative set-theoretic comonoids and the category of presheafs on the category of finite sets and injections.
4.3. Set-theoretic bimonoids and Hopf monoids
A set-theoretic bimonoid is, by definition, a set-theoretic monoid and comonoid such that the diagram
| (104) |
commutes for every choice of sets as in (53). The top map simply interchanges the two middle terms. The condition may be reformulated as follows:
| (105) |
for all and . Note that the set-theoretic analogues of (65) and (66) hold automatically.
A set-theoretic Hopf monoid is a set-theoretic bimonoid such that the monoid is a group. Its antipode is the map
| (106) |
4.4. The Hadamard product
The Hadamard product of two set species and is the set species defined by
where on the right-hand side denotes the cartesian product of sets.
Set-theoretic monoids are preserved under Hadamard products. The same is true of set-theoretic comonoids, bimonoids and Hopf monoids.
4.5. Linearization
Given a set , let
denote the vector space with basis , and the vector space of scalar functions on , respectively. We have .
Convention 5.
We use
to denote, respectively, the canonical basis of and the basis of of characteristic functions (). Thus, and are dual bases.
The linearization functors
are covariant and contravariant, respectively. Composing a set species with the linearization functors we obtain vector species and .
Linearization transforms cartesian products into tensor products. Therefore, if is a set-theoretic monoid, then the species is a monoid whose product is the linear map
for every and . If in addition is finite, then the species is a comonoid whose coproduct is the linear map
for every . If is a set-theoretic comonoid, then is a comonoid whose coproduct is the linear map
and is a monoid whose product is the linear map
Similar remarks apply to bimonoids and Hopf monoids. If is a Hopf monoid in set species, then is a group algebra, and the antipode of exists by item (ii) in Proposition 1. (Co)commutativity and Hadamard products are also preserved under linearization.
5. Connected Hopf monoids
The study of a Hopf monoid necessarily involves that of the Hopf algebra . A special yet nontrivial class of Hopf monoids consists of those for which this Hopf algebra is isomorphic to , the simplest Hopf algebra. These are the connected Hopf monoids.
5.1. Connected species and Hopf monoids
A species is connected if
A set species is connected if is a singleton. In this case, the species is connected.
Connectedess is preserved under duality and Hadamard products.
In a connected monoid, the map is an isomorphism , and by (44) the resulting maps
are identities. Thus, to provide a monoid structure on a connected species it suffices to specify the maps when and are nonempty.
A dual remark applies to connected comonoids . It can be expressed as follows:
| (107) |
where denotes the element of such that .
Proposition 6.
A connected -bimonoid is necessarily a -Hopf monoid.
This follows from item (ii) in Proposition 1. The special consideration of formulas (67) and (68) in the connected situation is the subject of Sections 5.2 and 5.3 below.
Proposition 7.
Let be a connected -bimonoid. Let , and . Then
| (108) |
5.2. Milnor and Moore’s antipode formulas
Suppose is a connected -bimonoid. Define maps and by induction on the finite set as follows. Let
be the identity of , and for nonempty ,
| (109) | ||||
| (110) |
Proposition 8.
We have that
and this morphism is the antipode of .
5.3. Takeuchi’s antipode formula
Takeuchi’s formula expresses the antipode in terms of the higher products and coproducts:
Proposition 9.
Let be a connected -Hopf monoid with antipode . Then
| (111) |
for any nonempty finite set .
The sum runs over all compositions of and the maps and are as in (45) and (47). Proposition 9 may be deduced from either Proposition 1 or 8, or by calculating the inverse of in the convolution algebra as
For more details, see [6, Proposition 8.13]. There is an analogous formula for the antipode of a connected Hopf algebra due to Takeuchi; see the proof of [72, Lemma 14] or [45, Lemma 5.2.10].
5.4. The antipode problem
Cancellations frequently take place in Takeuchi’s formula (111). Understanding these cancellations is often a challenging combinatorial problem. The antipode problem asks for an explicit, cancellation-free, formula for the antipode of a given -Hopf monoid . The problem may be formulated as follows: given a linear basis of the vector space , we search for the structure constants of on this basis. If is the linearization of a set-theoretic Hopf monoid , then we may consider the basis of . In this or other cases, we may also be interested in other linear bases of , and the corresponding structure constants.
5.5. The primitive part
Let be a connected comonoid. The primitive part of is the positive species defined by
| (112) |
for each nonempty finite set . An element is a primitive element of .
Proposition 10.
We have
| (113) |
The intersection is over all compositions of with more than one block and the map as in (47). The result follows from (112) by coassociativity.
Let be another connected comonoid and consider the Hadamard product (Section 3.1).
Proposition 11.
We have
| (114) |
This follows since the coproduct of is the tensor product of the coproducts of and .
On primitive elements, the antipode is negation:
Proposition 12.
Let be a connected -Hopf monoid and a primitive element of . Then
| (115) |
This follows from either of Milnor and Moore’s formulas (109) or (110), or also from (113) and Takeuchi’s formula (111).
Proposition 13.
Let be a connected bimonoid. Then is a Lie submonoid of under the commutator bracket (72).
Proof.
5.6. The indecomposable quotient
Let be a connected monoid. The indecomposable quotient of is the positive species defined by
| (116) | ||||
Primitives and indecomposables are related by duality: , and if is finite-dimensional, then .
Assume is finite-dimensional and let be a connected comonoid. Consider the internal Hom with the comonoid structure discussed in Section 3.2.
Proposition 14.
We have
5.7. The Lagrange theorem
Let be a monoid. We consider modules and ideals as in Section 2.6. Given a submonoid of , let denote the right ideal of generated by the positive part of .
Theorem 15.
Let be a connected Hopf monoid and a Hopf submonoid. Then there is an isomorphism of left -modules
In particular, is free as a left -module.
Theorem 15 is proven in [4, Theorem 2.2]. Similar results for ordinary Hopf algebras are well-known [58, Section 9.3]; these include the familiar theorem of Lagrange from basic group theory.
We remark that versions of this result should exist for certain nonconnected Hopf monoids, as they do for Hopf algebras, but we have not pursued this possibility.
6. The free monoid
We review the explicit construction of the free monoid on a positive species, following [6, Section 11.2]. The free monoid carries a canonical structure of -Hopf monoid. We briefly mention the free monoid on an arbitrary species.
There is a companion notion of cofree comonoid on a positive species which is discussed in [6, Section 11.4], but not in this paper.
6.1. The free monoid on a positive species
Given a positive species and a composition of , write
| (117) |
When is the unique composition of the empty set, we set
| (118) |
Define a new species by
A bijection transports a composition of into a composition of . The map
is the direct sum of the maps
In view of (118), the species is connected.
Every nonempty set admits a unique composition with one block; namely, . In this case, . This yields an embedding and thus an embedding of species
On the empty set, is (necessarily) zero.
Given and compositions and , there is a canonical isomorphism
obtained by concatenating the factors in (117). The sum of these over all and yields a map
This turns into a monoid. In fact, is the free monoid on the positive species , in view of the following result (a slight reformulation of [6, Theorem 11.4]).
Theorem 16.
Let be a monoid, a positive species, and a morphism of species. Then there exists a unique morphism of monoids such that
commutes.
The map is as follows. On the empty set, it is the unit map of :
On a nonempty set , it is the sum of the maps
where the higher product is as in (45).
When there is given an isomorphism of monoids , we say that the positive species is a basis of the (free) monoid .
6.2. The free monoid as a Hopf monoid
Let and a positive species. The species admits a canonical -Hopf monoid structure, which we denote by , as follows.
As monoids, . In particular, and are the same species. The comonoid structure depends on . Given , the coproduct
is the sum of the maps
Here , for each , and is as in (2). In the admissible case, we have written and .
The preceding turns into a -bimonoid. Since it is connected, it is a -Hopf monoid. We have
| (119) |
more precisely, maps into the primitive part under the embedding . When , is cocommutative, and is the Lie submonoid of generated by , see Corollaries 115 and 121.
We return to the situation of Theorem 16 in the case when the monoid is in fact a -Hopf monoid . Thus, we are given a morphism of species and we consider the morphism of monoids .
Proposition 17.
Suppose that
In other words, maps elements of to primitive elements of . Then is a morphism of -Hopf monoids.
This is a special case of [6, Theorem 11.10]; see also [6, Section 11.7.1]. It follows from here and [43, Theorem IV.1.2, item (i)] that the functors
| (120) |
form an adjunction (with being left adjoint to ).
The antipode problem for offers no difficulty.
Theorem 18.
The antipode of is given by
| (121) |
Here , for , and is as in (5). In particular, sends the summand of to the summand .
To obtain this result, one may first note that any is primitive in , so the result holds when by (115), then one may use the fact that the antipode reverses products (69). One may also derive it as a special case of [6, Theorem 11.38].
Remark.
A free monoid may carry other -Hopf monoid structures than the one discussed above. In particular, it is possible to construct such a structure from a positive comonoid structure on ; when the latter is trivial, we arrive at the -Hopf monoid structure discussed above. This is discussed in [6, Section 11.2.4] but not in this paper.
6.3. The free monoid on an arbitrary species
Let be an arbitrary species (not necessarily positive). The free monoid on exists [6, Example B.29]. It is obtained by the same construction as the one in Section 6.1, employing decompositions (in the sense of Section 1.11) instead of compositions. Theorem 16 continues to hold. One has
the free associative unital algebra on the vector space . Thus, is connected if and only if is positive.
If is a coalgebra, it is possible to turn into a -bimonoid as in Section 6.2, employing the notion of restriction of decompositions of Section 1.11. If , the resulting bialgebra structure on is not the standard one, and is not Hopf. Hence, is not a -Hopf monoid.
In the sections that follow, and in most of the paper, we restrict our attention to the case of positive .
6.4. Freeness under Hadamard products
Given positive species and , define a new positive species by
| (122) |
The sum is over all pairs of compositions of whose meet is the minimum composition.
The following result shows that the Hadamard product of two free monoids is again free and provides an explicit description for the basis of the product in terms of bases of the factors.
Theorem 19.
For any positive species and , there is a natural isomorphism of monoids
| (123) |
Theorem 19 is proven in [7, Theorem 3.8]. The isomorphism (123) arises from the evident inclusion via the universal property of in Theorem 16. It is shown in [7, Proposition 3.10] that this map is an isomorphism of -Hopf monoids
the analogous statement for does not hold.
Let and be arbitrary scalars.
Theorem 20.
Let be a connected -Hopf monoid. Let be a -Hopf monoid that is free as a monoid. Then the connected -Hopf monoid is free as a monoid.
6.5. Freeness for -bimonoids
The structure of a connected -Hopf monoid is particularly rigid, in view of the following result.
Theorem 21.
Let be a connected -Hopf monoid. Then there exists an isomorphism of -Hopf monoids
6.6. The free set-theoretic monoid
Let be a positive set species. Given a composition of , write
The set species defined by
carries a set-theoretic monoid structure, and it is free on . There is a canonical isomorphism of monoids
arising from
for and as above.
7. The free commutative monoid
We review the explicit construction of the free commutative monoid on a positive species, following [6, Section 11.3]. The discussion parallels that of Section 6, with one important distinction: we deal exclusively with Hopf monoids () as opposed to general -Hopf monoids. We briefly mention the free commutative monoid on an arbitrary species.
7.1. The free commutative monoid on a positive species
Given a positive species and a partition , write
| (124) |
It is not necessary to specify an ordering among the tensor factors above; the unordered tensor is well-defined in view of the fact that the tensor product of vector spaces is symmetric. Its elements are tensors
where for each .
Define a new species by
When is the unique partition of , we have . Thus, the species is connected.
Every nonempty admits a unique partition with one block; namely, . In this case, . This yields an embedding and thus an embedding of species
On the empty set, is (necessarily) zero.
Given and partitions and , there is a canonical isomorphism
in view of the definition of unordered tensor products. Explicitly,
The sum of these isomorphisms over all and yields a map
This turns into a monoid. The commutativity of the diagram
implies that the monoid is commutative. In fact, is the free commutative monoid on the positive species , in view of the following result (a slight reformulation of [6, Theorem 11.13]).
Theorem 22.
Let be a commutative monoid, a positive species, and a morphism of species. Then there exists a unique morphism of monoids such that
commutes.
The map is as follows. On the empty set, it is the unit map of :
On a nonempty set , it is the sum of the maps
where denotes the higher product as in (45) and is any composition of with support . ( is well-defined by commutativity of .)
When there is given an isomorphism of monoids , we say that the positive species is a basis of the (free) commutative monoid .
7.2. The free commutative monoid as a Hopf monoid
The monoid admits a canonical Hopf monoid structure. Given , the coproduct
is the sum of the maps
Note that is -admissible if and only if .
This turns into a cocommutative bimonoid. Since it is connected, it is a Hopf monoid. We have
more precisely, the primitive part identifies with under the embedding .
We return to the situation of Theorem 22 in the case when is in fact a commutative Hopf monoid . Thus, we are given a morphism of species and we consider the morphism of monoids .
Proposition 23.
Suppose that maps elements of to primitive elements of . Then is a morphism of Hopf monoids.
This is a special case of [6, Theorem 11.14].
Theorem 24.
The antipode of is given by
| (125) |
for any , where is the number of blocks of .
7.3. The free commutative monoid on an arbitrary species
Let be an arbitrary species (not necessarily positive). The free commutative monoid on exists. It is the Hadamard product
The second factor is the free commutative monoid (as in Section 7.1) on the positive part of . The first factor is the monoid associated to the free commutative algebra on the space as in (73). Theorem 22 continues to hold. One has ; in particular, is connected if and only if is positive.
If is a coalgebra, then acquires a nonstandard bialgebra structure and is a bimonoid. If , then and are not Hopf.
In the sections that follow, and in most of the paper, we restrict our attention to the case of positive .
7.4. The Hadamard product of free commutative monoids
Let and be positive species. Define a new positive species by
| (126) |
The sum is over all pairs of partitions of whose meet is the minimum partition.
Theorem 25.
For any positive species and , there is a natural isomorphism of Hopf monoids
| (127) |
Arguments similar to those in [7, Theorem 3.8 and Proposition 3.10] yield this result.
7.5. The free commutative set-theoretic monoid
Let be a positive set species. The free commutative set-theoretic monoid on is the set species defined by
where
There is a canonical isomorphism of monoids
7.6. The abelianization
Let be a positive species and consider the free Hopf monoid and the free commutative Hopf monoid on it. By freeness of the former (Theorem 16), there is a unique morphism of monoids
such that
commutes, where denotes either universal arrow.
It is easy to see that is the abelianization of . Since the elements of are primitive, Proposition 23 implies that is a morphism of Hopf monoids.
On a finite set , the morphism is the sum of the canonical isomorphisms
over all compositions of . The support is as in Section 1.9. (The map is not an isomorphism since a given partition supports many compositions.)
Since the support map does not preserve meets, the map
(where the isomorphisms are as in (122) and (126)) does not send to .
Abelianization is also meaningful in the set-theoretic context.
8. The free Lie monoid
Recall the notion of a Lie monoid from Section 2.9. We discuss two important universal constructions, that of the free Lie monoid on a positive species, and the universal enveloping monoid of a positive Lie monoid.
8.1. The free Lie monoid
We begin with some preliminary definitions. A bracket sequence on letters is a way to parenthesize letters. The concatenation of bracket sequences and is defined in the obvious manner. For instance, is the concatenation of and . The left bracket sequence is the one in which all brackets are to the left. For instance, is the left bracket sequence on letters.
A bracket composition of a finite nonempty set is a pair , where is a composition of and is a bracket sequence on letters. It is convenient to think of as a bracket sequence on the blocks of . For example, is a bracket composition.
Fix a positive species . We proceed to describe the free Lie monoid on . For any bracket composition , parenthesizing using yields a bracketed tensor product which we denote by . Note that there is a canonical identification
| (128) |
Now consider the positive species whose -component is
where the sum is over all bracket compositions of . The product (128) turns this space into a non-associative monoid. Let denote its quotient by the two-sided ideal generated by the relations:
For all , , and ,
| (129) |
and
| (130) |
The relations (129) and (130) imply that the induced product satisfies (70) and (71). So is a Lie monoid. In fact, by construction, it is the free Lie monoid on . It satisfies the following universal property.
Theorem 26.
Let be a Lie monoid, a positive species, and a morphism of species. Then there exists a unique morphism of Lie monoids such that
commutes.
Remark.
We mention that the free Lie monoid on an arbitrary species also exists. One has that is the free Lie algebra on the vector space . Thus, is positive if and only if is positive.
For any bracket composition , there is a map
| (131) |
constructed by replacing each bracket in by a commutator. For instance, if has three blocks and is the left bracket sequence, then
Note that among all tensors on the right, only one begins with . This property holds whenever is the left bracket sequence. An explicit description of the action of this map on the left bracket sequence is provided by Lemma 111 below.
By summing (131) over all and noting that the relations (129) and (130) map to zero, we obtain an induced map
| (132) |
Lemma 27.
The map (132) is injective.
Proof.
We first make a general observation. Repeated use of the Leibniz identity
changes any bracket sequence to a combination of left bracket sequences only, and antisymmetry can be used to get any specified factor to the first position.
Consider the component . Note that the bracket compositions in (129) or (130) have the same underlying set partition. So splits as a direct sum over partitions of . For each partition of , fix a block of . Using the above general observation, any element of in the -component can be expressed as a sum of elements in those for which the first block of is and is the left bracket sequence. Further, the image of any such (see (131)) is a linear combination of the (with having the same support as ) in which appears and it is the only composition which begins with . So the images of the different are linearly independent proving injectivity. ∎
The above proof shows the following. For each partition of , fix a block of . Then is isomorphic to the direct sum of those for which the first block of is and is the left bracket sequence. As a consequence, we have
| (133) |
where denotes the number of blocks of .
8.2. The universal enveloping monoid
Let be a positive Lie monoid. Let be the ideal of the monoid generated by the elements
| (134) |
for and , for all nonempty finite sets and compositions . The universal enveloping monoid of , denoted , is the quotient of by the ideal . It is a connected monoid.
Let
denote the quotient map, and let
be the composition of with the embedding of in . Since (134) equals in , the map is a morphism of Lie monoids when is viewed as a Lie monoid via the commutator bracket (72).
When the Lie bracket of is identically , we have and the map is the abelianization of Section 7.6.
Proposition 28.
Let be a monoid, a positive Lie monoid, and a morphism of Lie monoids. Then there exists a unique morphism of monoids such that the first diagram below commutes.
Moreover, the second diagram above commutes as well, where is the morphism of monoids afforded by Theorem 16.
Proof.
By Theorem 16, there is a morphism of monoids extending . By construction, its value on the element (134) is
Further, since is a morphism of Lie monoids,
and hence the above value is by (72). Therefore, factors through the quotient map to yield a morphism of monoids . The uniqueness of follows from that of and the surjectivity of . ∎
This explains the usage of “universal”. It is true but not obvious that is injective. This fact is a part of the PBW theorem (Theorem 119). This justifies the usage of “envelope”.
Recall from Section 6.2 that carries a canonical structure of a bimonoid.
Lemma 29.
The ideal of is also a coideal.
Proof.
As a consequence:
Proposition 30.
There is a unique bimonoid structure on for which the map is a morphism of bimonoids. Moreover, is cocommutative.
We return to the situation of Proposition 28 in the case when is in fact a Hopf monoid . Thus, we are given a morphism of Lie monoids and we consider the morphism of monoids . Combining Propositions 17 and 30, we obtain the following.
Proposition 31.
Suppose that maps elements of to primitive elements of . Then is a morphism of bimonoids.
It follows from here and [43, Theorem IV.1.2, item (i)] that the functors
| (135) |
form an adjunction (with being left adjoint to ). This may be composed with the adjunction
(arising from Theorem 26) and the result must be isomorphic to the adjunction (120). Hence, for any positive species ,
| (136) |
as bimonoids.
9. Examples of Hopf monoids
We discuss a number of important examples of connected Hopf monoids (and -Hopf monoids). In most cases at least part of the structure is set-theoretic. Starting with the simplest Hopf monoid (the exponential Hopf monoid), we build on more complicated examples involving combinatorial structures such as linear orders, set partitions, simple graphs, and generalized permutahedra. Many of these monoids are free or free commutative. An example involving groups of unitriangular matrices is included too. We discuss various morphisms relating them and provide formulas for the structure maps on one or more linear bases. Self-duality is studied and the antipode problem is solved in all but one instance.
Two important examples are discussed separately in Section 11. Additional examples can be found in [6, Chapters 12 and 13].
In this section we assume that the base field is of characteristic . This is only needed for certain statements regarding self-duality, certain basis changes, and on a few other occasions where rational numbers intervene.
9.1. The exponential Hopf monoid
Let be the set species defined by
on all finite sets . The set is a singleton canonically associated to the set . We refer to as the exponential set species. It is a set-theoretic connected Hopf monoid with product and coproduct defined by
The Hopf monoid axioms (Sections 4.2 and 4.3) are trivially satisfied. The Hopf monoid is commutative and cocommutative.
Let denote the linearization of . This agrees with (74). It is a connected Hopf monoid. We follow Convention 5 by writing for the unique element of the canonical basis of .
Let be the species defined by
It is positive. Note that if is a partition of , then
Moreover, in the former case, any subset of is -admissible. It follows that for all finite sets . This gives rise to a canonical isomorphism of Hopf monoids
In particular, is the free commutative monoid on the species and the primitive part is
As a simple special case of (125), we have that the antipode of is given by
The Hopf monoid is self-dual under .
9.2. The Hopf monoid of linear orders
For any finite set , let be the set of all linear orders on . If and is a bijection, then . In this manner, is a set species, called of linear orders. It is a set-theoretic connected Hopf monoid with product and coproduct defined by
We have employed the operations of concatenation and restriction from Section 1.4; other notions from that section are used below.
Consider the compatibility axiom (105). Given linear orders on and on , the axiom boils down to the fact that the concatenation of and agrees with the restriction to of .
The Hopf monoid is cocommutative but not commutative.
Let denote the linearization of . It is a connected Hopf monoid. The product and coproduct are
For the antipode of on the basis , set in (139) below.
For the dual Hopf monoid , we have
and
Note that if is a composition of ,
Since a set composition of into singletons amounts to a linear order on , we have for all finite sets . Moreover, for such , any subset of is -admissible. This gives rise to a canonical isomorphism of Hopf monoids
In particular, is the free monoid on the species .
By Corollary 121 below, the primitive part is
the free Lie monoid on . We denote this species by . It is the species underlying the Lie operad. (The species underlies the associative operad.) For more details, see [6, Section 11.9 and Appendix B].
Fix a scalar . Let denote the same monoid as , but endowed with the following coproduct:
where the Schubert cocycle is as in (6). Then is a connected -Hopf monoid. The following properties of the Schubert cocycle enter in the verification of coassociativity and of axiom (54), respectively.
| (137) | |||
| (138) |
There is an isomorphism of -Hopf monoids
As a special case of (121), we have that the antipode of is given by
| (139) |
where denotes the opposite linear order of . Other proofs of this result are discussed in [6, Example 8.16 and Proposition 12.3].
Generically, the -Hopf monoid is self-dual. In fact, we have the following result. Define a map by
| (140) |
The distance between linear orders and on is as in (7).
Proposition 32.
The map is a morphism of -Hopf monoids. If is not a root of unity, then it is an isomorphism. Moreover, for any , .
Proposition 32 is proven in [6, Proposition 12.6]. The invertibility of the map (140) follows from a result of Zagier; see [6, Example 10.30] for related information. A more general result is given in [6, Theorem 11.35].
The map sends to (and is an isomorphism of -Hopf monoids).
The isomorphism gives rise to a nondegenerate pairing on (for not a root of unity). The self-duality of the isomorphism translates into the fact that the associated pairing is symmetric.
9.3. The Hopf monoid of set partitions
Recall that denotes the set of partitions of a finite set . Below we make use of notions and notation for partitions discussed in Section 1.8.
The species of partitions is a set-theoretic connected Hopf monoid with product and coproduct defined by
We have employed the operations of union and restriction of partitions. The Hopf monoid is commutative and cocommutative.
Let denote the linearization of . It is a connected Hopf monoid. The product and coproduct are
The monoid is the free commutative monoid on the positive part of the exponential species: there is an isomorphism of monoids
The isomorphism does not preserve coproducts, since when is not -admissible.
Theorem 33.
The antipode of is given by
For the dual Hopf monoid , we have
and
The Hopf monoid is self-dual. There are in fact several isomorphisms . We highlight two particular ones. Define maps by
| (141) |
Proposition 34.
The maps are isomorphisms of Hopf monoids. Moreover, and .
The assertion about is proven in [6, Proposition 12.48]. One verifies without difficulty that preserves products and coproducts. Its invertibility follows from a result of Dowling and Wilson [19, Lemma 1] (or from the considerations below).
Define a linear basis of by
The basis is determined by Möbius inversion. Let denote the linear basis of dual to the basis of . Equivalently,
Proposition 35.
We have
This is shown in [6, Section 12.6]. In addition, the maps in (141) satisfy
where the coefficient is as in (15). The second coefficient is the Möbius function value (17).
The expressions for the product and coproduct on the basis show that there is an isomorphism of Hopf monoids
It follows that the primitive part satisfies
with one-dimensional spanned by the element .
9.4. The Hopf monoid of simple graphs
A simple graph on is a collection of subsets of of cardinality . The elements of are the vertices and the subsets are the (undirected) edges.
Let be a simple graph on and . The restriction is the simple graph on consisting of those edges of connecting elements of . Write . We say that is -admissible if no edge of connects an element of to an element of .
Given and simple graphs on and on , their union is the simple graph on whose edges belong to either or .
Let denote the set of simple graphs with vertex set . The species of simple graphs is a set-theoretic connected Hopf monoid with product and coproduct defined by
The Hopf monoid is commutative and cocommutative.
We let denote the linearization of . It is a connected Hopf monoid. The product and coproduct are
Let denote the subspecies of spanned by connected simple graphs. The monoid is the free commutative monoid on : there is an isomorphism of monoids
Here is the partition into connected components of . The isomorphism does not preserve coproducts, since when is not -admissible.
There is a pleasant solution to the antipode problem for . In order to state it, we set up some terminology.
Given a simple graph on and a partition , let
This is a simple graph on . Let also be the simple graph on such that there is an edge between if there is at least one edge in between an element of and an element of . In other words, is obtained from by identifying all vertices in each block of and removing all loops and multiple edges that may arise as a result. The lattice of contractions of is the set
An orientation of the edges of is acyclic if it contains no oriented cycles. Let denote the number of acyclic orientations of .
Theorem 36.
The antipode of is given by
| (142) |
Note that if , then can be recovered from as the partition into connected components. Hence there are no cancellations in the antipode formula (142).
Theorem 36 is given in [1] and also (in the setting of Hopf algebras) in work of Humpert and Martin [33, Theorem 3.1]. It is explained in these references how Stanley’s negative one color theorem [69, Corollary 1.3] may be derived from this result. It is possible to proceed conversely, as we outline after Proposition 38 below.
For the dual Hopf monoid , the product is
The sum is over those simple graphs for which and . In other words, an edge of is either an edge of , or an edge of , or connects an element of with an element of . The coproduct is
The Hopf monoid is self-dual. The following result provides a canonical duality. The complement of a simple graph on is the simple graph with the complementary edge set. Define a map by
| (143) |
The simple graphs in the sum are those without common edges with .
Proposition 37.
The map is an isomorphism of Hopf monoids. Moreover, .
One verifies that preserves products and coproducts. It is triangular with respect to inclusion, hence invertible.
Define a linear basis of by
The basis is determined by Möbius inversion. Let denote the linear basis of dual to the basis of . Equivalently,
Proposition 38.
We have
| (144) | |||
| (145) | |||
| (146) |
Here denotes the number of connected components of . In addition, the map in (143) satisfies
Formulas (144) and (145) show that there is an isomorphism of Hopf monoids
The antipode formula (146) is then a special case of (125). This in turn can be used to derive the antipode formula (142) using Stanley’s negative one color theorem [69, Corollary 1.3].
It also follows that the primitive part satisfies
with spanned by the elements as runs over the connected simple graphs on .
The analogy in the analyses of the Hopf monoids and can be formalized. Let denote the complete graph on , and for a partition of , let
Define a map by
| (147) |
Proposition 39.
The map is an injective morphism of Hopf monoids. Moreover,
commutes.
The commutativity of the diagram translates into the fact that is an isometric embedding when and are endowed with the pairings associated to the isomorphisms .
9.5. The Hopf monoid of functions on unitriangular groups
Let be a fixed field, possibly different from the base field . Given a finite set and a linear order on , let denote the algebra of matrices
| , for all . |
The general linear group consists of the invertible matrices in , and the subgroup consists of the upper -unitriangular matrices
| , for all , whenever according to . |
Let . Given and , their direct sum is the matrix with entries
Let . The direct sum of an -unitriangular and an -unitriangular matrix is -unitriangular. The morphism of algebras
thus restricts to a morphism of groups
| (148) |
(The dependence of on is left implicit.)
Given , the principal minor indexed by is the matrix
If is -unitriangular, then is -unitriangular. Moreover, the map
is a morphism of groups if and only if is an -segment: if and according to , then also .
Let , , . Define a map
| (149) |
by
Note that is an initial segment for and is a final segment for . Thus is a morphism of groups.
Let and let and be the pairwise intersections, as in (53). We then have
| (150) |
and a similar statement for , , and .
Assume from now on that the field is finite. Thus, all groups of unitriangular matrices are finite. Define a vector species as follows. On a finite set ,
In other words, is the direct sum of the spaces of functions (with values in ) on all unitriangular groups on .
Let and , . From the morphism in (149) we obtain a linear map
Summing over all and , we obtain a linear map
| (151) |
Explicitly, given functions and ,
is the function given by
Similarly, from the map in (148) we obtain the components
(one for each ) of a linear map
| (152) |
Explicitly, given a function , we have
where
are functions such that
| (153) |
for all and .
Proposition 40.
The compatibility between the product and the coproduct follows from (150). The construction is carried out in more detail in [3].
We describe the operations on the basis of characteristic functions (see Convention 5). Let , . The product is
| (154) |
The coproduct is
| (155) |
To a unitriangular matrix we associate a graph on as follows: there is an edge between and if in and . For example, given nonzero entries ,
Let be the map defined by
| (156) |
Proposition 41.
The map is an injective morphism of Hopf monoids. If is the field with two elements, then is an isomorphism.
This result is given in [3, Proposition 8 and Corollary 9]. (In this reference, is identified with its dual .)
An injective morphism of Hopf monoids
is described in [3, Section 4.6]. The composite
maps to a certain Hopf submonoid of superclass functions on the unitriangular groups.
9.6. The Hopf monoid of generalized permutahedra
Consider the Euclidean space
endowed with the standard inner product
Let be a convex polytope in [76]. The faces of are then convex polytopes also. We write to indicate that is a face of .
Given a vector , let denote the subset of where the function
achieves its maximum value. Then is a face of .
Given a face of , its normal cone is the set
The collection of normal cones
is the normal fan of .
Let . The standard permutahedron is the convex hull of the points
The faces of are in bijection with compositions of . For , the corresponding face of has as vertices the bijections such that
The normal cone of this face is the set of vectors such that
| if and belong to the same block of , | |||
| if the block of precedes the block of in . |
When , the standard permutahedron is a regular hexagon. It lies on the plane . The intersection of its normal fan with this plane is shown next to it, below.
A polytope in is a generalized permutahedron if its normal fan is coarser than that of the standard permutahedron. In other words, each cone must be a union of cones in .
A generalized permutahedron is shown below, together with its normal fan.
Let denote the (infinite) set of generalized permutahedra in .
Fix a decomposition . This yields a canonical identification
Proposition 42.
If is a generalized permutahedron in and is another in , then is a generalized permutahedron in .
This holds because the normal fan of is coarser than that of . This allows us to define a map
Proposition 43.
Let be a generalized permutahedron in . Let be a vector such that
| if and belong both to or both to , | |||
| if and . |
Then there exist generalized permutahedra in and in such that
If or are empty, then . Otherwise, the conditions on express precisely that . Therefore, if is another vector in , then both and belong to the same cone in the fan , and hence . Thus, and are independent of the choice of , and we may define a map
It is shown in [1] that, with the above structure, the species is a set-theoretic connected Hopf monoid. It is commutative, but not cocommutative.
Let denote the linearization of . It is a connected Hopf monoid. The product and coproduct are
One of the main results in [1] is the following.
Theorem 44.
The antipode of is
| (157) |
Each face of appears once in the sum, with coefficient . The formula is thus cancellation free.
The antipode formula (142) for the Hopf monoid of simple graphs may be deduced from (157) by means of a morphism from to the quotient of in which generalized permutahedra with the same normal fan are identified.
Remark.
Generalized permutahedra are studied in depth by Postnikov [56], where they are defined as deformations of the standard permutahedron. The definition in terms of normal fans is from work of Postnikov, Reiner, and Williams [55, Proposition 3.2], where plenty of additional information is given. The same class of polytopes is studied in Fujishige [24]. The Hopf monoid of generalized permutahedra as defined above appears in [1]. Several related examples are discussed in that reference, which builds on the work of Billera, Jia, and Reiner on Hopf algebras [15].
10. Higher associativity and compatibility
Let be a bimonoid. The bimonoid axioms link the structure maps , , and of . We study the implications of these on the higher structure maps. A remarkable fact is that the four bimonoid axioms of Section 2.8 can be unified into a single axiom relating the higher product and coproduct. Associativity and unitality are similarly unified. The combinatorics of set decompositions underlies this study and the discussion necessitates the notions and terminology discussed in Section 1.11, including those of concatenation, refinement, splitting, and Tits product for set decompositions.
10.1. Higher associativity
Given a species , a finite set and a decomposition of , write
| (158) |
(This generalizes (117).) When (the unique decomposition of the empty set without subsets), we set .
Given and decompositions of and of , there is a canonical isomorphism
| (159) |
obtained by concatenating the factors in (158). If , then this is the canonical isomorphism
a similar remark applies if .
Assume now that is a monoid and is a decomposition of . We write
for the higher product maps of , as in (45). Recall that .
Given a bijection , let , , , and the map
The collection satisfies the following naturality condition. For any such and , the diagram
| (160) |
commutes. This follows from (27).
Let still be as above. Let be a decomposition of such that and a splitting of the pair . In other words, each is a decomposition of and . Define a map by means of the diagram
| (161) |
where the vertical isomorphism is an iteration of (159). If , then necessarily and , and we agree that the map is the identity of .
Recall that if and are compositions, then the splitting is uniquely determined by . In this situation we write instead of .
Higher associativity is the following result.
Proposition 45.
Let be a monoid. Let and be decompositions of with . Then, for any splitting of the pair , the diagram
| (162) |
commutes.
Considering on the one hand , , , and on the other , , , we see that (162) implies the associativity axiom (33). The unit axioms (44) also follow from (162); to obtain the left unit axiom one chooses , and . Conversely, (162) follows by repeated use of (33) and (44).
Thus, higher associativity encompasses both associativity and unitality. In fact, the preceding discussion leads to the following result.
10.2. Higher unitality
We continue to assume that is a monoid and is a decomposition of . Recall that denotes the composition of obtained by removing from those subsets which are empty. For any such , there is a canonical isomorphism
where
For each , let be the map defined by
where is the unit map of . We then define a map as follows.
The following result is higher unitality.
Proposition 47.
Let be a monoid. For any decomposition of , the diagram
| (163) |
commutes.
This result is in fact a special case of that in Proposition 45. To see this, note that for any decomposition of there is a canonical splitting of ; namely,
One then has that
and therefore (163) is the special case of (162) in which and .
We emphasize that, in view of the preceding discussion, higher unitality is a special case of higher associativity.
10.3. Higher commutativity
Let and be two decompositions of which consist of the same subsets, possibly listed in different orders. Let be any permutation such that
for each . The permutation may not be unique due to the presence of empty subsets. For any species , there is an isomorphism
obtained by reordering the tensor factors according to . Fix a scalar . Let
| (164) |
be the scalar multiple of the previous map by the factor with the distance function as in (4).
If and are compositions of , then is unique, and we write instead of .
We then have the following higher -commutativity.
Proposition 48.
Let be a -commutative monoid. For any decompositions and of and permutation as above, the diagram
| (165) |
commutes.
10.4. Higher coassociativity and counitality
Similar considerations to those in the preceding sections apply to a comonoid . In particular, given a splitting of the pair , there is a map
For any decomposition , there is a map
we have . If and are compositions, then there is a unique choice and we write instead of .
10.5. Higher compatibility
We turn to the compatibility between the higher product and coproduct of a -bimonoid. The Tits product plays a crucial role in this discussion.
Let and be two decompositions of and consider the situation in (9). We see from (10) that the Tits products and may only differ in the order of the subsets. Moreover, there is a canonical permutation ‘’ that reorders the subsets in into those in , namely .
Let be a species and let
be the map (164) for this particular choice of reordering. When and , the map coincides with the map in the top of diagram (54).
For any decompositions and of , there is a canonical splitting for . Namely,
| (166) |
the -th row in the matrix (9). Let ‘’ also denote this splitting.
We then have the following higher compatibility.
Proposition 50.
Let be a -bimonoid. For any decompositions and of , the diagram
| (167) |
commutes.
Consider the case and . Let , , , and be as in (53). Then , , and (167) specializes to the compatibility axiom (54).
Now consider and . Then , and following (161) we find that the maps and are as follows.
Also, and . It follows that (167) specializes in this case to the right diagram in the compatibility axiom (65).
Thus, all compatibility axioms (54)–(66) are special cases of (167). Conversely, the latter follows from the former.
Proposition 51.
The unit conditions have been subsumed under (162) and (167). Nevertheless, certain compatibilities involving the maps and are worth-stating.
Proposition 52.
Let be a -bimonoid. For any decomposition of , the diagram
| (168) |
commutes.
For the next result, we consider a special type of refinement for set decompositions. Let be a decomposition of . Let be a decomposition of refining and with the property that the nonempty subsets in are refined by into nonempty subsets, and each empty set in is refined by into a nonnegative number of empty sets. In this situation, there is a canonical choice of splitting for . Namely, we choose each to consist either of a maximal run of contiguous nonempty sets in , or of a maximal run of contiguous empty sets. We denote this splitting by .
Proposition 53.
Let be a -bimonoid. For any decompositions and as above, the diagrams
| (179) |
commute.
10.6. The connected case
The preceding discussion carries over to connected species. The statements are simpler. There are additional results that are specific to this setting.
First of all, note that a connected species can be recovered from its positive part by setting .
If is a connected monoid, then for any decomposition the map is invertible. Together with (163) this implies that the collection of higher product maps is uniquely determined by the subcollection indexed by compositions .
Proposition 54.
Let be a connected species equipped with a collection of maps
one map for each composition of a nonempty finite set . Then is a connected monoid with higher product maps if and only if naturality (160) holds and the diagram
| (180) |
commutes, for each compositions and of with . The monoid is commutative if and only if
| (181) |
commutes, for any compositions and with .
This follows from Propositions 46 and 48. Since splittings are unique for compositions, diagrams (162) and (165) simplify to (180) and (181).
For a connected comonoid , the map is invertible, and we have the dual statement to Proposition 54.
Proposition 55.
Let be a connected species equipped with two collections of maps
one map for each composition of a nonempty finite set . Then is a bimonoid with higher (co)product maps () if and only if the following conditions hold: naturality (160), higher associativity (180), their duals, and the diagram
| (182) |
commutes for any pair of compositions and of a nonempty finite set .
This follows from Proposition 50. The Tits products and in (182) are those for set compositions (Section 1.6).
Connectedness has further implications on the higher structure maps.
Proposition 56.
Let be a connected -bimonoid. For any decomposition of , the diagram
| (183) |
commutes.
Together with Proposition 52, this says that for a connected -bimonoid the maps and are inverse.
Proposition 57.
Let be a connected -bimonoid. For any compositions and of with , the diagram
| (184) |
commutes.
This follows from (182), since in this case and .
The following special cases are worth-stating.
Corollary 58.
Let be a connected -bimonoid. For any composition of , the diagrams
commute.
11. The bimonoids of set (de)compositions
We now discuss two important bimonoids and indexed by set compositions and set decompositions respectively. The former is connected and hence a Hopf monoid, while the latter is not a Hopf monoid. Both structures are set-theoretic. We also write down formulas for the higher (co)product maps. This is particularly compelling since the higher (co)product maps are also indexed by set decompositions (and the Tits product appears in the coproduct description).
We make use of notions and notation for (de)compositions discussed in Section 1. We work over a characteristic field.
11.1. The Hopf monoid of set compositions
Recall that denotes the set of compositions of a finite set .
The species of compositions is a set-theoretic connected Hopf monoid with product and coproduct defined by
| (186) | ||||||
We have employed the operations of concatenation and restriction of compositions.
The Hopf monoid is cocommutative but not commutative.
We let denote the linearization of . It is a connected Hopf monoid. The product and coproduct are
| (187) |
For the antipode of on the basis , set in Theorem 60 below.
For the dual Hopf monoid , we have
and
The monoid is free on the positive part of the exponential species: there is an isomorphism of monoids
for a composition of . The isomorphism does not preserve coproducts, since when is not -admissible.
Define a linear basis of by
| (188) |
The basis is determined by triangularity. In fact, one may show that
| (189) |
Proposition 59.
We have
| (190) | |||
| (191) | |||
| (192) |
There is an isomorphism of Hopf monoids
for a composition of . It follows that the primitive part of is the Lie submonoid generated by the elements . For instance,
are primitive elements.
Fix a scalar . Let denote the same monoid as , but endowed with the following coproduct:
where the Schubert cocycle is as in (2). Then is a connected -Hopf monoid.
Theorem 60.
The antipode of is given by
Generically, the -Hopf monoid is self-dual. In fact, we have the following result. Define a map by
| (193) |
Proposition 61.
The map is a morphism of -Hopf monoids. If is not an algebraic number, then it is an isomorphism. Moreover, for any , .
Proposition 61 is proven in [6, Proposition 12.26]. See [6, Section 10.15.2] for related information.
With , the map in (193) is not invertible. It satisfies
where is the basis of dual to the basis of . The coefficient is the Möbius function value (17). The support of a composition is defined in Section 1.9.
Regarding a linear order as a composition into singletons, we may view as a subspecies of . The analogy between the present discussion and that for linear orders in Section 9.2 is explained by the observation that in fact is in this manner a -Hopf submonoid of . Let be the inclusion. The diagram
Let be the map defined by
| (194) |
Proposition 62.
The map is a surjective morphism of Hopf monoids. Moreover, we have
and the diagram
commutes.
11.2. The bimonoid of set decompositions
Some of the considerations of Section 11.1 can be extended to all decompositions of finite sets, as opposed to only those into nonempty subsets (compositions).
Recall that denotes the set of decompositions of a finite set . The set-theoretic bimonoid structure on can be extended to by means of the same formulas as in (186): the product is given by concatenation and the coproduct by restriction of decompositions.
The set consists of the decompositions , where is a nonnegative integer. As with any set-theoretic bimonoid, this set is an ordinary monoid under and the map must be the diagonal. The latter statement is witnessed by the fact that for all . The product is concatenation, so as discussed in (19), is isomorphic to the monoid of nonnegative integers under addition. Since this is not a group, the set-theoretic bimonoid is not a Hopf monoid.
Let . Then is a bimonoid but not a Hopf monoid. The product and coproduct are as for :
where , and are decompositions.
The map defined by
| (195) |
is a surjective morphism of bimonoids. (It is set-theoretic.)
11.3. The higher (co)product in and
The following result describes the higher product and coproduct of the Hopf monoid of compositions, namely
Here is a composition of and the notation is as in (45) and (47); see also Section 10.1. It suffices to consider compositions because is connected (see Section 10.6).
Let . Given a composition refining , let . Define an element
In view of (8), the elements form a linear basis of as varies over the refinements of .
Proposition 63.
For any compositions of ,
| (196) |
For any pair of compositions and of ,
| (197) |
Now consider the bimonoid of decompositions. The higher product and coproduct of are given by essentially the same formulas as (196) and (197), but adjustments are necessary. We turn to them.
Let be a decomposition of . Let be a decomposition of such that and let be a splitting of . Define an element
In view of (22), the elements form a linear basis of as varies over decompositions satisfying and over all splittings of .
Now let and be two arbitrary decompositions of . Recall from (166) that there is a canonical choice of splitting for , denoted ‘’. Hence the element is defined.
Proposition 64.
Let and be decompositions of with . Then, for any splitting of ,
| (198) |
For any pair of decompositions and of ,
| (199) |
12. Series and functional calculus
Every species gives rise to a vector space of series, which is an algebra when the species is a monoid. For the exponential species, this identifies with the algebra of formal power series. When the species is a connected bimonoid, one can define group-like series and primitive series. The former constitutes a group and the latter a Lie algebra, and they are in correspondence by means of the exponential and the logarithm.
We work over a field of characteristic .
12.1. Series of a species
Let be a species. Define the space of series of as
the limit of the functor . Explicitly, a series of is a collection of elements
one for each finite set , such that
| (200) |
for each bijection . The vector space structure is given by
for and .
Condition (200) implies that each is an -invariant element of , and in fact there is a linear isomorphism
| (201) |
Let be the exponential species (74). We have an isomorphism of vector spaces
| (202) |
Explicitly, a series of corresponds to the morphism of species
Let be a morphism of species and a series of . Define by
Since commutes with bijections (23), is a series of . In this manner, defines a functor from species to vector spaces. Moreover, for any species and , there is a linear map
| (203) |
which sends to the series whose -component is
This turns into a braided lax monoidal functor. Such a functor preserves monoids, commutative monoids, and Lie monoids (see, for instance, [6, Propositions 3.37 and 4.13]). These facts are elaborated below.
Let be a monoid. The Cauchy product of two series and of is defined by
| (204) |
This turns the space of series into an algebra. The unit is the series defined by
| (205) |
In this case, is an algebra under the convolution product of Section 2.7, and (202) is an isomorphism of algebras
If the monoid is commutative, then the algebra is commutative.
If is a Lie monoid, then is a Lie algebra. The Lie bracket of two series and of is defined by
| (206) |
In view of (200), a series of is of the form
where is an arbitrary scalar sequence. The species is a Hopf monoid (Section 9.1) and hence is an algebra. If we identify with the formal power series
then the Cauchy product (204) corresponds to the usual product of formal power series, and we obtain an isomorphism of algebras
| (207) |
In this sense, series of are exponential generating functions.
Let and be series of and , respectively. Their Hadamard product is defined by
It is a series of . This gives rise to a linear map
| (208) |
In general, this map is not injective and not a morphism of algebras under the Cauchy product (when the are monoids). This occurs already for ; indeed, the Hadamard product (208) reduces in this case to the familiar Hadamard product (of exponential generating functions).
12.2. Exponential series
Let be a monoid. A series of is exponential if
| (209) |
for each , and
| (210) |
The unit series (205) is exponential.
Let be the set of exponential series of . Under (202), exponential series correspond to morphisms of monoids . Since is generated by , such a series is completely determined by the element
Conversely, for an element to give rise to an exponential series, it is necessary and sufficient that
| (211) |
where and is the unique bijection .
If the monoid is free, then the only element that satisfies (211) is , and the corresponding exponential series is . Thus,
in this case.
If the monoid is commutative, then any element satisfies (211). Moreover, exponential series are closed under the Cauchy product, and this corresponds to addition in . Thus, in this case,
In particular, is a group (a subgroup of the group of invertible elements of the algebra ). The inverse of is given by
These statements can be deduced from more general properties of convolution products (see the end of Section 2.8).
For the exponential species , we have
| (212) |
If corresponds to , then
| (213) |
Under the identification (207), corresponds to the formal power series
Suppose and are monoids. If and are exponential series of and , respectively, then is an exponential series of . In general, the map
| (214) |
is not injective and not a morphism of groups (when the are commutative monoids). This occurs already for ; in this case (214) identifies under (212) with the multiplication of . The Hadamard product
distributes over the Cauchy product:
Recall that for any species . If and , then the Hadamard product
corresponds to the series of given by
In this manner, series of arise in one-parameter families.
Let be a finite set-theoretic comonoid and consider the monoid as in Section 4.5. This monoid contains a distinguished exponential series defined by
| (215) |
12.3. Group-like series
Let be a comonoid. A series of is group-like if
| (216) |
for each , and
| (217) |
Note that it follows from (216) plus counitality that either (217) holds, or for every .
Let be the set of group-like series of . Under (202), group-like series correspond to morphisms of comonoids .
Let be a bimonoid. The unit series (205) is group-like. It follows from (54) that is closed under the Cauchy product (204). Thus, is an ordinary monoid (a submonoid of the multiplicative monoid of the algebra ). If is a Hopf monoid, then is a group. The inverse of is obtained by composing with the antipode of :
This follows from (67). (These statements can also be deduced from more general properties of convolution products; see the end of Section 2.8.)
Suppose the bimonoid is connected. It follows from (66) and (108) that if a series of is exponential, then it is group-like. Thus,
| (218) |
For the Hopf monoid , the maps and are inverse, and hence a series of is exponential if and only if it is group-like. Therefore, under the identification (207), group-like series of correspond to the formal power series of the form , where is an arbitrary scalar, and we have an isomorphism of groups
| (219) |
Suppose and are comonoids. If and are group-like series of and , respectively, then is a group-like series of , and we obtain a map
| (220) |
In particular, if is a group-like series of and is a scalar, then is another group-like series of . Group-like series thus arise in one-parameter families.
Let be a finite set-theoretic monoid and consider the comonoid as in Section 4.5. This comonoid contains a distinguished group-like series defined by
| (221) |
12.4. Primitive series
Let be a comonoid and and two group-like series of . A series of is -primitive if
| (222) |
for each , and
| (223) |
Let be a connected comonoid. In this case, possesses a distinguished group-like series determined by
| (224) |
A series of is -primitive if and only if
| (225) |
for each with , and
| (226) |
In this case, we simply say that the series is primitive.
Let be the set of primitive series of . Under (202), primitive series correspond to morphisms of species , where is the primitive part of (Section 5.5). In other words,
| (227) |
The primitive part of the exponential species is ; hence we have an isomorphism of abelian Lie algebras
The Hadamard product of an arbitrary series with a primitive series is primitive. We obtain a map
where the are connected comonoids.
Let be a bimonoid. The series of is group-like, regardless of whether is connected. A series of is -primitive if and only if (225) holds, and in addition
| (228) |
for nonempty. In particular, must be a primitive element of the bialgebra .
If in addition is connected, then , so -primitive series coincide with primitive series. Note also that, in this case, the last two conditions in (228) follow from counitality (the duals of (44)), and the first one is equivalent to (226).
Proposition 65.
Let be a bimonoid. Let , and be group-like series of , and a -primitive series of . Then the series and are - and -primitive, respectively.
Proof.
12.5. Complete Hopf algebras
Let be a Hopf monoid. In view of (201), the space of series of identifies with the completion of the graded space
(with respect to the filtration by degree). The summand is the invariant subspace of under the action of . In [6, Chapter 15], it is shown that carries a structure of graded Hopf algebra. One may thus turn the space into a complete Hopf algebra in the sense of Quillen [57, Appendix A.2]. Then and identify with the group of group-like elements and the Lie algebra of primitive elements of this complete Hopf algebra, respectively.
12.6. Series of the linear order species
Consider the Hopf monoid of linear orders from Section 9.2.
In view of (200), a series of is of the form
where is an arbitrary scalar sequence. If we identify with the formal power series
then the Cauchy product (204) corresponds to the usual product of formal power series, and we obtain an isomorphism of algebras
| (230) |
In this sense, series of are ordinary generating functions. Contrast with (207).
Since is free, the only exponential series is the unit series (205). The corresponding formal power series is the constant .
For a series as above we have
For to be group-like, we must have
for all , and . Therefore, each scalar gives rise to a group-like series defined by
and every group-like series is of this form. This gives rise to an isomorphism of groups
For to be primitive, we must have for every , and . Therefore, each scalar gives rise to a primitive series defined by
and every primitive series is of this form. This gives rise to an isomorphism of (abelian) Lie algebras
Note that, by (227),
since the species is the primitive part of . We deduce from the above and (201) the well-known result that
12.7. Formal functional calculus
Let be a monoid. Recall from (45) and (117) that given a composition of a finite set , the space and the map
are defined (see also Section 10.1). In this situation, given a series of , define an element by
Let
be a formal power series. Given a series of such that
| (231) |
define another series of by
| (232) |
for every finite set . In particular, . It follows that formal power series with operate in this manner on the space of series of satisfying (231).
Proposition 66.
If for a nonnegative integer , then . In particular, if , then , the unit series (205), and if , then .
Proof.
If is another formal power series, the sum and the product are defined. If in addition , the composition is defined.
Proposition 67 (Functional calculus).
Let and be formal power series and a series of satisfying (231).
-
(i)
If is a scalar, then .
-
(ii)
.
-
(iii)
.
-
(iv)
If , then .
-
(v)
If is a morphism of monoids, then .
12.8. Exponential, logarithm, and powers
Consider the formal power series
Let be a monoid. If is a series of such that
| (233) |
define
Then is a series satisfying (231), and for nonempty ,
| (234) |
If is a series of satisfying (231), then is a series satisfying (233), and more generally,
| (235) |
The following is a basic property of functional calculus.
Proposition 68.
Theorem 69.
For any monoid , the maps
| (238) |
are inverse bijections. Moreover, if is a connected bimonoid, these maps restrict to inverse bijections
| (239) |
Proof.
Remark.
Let be a scalar. Consider the formal power series
where
| (240) |
Let be a monoid. If is a series of satisfying (233), define
Then is a series satisfying (233), and for nonempty ,
| (241) |
12.9. Curves and derivatives
Let be a species. A (polynomial) curve on is a function
whose components are polynomial, for all finite sets . The component is defined by for all .
The derivative of a curve is defined by
where the latter is the usual derivative of a polynomial function. When is the field of real numbers, one may also speak of smooth curves defined on a real interval, and of their derivatives.
The derivative of a formal power series is the formal power series
| (244) |
Let be a curve on a monoid for which each series satisfies (231). Given a formal power series , the curve is defined (by means of functional calculus) as
for each .
Derivatives of curves and of power series are linked by the chain rule. This requires a commutativity assumption.
Proposition 70 (Chain rule).
Let be a curve as above. Suppose that
for all . Then
| (245) |
Proof.
The -components are zero, so we assume to be nonempty. Applying (232) to (244),
Substituting and calculating using (204),
| (246) |
Hypothesis (231) implies that . Thus, we may assume that in the previous sum.
On the other hand, the derivative of is
| (247) |
(Since is linear, it commutes with the derivative.) If , then
and
Commutativity allows us to move the differentiated factor in (247) to the right end, as we now verify. By hypothesis,
It follows that, for any ,
and hence
Substituting back in (247) we obtain
Letting , , and , and employing associativity, we obtain (246), as needed. ∎
Let be a comonoid and a curve on . When the series is group-like for every , we say that is a group-like curve.
Proposition 71.
Let be a group-like curve on a comonoid . For any , the series is -primitive.
Proof.
Corollary 72.
Let be a Hopf monoid and a group-like curve on . Fix , let and . Then the series
are -primitive. In particular, if is connected, then both series are primitive.
Proof.
Corollary 72 affords a construction of primitive series out of group-like curves.
| \psfrag{u}{u}\psfrag{g}{g}\psfrag{x}{\small x}\psfrag{y}{\small g^{-1}\ast x}\includegraphics[width=216.81pt]{curve-tan} |
In the next section we study the result of this construction when applied to certain special curves.
12.10. Two canonical curves
Let be a series of a species . There is a curve on defined by
| (248) |
for every finite set . It is clearly polynomial.
Let be a monoid and assume that satisfies (233). There is then another curve on defined by
| (249) |
Formulas (240) and (241) show that is polynomial. We say that is the one-parameter subgroup generated by .
We have
This is illustrated below.
| \psfrag{0}{\tiny(c=0)}\psfrag{1}{\tiny(c=1)}\psfrag{u}{u}\psfrag{t}{t}\psfrag{A}{\langle t\rangle}\psfrag{B}{\{t\}}\includegraphics[width=158.99377pt]{two-curves} |
We proceed to apply the construction of Corollary 72 to these curves.
We first describe the derivative of the curve . The number operator is the morphism of species defined by
| (250) |
Note that
| (251) |
Corollary 73.
Let be a group-like series of a Hopf monoid . Then the series
are -primitive. In particular, if is connected, then both series are primitive.
Proof.
We turn to the derivative of the one-parameter subgroup .
Lemma 74.
Let be a series of a monoid satisfying (233). Then, for any ,
| (252) |
Proof.
Let be the curve defined by
We have
Since , we have , and the chain rule (245) applies. We deduce that
For the same reason, . ∎
Let us apply the construction of Corollary 72 to the curve , where is a group-like series of a connected Hopf monoid . First of all, the curve is group-like by (243). Lemma 74 implies that . The conclusion is then that is a primitive series of . We thus recover a part of Theorem 69.
The curves enjoy special properties, as discussed next.
Proposition 75.
Proof.
Assume (i) holds. Fixing and taking derivatives at we obtain (ii). Assume the latter holds. If , then is a solution of the differential equation in (ii), according to (252). Since , uniqueness of solutions of such equations guarantees that , proving that (iii) holds. Finally, the latter implies (i) by the first identity in (242).
Note also that, in the situation of Proposition 75, for all .
12.11. Series of the internal Hom species
Recall the functors and from Section 3.2. Let and be species. A series of the species is precisely a morphism of species from to :
| (253) |
This follows by comparing (23) and (75), or by using (76) and (202). We may also recover (202) by setting in (253). Setting we obtain
If is a monoid and is a comonoid, then is a monoid (Section 3.2) and hence is an algebra under the Cauchy product (204). On the other hand, the space is an algebra under the convolution product of Section 2.7. Building on (253), we have that these two are the same algebra: . In particular, the unit series (205) coincides in this case with the morphism (50).
Let and be connected bimonoids. Assume that is finite-dimensional. As explained in Section 3.2, is then a finite-dimensional connected bimonoid. We then have, by (218),
This inclusion can be refined as follows. We let
denote the subset of consisting of morphisms of monoids. We similarly use for comonoid morphisms.
Proposition 76.
For and as above, we have
and
Proof.
Let and be connected bimonoids with finite-dimensional, and a morphism of species. Proposition 76 implies that if is a morphism of either monoids or comonoids, then it is a group-like series of . The same conclusion holds if is an antimorphism of either monoids or comonoids. For this one may use a similar argument to that in the proof of Proposition 76, using the second identity in (108). In particular, both the identity and the antipode of are group-like series of .
Let us denote the antipodes of , , and by , , and , respectively.
Corollary 77.
Let and be as above. If is a morphism of monoids, then . If is a morphism of comonoids, then .
Proof.
Since in both cases is a group-like series of , the series is its inverse with respect to convolution. But we know from Section 2.8 that this inverse is as stated. ∎
The previous result can also be shown using Takeuchi’s formula (111) and Corollary 58: For a morphism of monoids,
Takeuchi’s formula is used in the first and last steps, is used twice, and is used once.
Remark.
We now show that under the - correspondence of Section 12.8
the space corresponds with (provided is cocommutative), and the space corresponds with (provided is commutative). These results hold in greater generality, as stated below. In particular, we no longer require finite-dimensionality.
Let be a monoid and be a comonoid. Then and are monoids under convolution. The and maps below act on the spaces of series of these monoids, namely and .
Theorem 78.
Let be a cocommutative comonoid and a connected bimonoid. The maps and restrict to inverse bijections
| (254) |
Let be a connected bimonoid and a commutative monoid. The maps and restrict to inverse bijections
| (255) |
We provide two proofs of this result, each one with its own advantage. The first one employs the higher forms of the bimonoid axioms of Section 10 and the combinatorics of the Tits algebra. It extends to more general settings (beyond the scope of this paper, but hinted at in Section 1.10) where the Tits product is defined. The second proof relies on functional calculus and extends to a context such as that of complete Hopf algebras. It is essentially the same proof as those in [57, Appendix A, Proposition 2.6] and [66, Theorem 9.4].
First proof of Theorem 78.
In view of (238), it suffices to show that and map as stated. We show that if is a morphism of species, then is a morphism of comonoids. The remaining verifications are similar.
We have to show that
for . We may assume that and are nonempty.
Let denote the composition . In the following calculation, we make use of the higher compatibility (182) for .
Since maps to the primitive part, unless . In this case, letting and , we have
and also
Therefore,
Since is cocommutative,
by the dual of (181). The number of terms in the last sum is
Therefore,
as needed. ∎
We remark that a morphism of species is the same as a derivation when is endowed with the trivial -bimodule structure. Dually, a morphism of species is the same as a morphism of species such that
| (256) |
where is the unit in the convolution algebra , as in (50). This says that is a coderivation from viewed as a trivial -bicomodule to .
Second proof of Theorem 78.
As in the first proof, we start from and show that is a morphism of comonoids. We deduce this fact by calculating
The exponentials in the middle terms are calculated in the convolution algebra . The Cauchy product of (co)monoids is as in (46) (with ).
The first equality holds by functoriality of the convolution algebra (51) and naturality of functional calculus (item (v) in Proposition 67), since is a morphism of monoids. The third holds by the same reason, since is a morphism of comonoids (this uses that is cocommutative). The second holds by (256). The fourth holds by (236), since and commute in the convolution algebra by (52). We complete the proof by justifying the last equality.
Proposition 79.
Let be a connected bimonoid and a morphism of species with . Then and agree when restricted to the primitive part , and also when followed by the canonical projection to .
If is a morphism of comonoids, then it preserves the primitive part, and we deduce from Proposition 79 that and restrict to the same map on .
If is a morphism of monoids, then and induce the same map on the indecomposable quotient . Specializing further, and combining with Theorem 78, we obtain the following result.
Corollary 80.
Let be a connected bimonoid. If is cocommutative, then maps to and is in fact a projection from onto . If is commutative, then factors through and splits the canonical projection .
We provide another (though similar) proof of this result in Corollary 101.
13. The characteristic operations (Hopf powers)
The considerations of Section 10 show that the combinatorics of set decompositions is intimately linked to the notion of bimonoid, with set compositions playing the same role in relation to connected bimonoids. On the other hand, in Section 11 we discussed two particular bimonoids and which themselves are built out of set (de)compositions. This double occurrence of set (de)compositions acquires formal meaning in this section, where we show that elements of give rise to operations on connected bimonoids, and elements of to operations on arbitrary bimonoids.
13.1. The characteristic operations on a bimonoid
Let be a bimonoid and . Given an element , write
for some , with running over the decompositions of (and all but a finite number of equal to zero). Define an element by
| (257) |
Here and denote the higher product and coproduct of . We refer to as the characteristic operation of on .
In particular, for a decomposition of , we have
| (258) |
We may take , and (257) results in an operation on each space . According to the following result, this is simply the linearization of the Tits product of Section 1.11.
Proposition 81.
For any decompositions and of ,
| (259) |
Endow with the corresponding algebra structure (the monoid algebra of the monoid of compositions under the Tits product). The unit element is and the product is as in (259). We call the Tits algebra of decompositions.
Consider the map
| (260) |
The result below shows that, when is cocommutative, is a left action of the algebra on the space (and a right action when is commutative).
Theorem 82.
The following properties hold.
-
(i)
For any ,
(261) -
(ii)
If is cocommutative, then for any and ,
(262) -
(iii)
If is commutative, then for any and ,
(263)
Proof.
The following result links the bimonoid structures of and through the characteristic operations. First, we extend (260) to a map
for any decomposition , by defining
| (264) |
for and . Let be a decomposition such that . It follows that
| (265) |
for all splittings of , and .
Theorem 83.
Let be a bimonoid and a decomposition of . The following properties are satisfied.
-
(i)
For any and ,
(266) -
(ii)
If is commutative, then for any and ,
(267) -
(iii)
If is cocommutative, then for any and ,
(268)
Proof.
We prove (266). We may assume for a splitting of , . We have
We used (265), higher (co)associativity (162) and formula (196).
The remaining identities can be proven by similar arguments. ∎
Remark.
The conditions in Theorem 83 may be interpreted as axioms for a module over a ring in the setting of -monoidal categories, with (266) and (267) playing the role of right and left distributivity of multiplication over addition. We do not pursue this point here. Related results are given by Hazewinkel [31, Section 11], Patras and Schocker [54, Corollary 22], and Thibon [73, Formula (52)], among others.
Let be a bimonoid and . Let be the map defined by
| (269) |
Proposition 84.
The map is a morphism of monoids. If is cocommutative, then each
is a morphism from the Tits algebra to the algebra of linear endomorphisms (under ordinary composition). If is commutative, then it is an antimorphism.
13.2. The characteristic operations on arbitrary bialgebras
Let be an arbitrary bialgebra over . Let denote the structure maps. We employ Sweedler’s notation in the form
for .
For each integer , the higher product
is well-defined by associativity. One also defines
The higher coproducts are defined dually.
Given and , define an element by
In the recent Hopf algebra literature, the operations are called Hopf powers [46] or Sweedler powers [36, 37]. The term characteristic is used in [28, Section 1] and [50, Section 1] for these operations on graded bialgebras. They enjoy the following properties. Let and .
-
(i)
.
-
(ii)
If is either commutative or cocommutative, then .
-
(iii)
.
-
(iv)
If is commutative, then .
-
(v)
If is cocommutative, then .
These properties are special cases of those in Theorems 82 and 83. They arise by choosing (as in Section 2.10) and . In this case, , , and . For instance, (ii) follows from (262) in view of the fact that the Tits product in corresponds to multiplication in . More general properties follow by choosing .
13.3. The characteristic operations on a connected bimonoid
Let be a connected bimonoid and . Given an element , formula (257) defines an element (the sum is now over the compositions of ). As before, we refer to this element as the characteristic operation of on .
The set of compositions of is a monoid under the Tits product (10) and is the algebra of this monoid. We use to denote its product, as in (259). We call the Tits algebra of compositions.
The results of Proposition 81 and Theorem 82 continue to hold for the characteristic operations of on , with the same proofs. In particular, when is cocommutative, defines an action of the algebra on the space .
To the result of Proposition 81 we may add the following.
Proposition 85.
For any compositions and of ,
| (270) |
In particular, if and have the same support, then .
Consider the morphism of bimonoids of (195). As noted in (20), the map preserves Tits products. The following result shows that, when specialized to connected bimonoids, the characteristic operations factor through .
Proposition 86.
Let be a connected bimonoid. For any and ,
Proof.
On connected bimonoids, the characteristic operations enjoy additional properties to those in Theorem 83.
Proposition 87.
Let be a connected bimonoid and a composition of . The following properties are satisfied.
-
(i)
For any and ,
(271) -
(ii)
For any and ,
(272) -
(iii)
For any and ,
(273)
Proof.
Let be a connected bimonoid. By Proposition 86, the map factors through . We also use to denote the resulting map
Recall from Section 3.2 that is a monoid, and moreover a Hopf monoid if is finite-dimensional.
Proposition 88.
The map is a morphism of monoids, and moreover of Hopf monoids if is finite-dimensional. If is cocommutative, then each
is a morphism from the Tits algebra to the algebra of linear endomorphisms (under ordinary composition). If is commutative, then it is an antimorphism.
Proof.
Remark.
Connected graded Hopf algebras also carry characteristic operations. In this context, the role of the Hopf monoid is played by the Hopf algebra of noncommutative symmetric functions, and that of the Tits algebra by Solomon’s descent algebra of the symmetric group. The latter is the invariant subalgebra of under the action of and hence has a linear basis indexed by compositions of the integer .
Noncommutative symmetric functions are studied in a series of papers starting with [26] by Gelfand et al and including [73] by Thibon. The descent algebra (of a finite Coxeter group) is introduced by Solomon in [67]. The theory of characteristic operations on graded Hopf algebras appears in work of Patras [51]. It also occurs implicitly in a number of places in the literature. The operation of Solomon’s descent algebra on the Hopf algebra of noncommutative symmetric functions is considered in [26, Section 5.1].
13.4. Primitive operations
We study the effect of operating on a cocommutative connected bimonoid by primitive elements of . We write
| (274) |
for a typical element of .
Theorem 89.
Let and be as above, and .
-
(i)
If , then .
-
(ii)
If in addition , then .
Proof.
It is convenient to restate the above result in terms of the map of (269).
Theorem 90.
Let and be as in Theorem 89.
-
(i)
If , then .
-
(ii)
If in addition , then is a projection from onto .
Remark.
The above result has a converse:
Proposition 91.
Let be an element such that
for every cocommutative connected bimonoid . Then
Proof.
Since is a cocommutative connected bimonoid, we may apply the hypothesis to . Hence for all compositions . But in view of (259) we have , so . ∎
We may consider the operation of on itself.
Corollary 92.
Let be as in (274). If is a primitive element of , then .
Proof.
This follows from item (ii) in Theorem 89 by letting , . ∎
Thus, a primitive element in the Hopf monoid either satisfies , or (if ) is a quasi-idempotent in the Tits algebra . If , it is an idempotent.
The link between the Hopf monoid structure of and the algebra structure of is further witnessed by the following result.
Corollary 93.
The primitive part is a right ideal of the Tits algebra . Moreover, if as in (274) is any primitive element with , then is the right ideal of generated by .
Proof.
Applying statement (i) in Theorem 89 with and an arbitrary element of , we deduce that the right ideal generated by is contained in . Applying item (ii) to an arbitrary element of , we deduce , and hence the converse inclusion. ∎
The following is a necessary condition for an element of to be primitive.
Proposition 94.
Proof.
Denote the left-hand side of (275) by . Since , we have . Moreover, for any ,
(Let be any face with support . Then the coefficient of in is the left-hand side above, and it is zero since is primitive.) These conditions imply as required. ∎
13.5. The cumulants of a connected bimonoid
Let be a finite-dimensional connected bimonoid. Recall from (124) that
for each partition of . The cumulants of are the integers defined by
| (276) |
or equivalently, by
| (277) |
where denotes the Möbius function of the partition lattice. The -th cumulant is
where and is the partition of with one block. Thus,
| (278) |
One can deduce from (16) that
| (279) |
Proposition 95.
For any finite-dimensional cocommutative connected bimonoid , the dimension of its primitive part is
| (280) |
Proof.
Let as in (274) be any primitive element of with . (As mentioned above, such elements exist.) Then is a projection onto , by Theorem 90. Hence the dimension of its image equals its trace, and
Since is idempotent, the trace of equals the dimension of its image. Now, and by (185). Hence the dimension of the image of equals the dimension of . So
Since , we have
Now, by Proposition 94 and (278),
It follows from (279) and (280) that the integers are nonnegative, a fact not evident from their definition. Here are some simple examples, for the Hopf monoids , and of Section 9.
Remark.
Suppose the integer is the -th moment of a random variable . Then the integer is the -th cumulant of in the classical sense [22, 23]. Proposition 95 implies that if is cocommutative, then all cumulants are nonnegative. For example, is the -th Bell number. This is the -th moment of a Poisson variable of parameter . Also, is the -th moment of an exponential variable of parameter , and is the -th moment of the Dirac measure .
13.6. Group-like operations
Let be a series of and a bimonoid. Building on (269), the operation is defined by
If is connected, the same formula defines the operation of a series of on .
Theorem 96.
Let be a group-like series of . If is (co)commutative, then is a morphism of (co)monoids.
Proof.
The universal series of is defined by
| (281) |
for all finite sets . In particular, .
Lemma 97.
The universal series is group-like.
Proof.
Indeed, by (187), . ∎
By (261), the universal series operates on as the identity:
| (282) |
The terminology is justified by the following results.
Theorem 98.
Let be a series of a monoid . Then there exists a unique morphism of monoids such that
| (283) |
Moreover, is given by
| (284) |
for any decomposition of
Proof.
This is a reformulation of the fact that is the free monoid on the species . ∎
Theorem 99.
Let be a group-like series of a bimonoid . Then the unique morphism of monoids such that
| (285) |
is in fact a morphism of bimonoids. Moreover, is given by
| (286) |
Proof.
From Theorem 98 we deduce a bijection between the set of series of and the set of monoid morphisms from to given by
The inverse maps a series to the morphism in (284).
The set is an algebra under the Cauchy product (204) and the space of species morphisms is an algebra under the convolution product (49). These operations do not correspond to each other. Indeed, the subset is not closed under either addition or convolution.
Similarly, from Theorem 99 we deduce a bijection between the set of group-like series of and the set of bimonoid morphisms from to given by
| (287) |
If is commutative, then is a group under convolution, and the above bijection is an isomorphism of monoids. This follows from (267).
The results of this section hold for group-like series of in relation to connected bimonoids . The role of the universal series is played by . As in (281), we have
| (288) |
for nonempty , and (where is the composition of with no blocks).
Remark.
Theorem 99 is analogous to [2, Theorem 4.1]. The latter result is formulated in dual terms and for graded connected Hopf algebras. In this context, the expression for the morphism in terms of the characteristic operations appears in [32, Proposition 3.7]. Many of these ideas appear in work of Hazewinkel [31, Section 11]. (Warning: Hazewinkel uses curve for what we call group-like series.) One may consider the bijection (287) in the special case . The analogue for graded connected Hopf algebras is then a bijection between endomorphisms of the Hopf algebra of noncommutative symmetric functions, and group-like series therein. This is used extensively in [38] under the name transformations of alphabets.
14. Classical idempotents
This section introduces a number of idempotent elements of the Tits algebra of compositions. We start by studying the first Eulerian idempotent. This is in fact a primitive element of and it operates on cocommutative connected bimonoids as the logarithm of the identity.
We then introduce a complete orthogonal family of idempotents for the Tits algebra. The family is indexed by set partitions. We call these the Garsia-Reutenauer idempotents. The idempotent indexed by the partition with one block is the first Eulerian. Lumping according to the length of the partitions yields the higher Eulerian idempotents.
In the Tits algebra of compositions, we define elements indexed by integers , whose characteristic operation on a connected bimonoid is the -th convolution power of the identity. There is a simple expression for these elements in terms of the higher Eulerian idempotents which diagonalizes their operation on .
We introduce the Dynkin quasi-idempotent and describe its operation on connected bimonoids. We establish the Dynkin-Specht-Wever theorem for connected Hopf monoids.
14.1. The first Eulerian idempotent
We proceed to define a primitive series of . We call it the first Eulerian idempotent and denote it by .
Set , and for each nonempty finite set define an element by
| (289) |
Recall the group-like series of (288). It is given by . It follows that
| (290) |
with the logarithm of a series as in (234).
Proposition 100.
The series of is primitive and the element is an idempotent of the Tits algebra .
Proof.
The primitivity of the may be deduced in other ways. We may derive it from (291) and the coproduct formula (191), using the fact that no proper subset of is -admissible. Alternatively, applying (270) with , we have for , so is primitive since is injective by Corollary 58.
We turn to the characteristic operation (257) of on connected bimonoids.
Corollary 101.
For any connected bimonoid ,
| (292) |
Moreover, if is cocommutative, then is a projection onto the primitive part .
Proof.
Remark.
Recall from (275) that for an element of to be primitive, the sum of the coefficients of compositions with support must be . So a natural way to construct a primitive is to let these coefficients be equal. Since there are such compositions, each coefficient must be
by (17). This is precisely the coefficient used to define in (289). This observation may be used to extend the definition of the first Eulerian idempotent to more general settings (beyond the scope of this paper, but hinted at in Section 1.10).
Remark.
The first Eulerian idempotent belongs to Solomon’s descent algebra (and hence to the symmetric group algebra). As such, it appears in work of Hain [29]. Its action on graded Hopf algebras is considered by Schmitt [66, Section 9]. Patras and Schocker [54, Definition 31] consider this element in the same context as here. It is the first among the higher Eulerian idempotents. These elements are discussed in Section 14.3 below, and additional references to the literature (which also pertain to the first Eulerian idempotent) are given there.
14.2. The Garsia-Reutenauer idempotents
Fix a nonempty set . The element is a part of a family of elements indexed by partitions of : For each , let be the element defined by
| (293) |
We call these elements the Garsia-Reutenauer idempotents. It follows from (189) that
| (294) |
Setting recovers the first Eulerian: .
Theorem 102.
For any nonempty set , we have
| (295) | |||
| (296) |
Theorem 102 states that the family , as runs over the partitions of , is a complete system of orthogonal idempotents in the Tits algebra . (They are moreover primitive in the sense that they cannot be written as a sum of nontrivial orthogonal idempotents.)
Proposition 85 and (293) imply that for any composition with support ,
| (297) |
Note from (18) and (189) that whenever and have the same support; so we also have . One can now deduce that is an idempotent and
| (298) |
Remark.
The Garsia-Reutenauer idempotents (indexed by set partitions) possess analogues indexed by integer partitions which play the same role for graded Hopf algebras as the do for Hopf monoids. They are elements of Solomon’s descent algebra and appear in [25, Theorem 3.2] and [61, Section 9.2].
Brown [17, Equations (24) and (27)] and Saliola [62, Section 1.5.1] or [63, Section 5.1] construct many families of orthogonal idempotents for the Tits algebra of a central hyperplane arrangement. The Garsia-Reutenauer idempotents constitute one particular family that occurs for the braid arrangement. Brown worked in the generality of left regular bands; for further generalizations, see the work of Steinberg [70].
14.3. The higher Eulerian idempotents
Let be any integer. Set , and for each nonempty set define
| (299) |
The sum is over all partitions of of length . This defines the series of . We call it the -th Eulerian idempotent. The series is the first Eulerian idempotent as defined in Section 14.1. By convention, we set to be the unit series (205).
The Cauchy product (204) of the series with itself can be computed using (190) and (291):
The last step used (293) and (299). More generally, by the same argument, we deduce
| (300) |
The right-hand side involves the -fold Cauchy power of .
Recall from Corollary 101 that the characteristic operation of on a connected bimonoid is . Since is a morphism of monoids (Proposition 88), it preserves Cauchy products of series and we obtain the following result.
Proposition 103.
The characteristic operation of on a connected bimonoid is
| (301) |
Remark.
The higher Eulerian idempotents appear in the works of Gerstenhaber and Schack [27, Theorem 1.2], [28, Section 3], Hanlon [30], Loday [40, Proposition 2.8], Patras [49, Section II.2], and Reutenauer [60, Section 3], [61, Section 3.2], among other places. A related idempotent (the sum of all higher Eulerian idempotents except the first) goes back to Barr [9]; see [27, Theorem 1.3].
14.4. The convolution powers of the identity
In the Tits algebra of decompositions , define for any nonnegative integer ,
| (302) |
The sum is over all decompositions of of length . The resulting series of is denoted .
Proposition 104.
We have
| (303) |
Proof.
Let be a decomposition of of length , and one of length . Then their Tits product has length . If we arrange the blocks of in a matrix as in (9), we recover by taking the union of the blocks in each row of this matrix, and we similarly recover from the columns. Thus, the Tits product sets up a bijection between pairs of decompositions of lengths and , and decompositions of length . The result follows. ∎
It is clear from the definitions (257) and (302) that the characteristic operation of is the -th convolution power of the identity:
Proposition 105.
For any bimonoid, we have
| (304) |
We deduce that when composing convolution powers of the identity of a bimonoid that is either commutative or cocommutative, the exponents multiply:
| (305) |
This follows by applying to (303), using (304), and the fact that in this situation
either preserves or reverses products (Proposition 84).
Under the morphism , the element maps to
| (306) |
The sum is over all compositions of ; the binomial coefficient accounts for the number of ways to turn into a decomposition of of length by adding empty blocks. Since preserves Tits products, it follows from (303) that
| (307) |
It follows from (304) that on a connected bimonoid,
| (308) |
Comparing with (288), we see that , and more generally
| (309) |
The latter is the Cauchy product in the algebra of series of .
Theorem 106.
We have
| (310) |
Proof.
Observe that (306) defines for any integer . In particular, for ,
| (311) |
By polynomiality, the identities (307)–(310) continue to hold for all scalars and . In particular, from (310) we deduce that
| (312) |
and from (308) that this element operates on a connected bimonoid as the antipode:
| (313) |
Together with (311), this yields another proof of Takeuchi’s formula (111).
Remark.
The elements belong to Solomon’s descent algebra (and hence to the symmetric group algebra). In connection to the higher Eulerian idempotents, they are considered in [28, Section 1], [40, Definition 1.6 and Theorem 1.7], and [49, Section II]. They are also considered in [11, Lemma 1] and [14, Proposition 2.3] in connection to riffle shuffles. The connection to convolution powers of the identity of a graded Hopf algebra is made in [28, Section 1], [41, Section 4.5] and [50, Section 1]. A discussion from the point of view of -rings is given in [52, Section 5].
14.5. The Dynkin quasi-idempotent
Fix a nonempty finite set . Given a composition of , let
denote its last block.
For each , define an element by
| (314) |
The sum is over all compositions of whose last block contains . The sum of these elements defines another element of
| (315) |
We call it the Dynkin quasi-idempotent. It follows that
| (316) |
where denotes the size of the last block of . The elements define a series of . By convention, . The following result shows that this series is primitive.
Proposition 107.
The elements (and hence ) are primitive elements of .
Proof.
Let be a decomposition with . Let and . Then by (187), the coefficient of in is
The sum is over all quasi-shuffles of and such that . (Quasi-shuffles are defined in Section 1.2.) We show below that this sum is zero.
We refer to the which appear in the sum as admissible quasi-shuffles. If does not belong to either or , then there is no admissible and the sum is zero. So suppose that belongs to (say) . For any admissible quasi-shuffle in which appears by itself, merging this last block of with the next block (which exists and belongs to ) yields another admissible quasi-shuffle say . Note that can be recovered from . Admissible quasi-shuffles can be paired off in this manner and those in a pair contribute opposite signs to the above sum. ∎
Corollary 108.
For each , the element is an idempotent of the Tits algebra . The element is a quasi-idempotent of the Tits algebra:
For any cocommutative connected bimonoid , the characteristic operations and are projections from onto .
The operations of and on can be described in terms of the antipode of ; these results do not require cocommutativity.
Proposition 109.
For any connected bimonoid and any ,
| (317) |
Proof.
According to (314), we have
where the sum is over those compositions of for which lies in the last block. Given such , let be its last block and let be the union of the other blocks. Note that may be empty. Then consists of a composition of followed by . By (co)associativity (180),
Therefore,
by Takeuchi’s formula (111). ∎
Corollary 110.
Proof.
In view of (317),
It follows from (313) and (318) that
| (319) |
In other words, the Dynkin quasi-idempotent arises from the construction of Corollary 73 applied to the universal series.
Remark.
The Dynkin quasi-idempotent is classical. The analogue of formula (318) for graded Hopf algebras appears in work of Patras and Reutenauer [53, Section 3] and (less explicitly) in work of von Waldenfels [75]. The analogue of (319) is the definition of the Dynkin quasi-idempotent by Thibon et al [38, Section 2.1], [73, Section 2.1]. (In these references, this element is also called the power sum of the first kind.) Aubry studies the operator in [8, Section 6]. The operators are considered in unpublished notes of Nantel Bergeron; related ideas appear in Fisher’s thesis [21]. The idempotents are specific to the setting of Hopf monoids.
14.6. The Dynkin-Specht-Wever theorem
Recall the bimonoid of linear orders (Section 9.2). We proceed to describe the characteristic operation of the Dynkin quasi-idempotent on . In the discussion, always denotes a nonempty set.
To begin with, note that the commutator bracket (72) of satisfies
| (320) |
for any , and linear orders on and on .
Lemma 111.
Let be a linear order on . Then
| (321) |
Proof.
Expanding using (320) we obtain a sum of elements of the form . The linear orders are characterized by the following property: In , any either precedes all of or succeeds all of them. Write , where consists of those for which preceeds . By convention, , so the sign in front of is the parity of . Also note that , that is, in , the elements of appear first but reversed from the order in which they appear in , followed by the elements of appearing in the same order as in . ∎
Recall the element from (314).
Proposition 112.
Let be a linear order on . Then, for any ,
| (322) |
Proof.
Put . By (317),
The second step used the explicit formulas for the product, coproduct and antipode of . The claim in the case follows from Lemma 111. Now suppose . Then may appear either in or in . In the former case, is the last element of , while in the latter, is the first element of . Thus, for two decompositions and which differ only in the location of , we have and . Thus . ∎
Theorem 113.
For any linear order on ,
| (323) |
This result shows that operates on as the left bracketing
This operator on (rather than the element ) is sometimes taken as the definition of the Dynkin quasi-idempotent.
As a special case of Corollary 108, we have the following result. Recall that .
Corollary 114.
The image of the left bracketing operator on is . In addition, is the eigenspace of eigenvalue of this operator.
This is the analogue for Hopf monoids of the classical Dynkin-Specht-Wever theorem [59, Theorem 2.3].
We turn to the left bracketing operator on an arbitrary connected bimonoid .
Given a composition of a finite set and elements . Recall that denotes the higher product of (45). Let us write
Thus, is the product of the in the order specified by . The left bracketing of the is the element defined by
It is obtained by iterating the commutator bracket (72) of .
Let denote the first block of .
| (324) |
where the sum is over all decompositions for which is -admissible and . (Note that may be empty.) If the are primitive, then, for any ,
| (325) |
Under the same hypothesis,
| (326) |
Recall the cocommutative Hopf monoid from Section 6.2.
Corollary 115.
The primitive part is the smallest Lie submonoid of containing .
Proof.
Recall from Proposition 13 that is a Lie submonoid of . Moreover, , so it only remains to show that every primitive element belongs to the Lie submonoid generated by . Accordingly, pick a primitive element . The elements span , as the vary. So express as a linear combination of these elements. Now apply to this equation and use Corollary 108 and (326). This expresses (and hence ) as a linear combination of the . ∎
Remark.
There is a “right version” of the Dynkin quasi-idempotent. It is defined as follows. Let be as in (314), where the sum is now over all set compositions whose first block contains . Let be the sum of these over all :
This defines another primitive series of . It satisfies
| (327) |
Its characteristic operation on a product of primitive elements is given by “right bracketing”.
15. The Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems
We establish analogues of the Poincaré-Birkhoff-Witt (PBW) and Cartier-Milnor-Moore (CMM) theorems for connected Hopf monoids. These results appeared in the work of Joyal [35] and Stover [71], with a precursor in the work of Barratt [10]. We provide here an approach based on the Garsia-Reutenauer idempotents. For connected Hopf algebras, proofs of these classical results can be found in [44, Section 7] and [57, Appendix B].
Let be a field of characteristic .
15.1. The canonical decomposition of a cocommutative connected bimonoid
Let be a cocommutative connected bimonoid. Let be any nonempty set. Consider the characteristic operation of the first Eulerian idempotent on . By Corollary 101,
| (328) |
More generally, the characteristic operation of the Garsia-Reutenauer idempotent on is related to the primitive part of as follows.
Proposition 116.
For any composition of with support , the map restricts to an isomorphism
Proof.
Write . Since ,
for any . This used (190), (258) and (273). Now by Corollary 58, is the identity, so is surjective (onto ) and is injective, and by (291) and (328),
One can deduce from here that the maps and restrict to inverse isomorphisms:
We know from (298) that and are canonically isomorphic under . Further,
The last step used (297). The claim follows. ∎
Theorem 117.
For any cocommutative connected bimonoid ,
| (329) |
and, if is finite-dimensional, then
| (330) |
with as in (277).
Proof.
Theorem 118.
For any cocommutative connected bimonoid , there is an isomorphism of comonoids
| (331) |
Explicitly, the isomorphism is given by the composite
| (332) |
where, for with , the map is the canonical isomorphism
| (333) |
Proof.
Let be the (isomorphic) image of inside the direct sum in (332). Consider the composite map
In both maps ranges over all compositions with support . The first map is an isomorphism by Proposition 116. Using (293) and (297), one can deduce that the composite is the identity:
So the second map is an isomorphism as well. It follows that the image of (332) is precisely . Combining with the decomposition (329) establishes the isomorphism (331).
To check that this is a morphism of comonoids, we employ the coproduct formula of given in Section 7.2. The calculation is split into two cases, depending on whether is -admissible or not. Both cases can be handled using the higher compatibility axiom (167) for arbitrary and . We omit the details. ∎
15.2. PBW and CMM
Let be any positive species. For any partition , consider the map
where is as in (333). Summing over all yields a map
| (334) |
It is straightforward to check that this is a morphism of comonoids.
Now let be a positive Lie monoid. Composing the above map with the quotient map yields a morphism
| (335) |
of comonoids.
Theorem 119 (PBW).
For any positive Lie monoid , the map (335) is an isomorphism of comonoids. In particular, the canonical map is injective.
Proof.
The first step is to construct a surjective map fitting into a commutative diagram
The map on each summand is defined by an induction on the rank of . We omit the details. The existence of shows that (335) is surjective.
To show that (335) is injective, we need to show that , the kernel of , is contained in the kernel of . Suppose is a surjective map of Lie monoids, and suppose that this result holds for . Then the result holds for as well. To see this, consider the commutative diagram
The key observation is that the map is surjective. (This can be deduced using the generating relations (134).) Then the top composite being zero implies that the bottom composite is also zero as required.
Next note that for any positive Lie monoid , the canonical map (defined using the Lie structure of ) is a surjective morphism of Lie monoids. So it suffices to prove injectivity for free Lie monoids. Accordingly, let . The map (335) is the composite
The second map is an isomorphism by Theorem 118. The first map is induced by . In the free Lie monoid case, by (136). So by Lemma 27, this map (and hence the induced map) is injective. ∎
Theorem 120 (CMM).
The functors
form an adjoint equivalence.
In other words, the functors form an adjunction for which the unit and counit are isomorphisms
| (336) |
Proof.
The adjunction follows from (135), since (connected) cocommutative bimonoids form a full subcategory of the category of (connected) bimonoids.
The isomorphism (331) can be expressed as the composite
The first map is obtained by setting in (335) and hence is an isomorphism by Theorem 119. Hence the second map which is as in (336) is also an isomorphism.
By the adjunction property,
is the identity. From the above, we know that the second map is an isomorphism. Hence so is the first. Now by Theorem 119, a Lie monoid embeds in its universal enveloping monoid. So this map restricts to an isomorphism as required. ∎
Corollary 121.
There is an isomorphism of Lie monoids
| (337) |
More precisely, the image of the map (132) identifies with the primitive part of . In particular, is the Lie submonoid of generated by . This latter result was obtained by different means in Corollary 115.
Remark.
Theorem 120 is due to Stover [71, Proposition 7.10 and Theorem 8.4]. This result does not require characteristic . The map (335) in the PBW isomorphism is the analogue of Quillen’s map in the classical theory [57, Appendix B, Theorem 2.3]. Theorem 119 is due to Joyal [35, Section 4.2, Theorem 2]. He deduces it from the classical PBW; he does not mention comonoids though. Stover [71, Theorem 11.3] uses a different map to show that and are isomorphic; his map is not a morphism of comonoids.
16. The dimension sequence of a connected Hopf monoid
In this section, all species are assumed to be finite-dimensional.
We consider three formal power series associated to a species :
They are the exponential, type, and ordinary generating functions, respectively. In the second function, the coefficient of is the dimension of the space of coinvariants of under the action of .
For example,
These results are given in [12]. More generally, for any positive species ,
These follow from [12, Theorem 1.4.2].
As for the ordinary generating function, note that for any species
The following results assert that certain power series associated to a connected Hopf monoid have nonnegative coefficients. They impose nontrivial inequalities on the dimension sequence of .
Theorem 122.
Let be a connected -Hopf monoid. Then
Theorem 122 is given in [7, Theorem 4.4]. The proof goes as follows. By Theorem 20, is free. Hence there is a positive species such that
and this gives the result. It can be stated as follows: the Boolean transform of the dimension sequence of is nonnegative.
Theorem 123.
Let be a connected Hopf monoid. Then
This result is given in [4, Corollary 3.4]. For the proof one applies the dual of Theorem 15 to the surjective morphism of Hopf monoids
The same argument yields
but as shown in [7, Section 4.6], this result follows from Theorem 122.
Theorem 124.
Let be a nontrivial set-theoretic connected Hopf monoid and . Then
This is given in [4, Corollary 3.6]. Nontrivial means that . In this case one must have for all . Then surjects onto canonically (as Hopf monoids) and one may again apply Theorem 15 to conclude the result. The first condition states that the binomial transform of the sequence is nonnegative. The second that the sequence is weakly increasing.
Theorem 125.
Let be a cocommutative connected Hopf monoid. Then
Theorem 118 implies , whence the result.
Remark.
Let be a connected Hopf monoid. Suppose the integer is the -th moment of a random variable (possibly noncommutative) in the classical sense. Theorem 125 may be rephrased as follows: if is cocommutative, then the classical cumulants of are nonnegative. See the remark at the end of Section Remark. There is a notion of Boolean cumulants and another of free cumulants [39, 68]. Lehner obtains expressions for these as sums of classical cumulants [39, Theorem 4.1]; hence their nonnegativity holds for cocommutative . However, Theorem 122 yields the stronger assertion that the Boolean cumulants are nonnegative for any connected Hopf monoid . We do not know if nonnegativity of the free cumulants continues to hold for arbitrary . For information on free cumulants, see [47].
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