Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

The calculations we perform on the higher cyclic Hoshschild complex which lead to the formality proof as well as the construction of this complex itself are inspired by the notion of pre-Calabi-Yau algebra, which was introduced by Kontsevich, Vlassopoulos [26] (see also the talk [25]), Seidel [31], and Tradler, Zeinalian [36]. It turned out that this structure is present in many different areas, including topology of compact manifolds with boundary, algebraic geometry, symplectic geometry. For example, Fano varieties are endowed with a pre-Calabi-Yau structure, open Calabi-Yau manifolds have this structure, from the HMS conjecture it is expected that the Fukaya wrapped category of an open symplectic manifold endowed with a pre-Calabi-Yau structure. Pre-Calabi-Yau structure produces a generalization (for arbitrary genus) of the Tamarkin-Tsygan calculus. Big class of examples comes from the notion of algebra (category) of finite type, introduced by Toën and Vaquié [35]. For these algebras $d$-pre-Calabi-Yau structure on $A$ produces a $(2-d)$-shifted derived Poisson structure on the moduli stack of finite-dimensional $A$-modules.

The philosophy presented in the Kontsevich and Soibelman paper [21] says that it is natural to construct a formal noncommutative geometry where the role of manifold played by $A_{\infty}$-algebra, and the role of Poisson structure, by pre-Calabi-Yau algebra. There were numerous attempts in the past to introduce a notion of noncommutative Poisson structure. First natural thought in this direction is to try and define it exactly like in commutative case, as a Lie bracket $\{.,.\}:A\times A\to A$ which satisfies the Leibnitz rule $\{a,bc\}=\{a,b\}c+b\{a,c\}$. However this definition would give a very restricted class of objects. As it was shown by Farkas and Letzter [12] the only such Poisson bracket on noncommutative prime ring is the bracket $[a,b]=ab-ba.$ So this definition turned out to be not what one would hope for. Another notion of noncommutative Poisson bracket was suggested by Xu [40] and by Block and Getzler [2]. It had the property that if $A$ endowed with this noncommutative Poisson structure, then on the centre of $A$, ${\cal Z}(A)$ there is an induced commutative Poisson structure, but the definition did not ensure that there will be an induced Poisson structure on representation spaces of $A$ or on their moduli. It was desirable to find a notion of noncommutative Poisson structure, which would behave well on the testing ground of representation spaces. This was achieved in the definition of double Poisson bracket by Van den Bergh [39]. The bracket was ’thickened’, that is it was defined as a map $\{\!\{.,.\}\!\}:A\otimes A\to A\otimes A$ , satisfying axioms which are certain generalisation of the Leibnitz and the Jacobi identities on $A\otimes A$. In the work of Crawley-Boevey [10] the definition of noncommutative ($H_{0}$) Poisson structure was given as a Lie bracket on zero Hochschild homology $H_{0}(A)=A/[A,A]$, which can be lifted to a derivation on $A$. We put these latter developments into the context of pre-Calabi-Yau structures (or rather associated cohomology) and embed noncommutaive bivector fields into a calculus of ’highly noncommutative’ words. This gives a new perspective, kind of panoramic aerial view on what is going on, how noncommutative notions involving ’thickening’ or factorizations arise. The attempts to reach a good behaviour on representation spaces was based on the philosophy introduced by Kontsevich and Rosenberg [24, 27], we follow the ideas of this paper throughout the text and pursue some aspects of those, as well as of their reincarnations, for example, in Kontsevich and Soibelman [21].

We give several equivalent definitions of pre-CY structures, one of them in terms of higher cyclic Hochshild complex. Amongst the advantages of this definition is that it works not only for finite dimensional algebras, or algebras with finite dimensional graded components. We show in section 4 the reason why this definition contains as a particular case the double Poisson bracket. Roughly speaking, a pre-Calabi-Yau structure is a solution of the the Maurer-Cartan equation with respect to generalized necklace bracket in the higher cyclic Hochschild complex (for precise definition see section 3).

We study the higher cyclic Hochschild complex (${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$), its homologies and Lie structure. One of the tools we use for the combinatorial description of Lie structure, and for the proof of purity of this complex is a calculus of noncommutative cyclic words with labels, which we introduce. We start with the free associative algebra $A=\mathbb{K}\langle X\rangle$ with finite number of generators $X=\{x_{1},...,x_{r}\}$ and ’labels’ $\delta_{1},...,\delta_{r}$, $\xi$. Monomials from free algebra $\mathbb{K}\langle X\rangle$ are written cyclically on the circle and separated by labels. These generalised cyclic monomials with labels, which we call $\xi\delta$-monomials, represent operations on tuples of monomials from $A$ and form a convenient basis in the small subcomplex of the higher cyclic Hochschild complex. The $\xi\delta$-monomials can be considered as highly noncommutative words, which can be multiplied not only from the right or from the left, but from $r$ sides, where $r$ is a number of $\delta$th in the monomial. The result on formality of the higher cyclic Hochschild complex we got provides $L_{\infty}$ quasi-isomorphism between higher cyclic Hochschild complex and its cohomology. Thus the consideration of homology of the complex instead of the complex itself for the notion of noncommutative Poisson structures and of noncommutative polyvector fields becomes justified.

Whenever we are working with the higher cyclic Hochschild complex itself, without embracing in further word combinatorics, we can speak of an arbitrary smooth algebra $A$, as defined in [27], i.e. a finitely generated algebra $A$, with kernel of multiplication being a projective $A$-bimodule.

To deal with the higher cyclic Hochschild complex ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}(A)=C^{(\bullet)}(A)$ we choose a small subcomplex $\xi^{(\bullet)}$, quasi-isomorphic to $C^{(\bullet)}(A).$ We specify a particular embedding of the subcomplex $\zeta^{(\bullet)}$ into $C^{(\bullet)}(A)$ (section 5) by choosing a basis of $\xi\delta$-monomials in $\xi^{(\bullet)}$ and describing an element of $C^{(\bullet)}$ (which we call an ’operation’) corresponding to a $\xi\delta$-monomial. The element of $\prod\limits_{r_{1},...,r_{n}\geqslant 0}{\rm Hom}\,(\mathop{\otimes}\limits_{i=1}^{n}A^{\otimes r_{i}},A^{\otimes n})$, corresponding to $\xi\delta$-monomial with $m=\sum r_{i}$ occurrences of labels $\delta_{i},i=\bar{1,r}$, and $n-m$ occurrences of label $\xi$ is schematically shown in the following picture.

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The black arches are input monomials from $A$ and green arches are output monomials, consisting of parts of inputs and parts of $\xi\delta$-monomial defining the operation. We suppose orientation is clockwise everywhere. One can read the output monomials following this orientation. In this particular picture we see the $\xi\delta$-monomial which encodes an operation $\Phi:A^{\otimes 3}\to A^{\otimes 5}$ (more precisely, $\Phi:A\otimes A^{0}\otimes A\otimes A^{0}\otimes A\to A^{\otimes 5})$.

The particular case of operation encoded by $\xi\delta$-monomial with two $\delta$th and no $\xi$th correspond to the Poisson double bracket. Indeed, the operation $A\otimes A\to A\otimes A$, obtained from such a $\xi\delta$-monomial automatically satisfies the double Leibnitz rule. The double Jacobi identity comes from the Maurer-Cartan equation on the elements of small subcomplex $\zeta^{(\bullet)}$ of the higher Hochschild complex. This will be explained more precisely in section 4. We show there how the double Poisson bracket invented by Van den Bergh [39] as a structure which induce a Poisson bracket on representation space of algebra, appear as a particular pre-Calabi-Yau structure. In [17] we gave a detailed proof of the following fact. Any pre-Calabi-Yau structure with $m_{4}=0$ on arbitrary associative algebra gives rise to a double Poisson bracket according to the formula [17]:

$$(*)\quad\quad\quad\langle g\otimes f,\{\!\{b,a\}\!\}\rangle:=\langle m_{3}(a,f,b),g\rangle,$$

Moreover, an arbitrary double Poisson bracket can be obtained from pre-Calabi-Yau structure of special type, with only second and third multiplications $m_{2}$ and $m_{3}$ present. We comment here on the main idea behind this earlier direct proof from the point of view of the definition of pre-Calabi-Yau structure via higher cyclic Hochschild complex.

Coming back to the general situation, we describe the generalised necklace bracket which endows the higher cyclic Hochschild complex with a graded Lie algebra structure. In section 6 we show how this bracket works in terms of $\xi\delta$-monomials. By this we not only prove that the small subcomplex $\zeta^{(\bullet)}_{A}$ is a Lie subalgebra in ${\bf g}=(C^{(\bullet)}_{A}(A),[,]_{g.n})$, but also give a concrete combinatorial formula for this bracket on $\xi\delta$-monomials. We prove that the bracket $[A,B]$ of two $\xi\delta$-monomials $A,B\in\zeta^{(\bullet)}_{A}$ is $[A,B]=A\circ B-B\circ A$, where $A\circ B$ is a linear combination of $\xi\delta$-monomials obtained from $\xi\delta$-monomials $A$ and $B$ by all possible gluings of $\delta_{j}$ of $A$ and $x_{j}$ of $B$, as shown in the following picture.

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Namely, we glue all $\delta_{j}$ from $\xi\delta$-monomial $A$ to a corresponding $x_{j}$ from $B$, then cut at the place of gluing, and open up to obtain one new $\xi\delta$-monomial ($x_{j}$ and $\delta_{j}$ disappear).

The choice of the basis of $\xi\delta$-monomials in the subcomplex $\xi^{(\bullet)}$ of the higher cyclic Hochschild complex allows, among other things, an easy interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure. Namely, the $\xi\delta$-monomial produce an obvious formal analogue of polyvector field, which in turn create a Poisson structure on the representation space of $A$, via kind of Schouten bracket.

Let us note also the following consequence of our result. In view of quite obvious connection between antisymmetric solutions of Yang-Baxter equation and double Poisson brackets, which was noticed first in [32], this our description of all double Poisson brackets in terms of $\xi\delta$-words, provides at the same time a description of all antisymmetric solutions of the Yang-Baxter equation on the vector space $X$.

In section 7 we concentrate on homological properties of the higher cyclic Hochschild complex and prove its homological purity. We again use the small quasi-isomorphic subcomplex $\zeta^{(\bullet)}_{A}$ introduced in section 5. From the expression of the differential in the whole dualised bar complex $C^{(\bullet)}_{A}(A)$, which we spell out in section 5, we get a differential in $\zeta=\zeta^{(\bullet)}_{\mathbb{K}}$.

Note that while the elements of higher cyclic Hochschild complex are defined as elements of ${\rm Hom}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes r_{i}},A^{\otimes N})^{{\mathbb{Z}}_{N}}$, invariant under ${\mathbb{Z}}_{N}$-acton, our homology calculations are reduced to a related non-${\mathbb{Z}}_{N}$-invariant complex $\tilde{\zeta}$, corresponding to operations with fixed point. This is possible since the differential commutes with the cyclisation procedure (see Lemma 7.21).

The complex with the fixed point $\tilde{\zeta}=\oplus\tilde{\zeta}^{n}_{N}$, where $\tilde{\zeta}^{n}=\{$ monomials $u\in\mathbb{K}\langle\xi,x_{i},\delta_{i}\rangle$, starting from $\xi$ or $\delta_{i}$, such that $\hbox{\tt\rm deg}\,_{\delta}u=n,\hbox{\tt\rm deg}\,_{\delta,\xi}u=N\}_{\mathbb{K}}$ has natural bigrading by $\delta$-degree, and by degree with respect to $\xi$ and $\delta_{i}$th, $i=\bar{1,r}$, the latter we call weight. Essential for our considerations is the cohomological grading by $\xi$-degree: $\zeta=\oplus\zeta(l),$ where $\zeta(l)=\mathop{\oplus}\limits_{N-n=l}\zeta^{n}_{N}$.

In other words, Theorem1.1 holds for path algebra $P{\mathcal{Q}}$ of an arbitrary quiver ${\mathcal{Q}}$, except for the quiver with one vertex and one loop.

This purity result is obtained via use of $\xi\delta-monomials$ as a basis of qiso subcomplex and of the Gröbner bases theory in the ideals of free algebra generated by the element defined by the differential of this complex. Similar techniques and the Gröbner bases theory in the ideals of path algebras are used for the case when $A$ is the path algebra of a quiver $A=P{\mathcal{Q}}.$

As a consequence of purity result we are able to deduce $L_{\infty}$-formality for this complex over $A=\mathbb{K}\langle x_{1},...,x_{r}\rangle,r\geqslant 2$ or $A=P{\mathcal{Q}},|{\mathcal{Q}}_{0}|\geqslant 2$, using standard arguments related to pertrubation theory, similar to the ones appeared in [11].

Since there are more $L_{\infty}$-morphisms between given DGLAs, than just DGLA morphisms, the notion of $L_{\infty}$-formality is weaker than formality. But it is exactly what is needed for deformation theory. One of the main points in [23] emphasise that what really determines the deformation functor, is not just qiso class of a DGLA, but its qiso class as $L_{\infty}$-algebra. Thus the best thing one can achieve in understanding the deformation theory is to prove $L_{\infty}$-formality of corresponding DGLA.

Thus the line of our study related to the deformation theory got its best possible outcome in case of free algebras. The $L_{\infty}$-formality holds also for such smooth (quasi-free) algebras, as path algebras of quivers, but certainly not for an arbitrary associative algebras concentrated in degree zero, or for ${\mathbb{Z}}$-graded free algebras.

The typical example of an algebra in this paper is a free associative algebra $A=\langle x_{1},...,x_{r}\rangle$, the most noncommutative algebra possible. We develop elements of noncommutative geometry based on this algebra following the spirit of [27, 24]. For example, we adopt the ideology introduced and developed in these papers, which says that noncommutative structure should manifest as a corresponding commutative structure on representation spaces. We develop the ideas of these papers further and introduce a calculus of ’highly noncommutative monomials’: monomials which can be multiplied not only from the right and from the left, but from any number of specified directions, we call them $\xi\delta$-monomials.

Throughout the text $A$ will be an associative unital algebra over the field $\mathbb{K}$ of characteristic zero, if not specified otherwise. Denote by $A-mod-A$ the category of all $A$-bimodules, which is the same as $A^{e}$-modules, i.e. modules over the enveloping $A^{e}=A\otimes A^{op}$. We consider mainly Homs of $A$-bimodules or $A^{\otimes N}$-bimodules which we denote ${\rm Hom}\,_{A-mod-A}$ or ${\rm Hom}\,_{A^{\otimes N}-mod-A^{\otimes N}}$ respectively.

To give a definition of pre-Calabi-Yau structure as it was originally defined in [25], [26], [31] we start with reminding the definition of $A_{\infty}$-algebra, or strong homotopy associative algebra introduced by Stasheff [34].

First note, that there are two accepted conventions on grading of an $A_{\infty}$-algebra. They differ by a shift in numeration of graded components. In one convention, we call it shifted convention, each operation has degree 1. While the other, which we call a naive convention is determined by making the binary operation to have degree 0, hence the degrees of operations $m_{n}$ of arity $n$ become $2-n$. If the degree of element $x$ in naive convention is ${\rm deg}x=|x|$, then shifted degree in $A^{sh}=A[1]$, which fall into shifted convention, will be ${\rm deg^{sh}}x=|x|^{\prime}$, where $|x|^{\prime}=|x|-1$, since $x\in A^{i}=A[1]^{i+1}$.

The formulae for the graded Lie bracket, Maurer-Cartan equations and cyclic invariance of the inner form are somewhat different in different conventions. We mainly use the shifted convention, but sometimes need the naive convention as well.

Let $A$ be a ${\mathbb{Z}}$ graded vector space $A=\mathop{\oplus}\limits_{n\in{\mathbb{Z}}}A_{n}$, and $C^{l}(A,A)$ be Hochschild cochains $C^{(}A,A)=\underline{{\rm Hom}\,}(A^{\otimes l},A)$, for $l\geqslant 0$, $C^{\bullet}(A,A)=\prod\limits_{k\geqslant 1}C^{l}(A,A).$

On $C^{\bullet}(A,A)[1]$ there is a natural structure of graded pre-Lie algebra, defined via composition:

$$\circ:C^{l_{1}}(A,A)\otimes C^{l_{2}}(A,A)\to C^{l_{1}+l_{2}-1}(A,A):$$

$$f\circ g(a_{1}\otimes...\otimes a_{l_{1}+l_{2}-1})=$$

$$\sum(-1)^{|g|\sum\limits_{j=1}^{i-1}|a_{j}|}f(a_{1}\otimes...\otimes a_{i-1}\otimes g(a_{i}\otimes...\otimes a_{i+l_{2}+1})\otimes...\otimes a_{l_{1}+l_{2}-1})$$

The operation $\circ$ defined in this way does satisfy the graded right-symmetric identity:

$$(f,g,h)=(-1)^{|g||h|}(f,h,g)$$

where

$$(f,g,h)=(f\circ g)\circ h-f\circ(g\circ h).$$

As it was shown in [14] the graded commutator on a graded pre-Lie algebra defines a graded Lie algebra structure.

Thus the Gerstenhaber bracket $[-,-]_{G}$:

$$[f,g]_{G}=f\circ g-(-1)^{|f||g|}g\circ f$$

makes $C^{\bullet}(A)$ into a graded Lie algebra. Equipped with the derivation $d={\rm ad}\,\,m_{2}$, $\,\,\,(C^{\bullet}(A),m_{2})$ becomes a DGLA, which is a Hochschild cochain complex.

Graphically the corresponding composition can be depicted as follows.

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With respect to the Gerstenhaber bracket $[-,-]_{G}$ we have the Maurer-Cartan equation

$$[m^{(1)},m^{(1)}]_{G}=\sum\limits_{p+q=k+1}\sum\limits_{i=1}^{p-1}(-1)^{\varepsilon}m_{p}(x_{1},\dots,x_{i-1},m_{q}(x_{j},\dots,x_{i+q-1}),\dots,x_{k})=0,$$

(2.1)

where

$$\varepsilon=|x_{1}|^{\prime}+{\dots}+|x_{i-1}|^{\prime},\qquad|x_{i}|^{\prime}=|x_{i}|-1={\rm deg}x_{i}-1$$

The Maurer-Cartan equation in the naive convention looks like:

$$[m^{(1)},m^{(1)}]=\sum\limits_{p+q=k+1}\sum\limits_{i=1}^{p-1}(-1)^{\varepsilon}m_{p}(x_{1},\dots,x_{i-1},m_{q}(x_{j},\dots,x_{i+q-1}),\dots,x_{k})=0,$$

(2.2)

where

$$\varepsilon=i(q+1)+q(|x_{1}|+{\dots}+|x_{i-1}|),$$

Equivalently, it can be formulated in a more compact way as a coderivation on the coalgebra of the bar complex of $A$.

In particular, for example, associative algebra with zero derivation $(A,m=m_{2}^{(1)})$ is an $A_{\infty}$-algebra. The component of the Maurer-Cartan equation of arity 3 says that the binary operation from this structure, the multiplication $m_{2}$ is associative:

$$(ab)c-a(bc)=dm_{3}(a,b,c)+(-1)^{\sigma}m_{3}(da,b,c)+(-1)^{\sigma}m_{3}(a,db,c)+(-1)^{\sigma}m_{3}(a,b,dc)$$

We can give now one of definitions of pre-Calabi-Yau structure (in the shifted convention).

The signs in this definition, written in shifted convention, are assigned according to the Koszul rule. Note, by the way, that in the naive convention, the cyclic invariance condition with respect to the natural non-degenerate pairing on $A\oplus A^{*}[1-d]$ from (II) sounds:

$$\langle m_{n}(\alpha_{1},...,\alpha_{n}),\alpha_{n+1}\rangle=(-1)^{n+|\alpha_{1}|^{\prime}(|\alpha_{2}|^{\prime}+...+|\alpha_{n+1}|^{\prime})}\langle m_{n}(\alpha_{2},...\alpha_{n+1}),\alpha_{1}\rangle.$$

The appearance of the arity $n$, which influence the sign in this formula, does not really follow the Koszul rule, this is the feature of the naive convention, and this is why the shifted one is preferable.

The cyclic invariance and inner form symmetricity in the shifted convention look like:

$$\langle m_{n}(\alpha_{1},...,\alpha_{n}),\alpha_{n+1}\rangle=(-1)^{|\alpha_{1}|^{\prime}(|\alpha_{2}|^{\prime}+...+|\alpha_{n+1}|^{\prime})}\langle m_{n}(\alpha_{2},...\alpha_{n+1}),\alpha_{1})\rangle.$$

(2.3)

$$\langle x,y\rangle=-(-1)^{|x|^{\prime}\,|y|^{\prime}}\langle y,x\rangle$$

(2.4)

Since the bilinear form on $A\oplus A^{*}$, which gives natural pairing, has degree zero (in non-shifted convention), the $A^{*}$ in the above definition is shifted by $1-d$ in order the corresponding cyclic Calabi-Yau structure on $A\oplus A^{*}$ is of degree $d$.

The most simple example of pre-Calabi-Yau structure demonstrates that this structure does exist on any associative algebra. Namely, the structure of associative algebra on $A$ can be extended to the associative structure on $A\oplus A^{*}[1-d]$ in such a way, that the natural inner form is (graded)cyclic with respect to this multiplication. This amounts to the following fact: for any $A$-bimodule $M$ the associative multiplication on $A\oplus M$ is given by $(a+f)(b+g)=ab+af+gb.$ In this simplest situation both structures on $A$ and on $A\oplus A^{*}$ are in fact associative algebras. More examples one can find, for example, in [16], [8].

Note that the notion of pre-Calabi-Yau algebra introduced in [26], [31], [36], as an $A_{\infty}$-atructure on $A\oplus A^{*}$, uses the fact that $A$ is finite dimensional, since there is no natural grading on the dual algebra $A^{*}={\rm Hom}(A,\mathbb{K})$, induced form the grading on $A$ in infinite dimensional case. One can reformulate it to give the general definition via the higher cyclic Hochschild complex (see [26], [25]), not requiring any finiteness conditions.

This reformulation is based on the fact that due to the cyclic invariance of the natural (evaluation) pairing on $A\oplus A^{*}$, any tensor of the type $C_{1}\otimes...\otimes C_{k}\otimes A$, where $C_{i}=A$ or $A^{*}$, can be considered as element of ${\rm Hom}\,_{\mathbb{K}}(C_{1}^{*}\otimes...\otimes{\mathbb{C}}_{k}^{*},A)$. Generally, in linear map from one tensor $C_{1}\otimes...\otimes C_{l}$ to another $C_{l+1}\otimes...\otimes C_{k+1}$ input from $A$ can be made into output from $A^{*}$ and vice versa. The obtained in this way new definition will be given in the next section. It is equivalent to the definition above, where the ${\rm Hom}(A,\mathbb{K})$ considered as graded Hom: $A^{*}=\oplus(A_{n})^{*}=\underline{{\rm Hom}}(A,\mathbb{K})$, in case the graded components of $A$ are finite dimensional.

We also should remark here that the theorem saying that pre-Calabi-Yau algebras give rise to TQFTs, analogous to the one proved in [36], for the definition 2.2, holds also for the definition via the higher cyclic Hochschild complex.

We start with the definition of the higher cyclic Hochschild cochains and generalised necklace bracket.

First let us consider higher Hochschild cochain complex ${\mathcal{H}}{\mathcal{H}}$:

$$C^{(N),n}(A):=\bigoplus\limits_{{r_{1},...,r_{N}\geqslant 0}\atop{r_{1}+...+r_{N}=n}}{\rm Hom}_{A^{\otimes N}-mod-A^{\otimes N}}(A^{\otimes i_{1}}\otimes...\otimes A^{\otimes i_{N}},A^{\otimes N}_{cycl})$$

The complex $C^{(N)}=\prod\limits_{n}C^{(N),n}$ is defined as $A^{\otimes N}$-bimodule ${\rm Hom}\,$ from the $N$th power of the bar complex ${\mathcal{B}}(A)$:

$$C^{(N)}={\rm Hom}_{A^{\otimes N}-mod-A^{\otimes N}}({\mathcal{B}}^{\otimes N},A^{\otimes N}_{cycl})$$

to the ${A}^{\otimes N}$-bimodule $A^{\otimes N}_{cycl}$ with the following bimodule structure. For any $x_{1}\otimes...\otimes x_{N}\in{A}^{\otimes N}_{cycl}$ and elements $a_{1}\otimes...\otimes a_{N},b_{1}\otimes...\otimes b_{N}\in{A}^{\otimes N}$,

$$(a_{1}\otimes...\otimes a_{N})\bullet(x_{1}\otimes...\otimes x_{N})\bullet(b_{1}\otimes...\otimes b_{N})=a_{1}x_{1}b_{2}\otimes...\otimes a_{N}x_{N}b_{1}.$$

Now we define the higher cyclic Hochshild complex ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$.

The differential is coming from the bar complex of $A^{\otimes N}$-bimodules, after it is dualised by
${\rm Hom}_{A^{\otimes N}-mod-A^{\otimes N}}(-,A_{cycl}^{\otimes N})$. It will be written precisely in section 5.

Elements of ${\rm Hom}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes r_{i}},A^{\otimes N}_{cycl})$ can be obviously interpreted as collections of $N$ operations with one output each. This interpretation is obtained if we first pass to the isomorphic complex over $\mathbb{K}$. Then elements of ${\rm Hom}_{\mathbb{K}}\prod\limits_{r_{1},...,r_{N}\geqslant 0}(A^{\otimes r_{i}-2},A_{cycl}^{\otimes N})$ are interpreted as a collection of $N$ operations.

In this interpretation it is easy to see that there is a natural ${\mathbb{Z}}_{N}$ group action on ${{\rm Hom}\,}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes r_{i}},A^{\otimes N}_{cycl})$, which cyclically permutes operations and assigns a sign $(-1)^{(d-1)(N-1)}$, according to the Koszul rule (and taking into account degree of operation).

The difference between ${\mathcal{H}}{\mathcal{H}}$ and ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ is that the latter consists only of elements invariant under this ${\mathbb{Z}}_{N}$ action, which we denote ${\rm Hom}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes r_{i}},A^{\otimes N}_{cycl})^{Z_{N}}.$ Let us introduce new notation: $C^{(N,d)}=C^{(N)}_{cycl}(A)\subset C^{(N)}(A)$, $\,\,C^{(N,d)}=C^{(N)}_{(d-1)(N-1)({\rm mod}2)},$ where $C^{(N)}_{0}$ are cochains symmetric under cyclic permutation of operations, and $C^{(N)}_{1}$ are antisymmetric cochains. By this we stress that ${\mathbb{Z}}_{N}$-invariant elements of ${\mathcal{H}}{\mathcal{H}}$, constituting ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$, consist of either symmetric or antisymmetric cochains, depending on $N$ and $d$. The fact that we should take only ${\mathbb{Z}}_{N}$-invariant elements comes from the condition of cyclic symmetry of the $A_{\infty}$-structure on $A\oplus A^{*}[1-d]$ w.r.t. the natural pairing.

We can point out at the difference between ${\mathcal{H}}{\mathcal{H}}$ and ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ also by saying that elements of ${{\rm Hom}\,}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes r_{i}},A^{\otimes N}_{cycl})$ have a fixed point, corresponding to the operation from which we start, however in ${{\rm Hom}\,}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes r_{i}},A^{\otimes N}_{cycl})^{{\mathbb{Z}}_{N}}$, after it is symmetrized, that is invariants are taken, the fixed (starting) point does not exist any more.

Denote by ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}=C^{(\bullet)}_{cycl}(A)=\mathop{\prod}\limits_{N\geqslant 1}C^{(N)}_{cycl}(A)$ the space of all higher cyclic Hochschild cochains. Further throughout the paper we frequently omit the subscript $cycl$, if it does not produce any confusion.

The space of all higher cyclic Hochschild cochains is denoted by $C^{(\bullet)}_{A}(A)$ or $C^{(\bullet)}_{\mathbb{K}}(A)$, depending on whether we are dealing with $A^{\otimes N}$ - bimodule Homs, or consider corresponding $\mathbb{K}$-module Homs. Sometimes we omit the $\mathbb{K}$ when it is clear from the context.

Note, that $C^{(1)}_{\mathbb{K}}(A)$ is the space of usual Hochschild cochains. To set up notations remind that the usual Hochschild cochains are:

$$C^{(1)}(A):=\prod_{r}{\rm Hom}_{A-mod-A}(A^{\otimes r},A):$$

$$A\to{\rm Hom}\,_{A-mod-A}(A,A)\mathop{\to}\limits^{D}{\rm Hom}\,_{A-mod-A}(A^{\otimes 2},A)\to...,$$

where $A$ is sitting in degree $0$, ${\rm Hom}\,(A,A)$ in degree $1$, etc. and

$$(D_{\mathbb{K}}h)(v_{1}\otimes...\otimes v_{n-1})=v_{1}h(v_{2}\otimes...\otimes v_{n-1})-h(v_{1}v_{2}\otimes v_{2}...\otimes v_{n-1})+...$$

$$(-1)^{n-2}h(v_{1}\otimes...\otimes v_{n-2}v_{n-1})+(-1)^{n-1}h(v_{1}\otimes...\otimes v_{n-2})v_{n-1}.$$

Now, when we start to define a Lie bracket on the higher cyclic Hochschild complex $C^{(\bullet)}_{A}(A)$, it becomes important which shifts of the grading on $A$ we chose, so we consider the higher cyclic Hochschild complex with the following shifts:

$${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}[1]=C^{(N)}_{cycl}(A)[1]:=\prod_{r_{1},...,r_{N}\geqslant 0}{{\rm Hom}\,}_{A^{\otimes N}-mod-A^{\otimes N}}(\mathop{\otimes}\limits_{i=1}^{N}A[1]^{\otimes r_{i}},A[1]^{\otimes N}_{cycl})^{{\mathbb{Z}}_{N}}[1].$$

The composition for the generalised necklace bracket can be graphically depicted as follows:

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Note that the (starting) point is not fixed in the elements of our complex (operations), thus generalized necklace bracket should produce also operations without a fixed point. Thus we need to clarify what means ’insertion’ of one operation into another in the cyclic situation of definition above: we should think of insertion of operations with fixed point according to the above rule, and then symmetrizing the result, by taking each resulting operation with all possible fixed points to the output.

Since the defined above composition $f\circ g$ makes ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}[1]=C_{cycl}^{(\bullet)}[1]$ into a graded pre-Lie algebra, the generalized necklace bracket obtained from it as a graded commutator, makes $C_{cycl}^{(\bullet)}[1]$ into a graded Lie algebra. We denote it by ${\bf g}=(C_{cycl}^{(\bullet)}(A)[1],[,]_{g.n}).$

Since in shifted by 1 situation operations have degree one, those elements of the ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}[1]$ which correspond to $d$-pre-Calabi-Yau structure should have degree $(d-1)(N-1)$. While reformulating the definition we take into account that in the definition 2.2 $A^{*}$ is shifted by $1-d$.

For the sake of clarity, simplicity, and since the main formality result holds only in this situation, we mainly consider here the grading, where $A$ is sitting in degree zero: $A_{0}=A$. This prompts us to deal with 2-pre-Calabi-Yau structures.

In the case $n=2,N=2$, which will be interesting for us in the next section, the definition says that we should take antisymmetric elements of ${\mathcal{H}}{\mathcal{H}}$ into ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex.

In this section we discuss a correspondence between particular part of pre-Calabi-Yau structure and the structure of double Poisson bracket invented by Van den Bergh [39] as a structure which produces the Poisson bracket on representation spaces.

There were many efforts to construct a reasonable notion of noncommutative Poisson bracket, which would according to the ideology of [24, 27] induce a kind of Poisson bracket on representation spaces or their moduli. First, the noncommutative Poisson bracket was defined in an obvious way: the same way as it is done in the commutative case, as a bracket on $A$, $\{-,-\}:A\to A$ which satisfy the Leibnitz rule: $\{a,bc\}=b\{a,c\}+\{a,b\}c$. But this notion turns out to be too restrictive, in [12] it was shown that defined this way bracket on noncommutative prime rings can be only commutator bracket $[a,b]=ab-ba$. There was a notion of ’noncommutative Poisson structure’ introduced in [40, 2], but it is only known that this bracket on the center of $A$ produces the usual commutative Poisson structure, it is unclear what it gives on moduli of representations. Then the attempts to introduce the notion of noncommutative Poisson bracket lead to a good definition in [39, 10]. In [39] the notion of the double Poisson bracket was defined as a map $\{\!\{\cdot,\cdot\}\!\}:A\otimes A\to A\otimes A$, satisfying the axioms which are certain generalization (thickening) of the usual Poisson axioms of anti-symmetry, Leibnitz and Jacobi identities. In [10] the noncommutative Poisson structure, called $H_{0}$-Poisson structure was defined as a Lie bracket on zero Hochschild homology of $A$: $H_{0}=A/[A,A]$, such that the map $\{\bar{a},-\}:A/[A,A]\to A/[A,A]$ is induced by a derivation $d_{a}:A\to A$. These explained many effects, for example, clarified the study of quasi-poisson structures [3]. There were further developments like [5], but we are trying here to continue the line of initial ideas from [24, 27].

We will put the earlier approaches in a more general framework, which explains a pattern of this generalisation (thickening) process from the perspective of the whole pre-Calabi-Yau structure. For example, the twisted structure of the diagonal bimodule $A^{\otimes N}$ will show how multiple derivations associated to a noncommutative polyvector fields interfere. We will see also in this section how the ’double’ definition comes as a particular part of general situation, coming from pre-Calabi-Yau structure. Namely, we will demonstrate that double bracket is defined by $\xi\delta$-words with two $\delta$th and no $\xi$th.

Remind, that double Poisson bracket is defined as a map $\{\!\{\cdot,\cdot\}\!\}:A\otimes A\to A\otimes A$ satisfying the following axioms:

Anti-symmetry:

$$\{\!\{a,b\}\!\}=-\{\!\{b,a\}\!\}^{op}$$

(4.1)

Here $\{\!\{b,a\}\!\}^{op}$ means the twist in the tensor product, i.e. if $\{\!\{b,a\}\!\}=\sum\limits_{i}b_{i}\otimes c_{i}$, then $\{\!\{b,a\}\!\}^{op}=\sum\limits_{i}c_{i}\otimes b_{i}.$

Double Leibniz:

$$\{\!\{a,bc\}\!\}=b\{\!\{a,c\}\!\}+\{\!\{a,b\}\!\}c$$

(4.2)

(here we use an outer bimodule structure on $A\otimes A:\quad a(b\otimes c)=ab\otimes c,(a\otimes b)c=a\otimes bc),$ and double Jacobi identity:

$$\{\!\{a,\{\!\{b,c\}\!\}\}\!\}_{L}+\tau_{(123)}\{\!\{b,\{\!\{c,a\}\!\}\}\!\}_{L}+\tau_{(132)}\{\!\{c,\{\!\{a,b\}\!\}\}\!\}_{L}$$

(4.3)

Here for $a\in A\otimes A\otimes A,\,\,$ and $\sigma\in S_{3}\,\,$

$$\tau_{\sigma}(a)=a_{\sigma^{-1}(1)}\otimes a_{\sigma^{-1}(2)}\otimes a_{\sigma^{-1}(3)}.$$

The $\{\!\{\,\}\!\}_{L}$ defined as

$$\{\!\{b,a_{1}\otimes...\otimes a_{n}\}\!\}_{L}=\{\!\{b,a_{1}\}\!\}\otimes a_{1}\otimes...\otimes a_{n}$$

The connection between the two structures is described by the following theorem.

The detailed proof of this theorem, taking into account signs and other details, was given in [17] (see also [19]) in terms of definition 2.2 of pre-Calabi-Yau structure.

We present here main idea of this proof, using definition 3.3 via higher cyclic Hochschild complex. It looks very transparent this way, which emphasises another advantage of this definition.

In terms of definition 3.3 the Maurer-Cartan equation on ’invariant’ with respect to the action of cyclic group elements from the higher cyclic Hochschild complex of particular kind, described in the theorem looks like:

0                 

0

Hence the Maurer-Cartan (in appropriate arity) is equivalent to the following equations:

0                                   

0

0                                   

0

These two are clearly exactly the double Leibnitz and double Jacobi identities respectively. The thing to be checked now is related to the following fact. The element of the higher cyclic Hochschild complex with two inputs and two outputs canonically corresponds (via the pairing on $A\oplus A^{*}$) to the maps $D:A\times A\to A\times A$ and $M:A\times A^{*}\times A\to A$. This correspondence is defined however only up to an arbitrary permutation of terms $A$. To establish an isomorphism between the two structures, we then need to choose appropriately this correspondence, which is done by formula (*) in [17]. After that axioms of double Poisson bracket can be checked, taking into account signs. Moreover we ensure that no other axioms appear from the Maurer-Cartan equation in case of the structure $(A\oplus A^{*},m_{2}+m_{3})$, hence all double Poisson brackets can be obtained from these structures, that is the map defined by (*) is a surjection. This means that structures of mentioned type are indeed in a bijective correspondence with the double Poisson brackets.

The question on better understanding of the structure needed to produce Poisson bracket on representation spaces recently received much attention. It is discussed, apart from mentioned paper [19], for example, in [4, 41]. Our work [17], [19] was also extended to the case when $A\neq A_{0}$, by means of the same correspondence given by formula (*), with signs added in [13].

Here we work with the smooth algebras [27], which are finitely generated 2-formally smooth algebras. By 2-formally smooth algebra we mean a formally smooth algebra in the sense of J. Cuntz and D. Quillen [9]:

First, consider a subcomplex $\zeta$ of the higher cyclic Hochschild complex, which we define as follows. Take a quotient complex

$$R_{\rm min}=[0\to\Omega\to A\otimes A]\twoheadrightarrow A$$

of the bar complex (considered as a complex of $A$-bimodules). Namely, $R_{\rm min}={\cal B}/{\cal F}$, where ${\cal B}$ is the bar complex

$${\cal B}=[\dots A\otimes A\otimes A\mathop{\to}\limits^{D_{3}}A\otimes A]\twoheadrightarrow A.$$

Denote its usual differential by $D_{A}$ or just $D$, when it is clear that we are talking about complex of $A$-bimodules. Obviously, ${\mathcal{B}}$ as well as ${\mathcal{R}}_{min}$ provide a free $A$-bimodule resolution for the diagonal bimodule $A$.