Action of vectorial Lie superalgebras on some split supermanifolds

The manuscript of this paper appeared as a preprint in proceedings of the “Seminar on Supersymmetries”, see http://staff.math.su.se/mleites/sos.html, and in Russian, see [52] in the list of references in the jubilee paper [AVGDZKLST*]. I updated the references; the ones I added are endowed with an asterisk. The abstract and comments are due to me. For a comprehensive description of simple Lie superalgebras of vector fields over algebraically closed fields of any characteristic, see  [BGLLS*]. D. Leites

The Lie superalgebra $W_{n}:=\mathop{\mathrm{Der}}\nolimits\Lambda_{\mathbb{C}}[\xi_{1},\ldots,\xi_{n}]$ consisting of all vector fields on the superpoint ${\mathcal{C}}^{0,n}$ is isomorphic, as shown in [S1], to the Lie superalgebra of vector fields, i.e., the global derivations of the structure sheaf, see [O], of a split complex supermanifold $\mathcal{CG}r_{n-1}^{n}$ determined by the tautological vector bundle of rank $n-1$ on the complex Grassmann manifold $\text{Gr}_{n-1}^{n}$. The Lie superalgebra $H_{n}$, the subsuperalgebra of $W_{n}$ consisting of Hamiltonian vector fields on the superpoint ${\mathcal{C}}^{0,n}$, is isomorphic to the Lie superalgebra of vector fields on a split complex supermanifold $\mathcal{CQ}^{n-2}$ associated with a vector bundle of rank $n-1$ orthogonal to the tautological bundle on the quadric $Q^{n-2}\subset{\mathbb{C}}P^{n-1}$. However, the method used in [S1], [S2] does not allow one to indicate explicitly these isomorphisms. In this paper I explicitly construct the $W_{n}$- and $H_{n}$-actions on the supermanifolds $\mathcal{CG}r_{n-1}^{n}$ and $\mathcal{CQ}^{n-2}$; I give a new version of the proof of the above results. D. Leites told me that the supermanifolds $\mathcal{CG}r_{n-1}^{n}$ and $\mathcal{CQ}^{n-2}$ are the simplest examples of what Manin called “curved” Grassmannians and “curved” quadrics, see [Ma*]. They were introduced in [KL*]. (Compare with the Grassmannians of linear subsuperspaces in a linear superspace, see [Ma*, Ch.4, § 3]. For the complete list of homogeneous superdomains associated with the known Lie superalgebras of polynomial growth, see [L*]. D.L.)

In this section, I superize a construction Serre introduced in [Se]. This enables us to interpret elements of a Lie superalgebra as Hamiltonian vector fields on the superpoint. Let $V$ be a purely odd vector space over ${\mathbb{C}}$, and $\overline{V}$ a second copy of the same space considered as purely even. The change of parity $V\longrightarrow\overline{V}$ will be denoted by $x\mapsto\overline{x}$ on every non-zero $x\in V$. Construct the Koszul complex of the ${\mathbb{Z}}$-graded algebra

which can be naturally considered as a free supercommutative superalgebra. There exists a unique derivation $d\in\mathop{\mathrm{Der}}\nolimits_{-1}A$ such that $dx=\overline{x}$ and $d\overline{x}=0$ for any $x\in V$. Obviously, $d^{2}=0$. Consider the $Z$-graded Lie superalgebra $W(V)=\mathop{\mathrm{Der}}\nolimits\Lambda(V)$. Any element $\delta\in W(V)$ can be uniquely extended to a derivation $\tilde{\delta}\in\mathop{\mathrm{Der}}\nolimits A$ such that $[\tilde{\delta},d]=0$, see [K]. The correspondence $\delta\mapsto\tilde{\delta}$ is a faithful linear representation of the Lie superalgebra $W(V)$ in $A$. Let $\omega\in S^{2}(\overline{V})$ be a nondegenerate bilinear form. Set

Then $H(\omega)$ is a ${\mathbb{Z}}$-graded subalgebra in $W(V)$ called the Lie superalgebra of Hamiltonian vector fields. Set

Clearly, $DH(\omega)$ is a ${\mathbb{Z}}$-graded subalgebra of $W(V)$, and $H(\omega)$ is its ideal. Hereafter we assume that $\dim V=n$. Set $W_{n}:=W(V)$, ${H_{n}:=H(\omega)}$, $DH_{n}:=DH(\omega)$ since these algebras are determined, up to an isomorphism, by $\dim V$. In $V$, select a basis $\xi_{1},\ldots,\xi_{n}$. Then the elements $x_{i}=\overline{\xi}_{i}=d\xi_{i}$, where $i=1,\ldots,n$, constitute a basis in $\overline{V}$ and

Obviously, $d=\sum\limits_{1\leqslant i\leqslant n}x_{i}\partial_{\xi_{i}}$. Notice that $\partial_{\xi_{i}}\in\mathop{\mathrm{Der}}\nolimits_{-1}A$ coincides with the extension $\tilde{\partial}_{\xi_{i}}$ of $\partial_{\xi_{i}}\in W(V)_{-1}$. {Lemma} 1) If $\tilde{\delta}(\omega)=\varphi\omega$ for some $\delta\in W(V)$, $\varphi\in A$, then $\varphi\in{\mathbb{C}}$. 2) There is the following semidirect sum decomposition:

Define some special supermanifolds associated with vector bundles over $Q^{n-2}\subset{\mathbb{C}}P^{n-1}$ and ${\mathbb{C}}P^{n-1}$. Let $\dim V=n$, and $P(V)$ the corresponding protective space. Let us assume that the nonzero elements of $V^{*}$ are odd and those of $\overline{V}{}^{*}$ are even. In $V$, select a basis $e_{1},\ldots,e_{n}$, and consider the dual bases $\xi_{1},\ldots,\xi_{n}$ and $x_{1},\ldots,x_{n}$ of $V^{*}$ and $\overline{V}{}^{*}$, respectively. In the notation of § 2 (applied to $V^{*}$ and $\overline{V}^{*}$) we have $x_{i}=\overline{\xi}_{i}$. The elements $x_{1},\ldots,x_{n}$ are homogeneous coordinates on $P(V)$; this means that the stalk ${\mathcal{F}}_{z}^{a}$ of the structure sheaf ${\mathcal{F}}^{a}$ of the algebraic variety $P(V)$ at $z\in P(V)$ is a subring of the field ${\mathbb{C}}(V)={\mathbb{C}}(x_{1},\ldots,x_{n})$ consisting of elements of the form $f/g$, where $f,g$ are homogeneous polynomials of the same degree in ${\mathbb{C}}[x_{1},\ldots,x_{n}]=S(\overline{V}^{*})$ and $g(z)\neq 0$. Consider the trivial vector bundle $P(V)\times V^{*}$ and its subbundle $E\subset P(V)\times V^{*}$ consisting of the pairs $({\mathbb{C}}x,y)$, where $x\in V\setminus\{0\}$ and $y\in\mathop{\mathrm{Ann}}\nolimits x=\{\alpha\in V^{*}\mid\alpha(x)=0\}$. Clearly, $E$ is an algebraic vector bundle of rank $n-1$ over $P(V)$ with the fiber

The map ${\mathbb{C}}x\mapsto\mathop{\mathrm{Ann}}\nolimits x$ identifies $P(V)$ with the Grassmann variety $\text{Gr}_{n-1}(V^{*})$, which consists of $(n-1)$-dimensional subspaces in $V^{*}$, and $E$ is the tautological bundle over this Grassmann variety. Let ${\mathcal{E}}^{a}\subset{\mathcal{F}}^{a}\otimes V^{*}$ be the sheaf of germs of the polynomial sections of $E$. The variety $P(V)$ can be endowed with two structures of a split complex algebraic supervariety: one is determined by the sheaf ${\hat{\mathcal{O}}{}^{a}={\mathcal{F}}^{a}\otimes\Lambda(V^{*})}$, and the other by its subsheaf ${\mathcal{O}}^{a}=\Lambda({\mathcal{E}}^{a})$. Besides, $\hat{\mathcal{O}}={\mathcal{F}}\otimes\Lambda(V^{*})$ and ${\mathcal{O}}=\Lambda({\mathcal{E}})\subset\hat{\mathcal{O}}$, where ${\mathcal{F}}$ is the sheaf of holomorphic functions on $P(V)$ and ${\mathcal{E}}$ is the sheaf of germs of holomorphic sections of $E$, determine two structures of a split complex analytic supermanifold associated with $P(V)$. Consider the superalgebra

and the derivation $d\in\mathop{\mathrm{Der}}\nolimits_{-1}A$ constructed in § 2 (with $V$ replaced by $V^{*}$ in these constructions). Let

be the localization of $A$ with respect to the multiplicative system $S(\overline{V}{}^{*})\setminus\{0\}$. The algebra $B$ has a natural supercommutative superalgebra structure, and $d$ can be uniquely extended to an odd derivation of $B$, which we will denote also by $d$. Clearly,

We have ${\mathcal{O}}_{z}^{a}=\hat{\mathcal{O}}{}_{z}^{a}\cap\mathop{\mathrm{Ker}}\nolimits D$ for any $z\in P(V)$.

Now, define a subsupermanifold of $(P(V),{\mathcal{O}}^{a})$ whose underlying manifold is the quadric $Q\subset P(V)$. Let $\omega$ be a non-degenerate quadratic function on $V$; it can be considered as an element of $S^{2}(\overline{V}{}^{*})$. Let $Q$ be the quadric in $P(V)$ given by the equation $\omega=0$ and $E^{\prime}=E|_{Q}$ the restriction of $E$ to $Q$. If we identify $V^{*}$ with $V$ with the help of the non-degenerate symmetric bilinear form corresponding to $\omega$, then $E^{\prime}$ is identified with the subbundle of $Q\times V$ orthogonal to the tautological line bundle over $Q$. Let ${\mathcal{F}}^{\prime a}$ and ${\mathcal{E}}^{\prime a}$ (resp. ${\mathcal{F}}^{\prime}$ and ${\mathcal{E}}^{\prime}$) be the sheaves of polynomial (resp. holomorphic) functions on $Q$ and polynomial (resp. holomorphic) sections of ${\mathcal{E}}^{\prime}$, respectively. Set ${{\mathcal{O}}^{\prime a}:=\Lambda({\mathcal{E}}^{\prime a}})$ and ${\mathcal{O}}^{\prime}:=\Lambda({\mathcal{E}}^{\prime})$. Let us give a description of ${\mathcal{O}}^{\prime a}$ similar to the above description of ${\mathcal{O}}^{a}$. For this, consider the superalgebra

where ${\mathbb{C}}[\hat{Q}]:=S(\overline{V}{}^{*})/\omega S(\overline{V}{}^{*})$ is the algebra of polynomial functions on the cone $\hat{Q}\subset V$ given by the equation $\omega=0$; the localization of $A^{\prime}$ is ${B^{\prime}:={\mathbb{C}}(\hat{Q})\otimes\Lambda(V^{*})}$. The algebras ${\mathcal{F}}{}^{\prime a}_{z}$ and ${\mathcal{O}}_{z}^{\prime a}$, where $z\in Q$, are embedded into $B^{\prime}$. The derivation $d$ transforms $\omega A$ into itself, and therefore determines an odd derivation $d^{\prime}$ of $A^{\prime}$ and $B^{\prime}$. Lemma 3 implies that

Let $M$ be a nonsingular complex algebraic variety, $E$ an algebraic vector bundle over $M$. Denote by ${\mathcal{F}}^{a}$, ${\mathcal{T}}^{a}$, ${\mathcal{E}}^{a}$ the structure sheaf on $M$, the tangent sheaf on $M$, and the locally free algebraic sheaf corresponding to $E$, respectively. We denote by the same letters without the superscript $a$ the corresponding analytic sheaves on $M$. In particular, $(M,{\mathcal{F}})$ is the complex analytic manifold corresponding to the algebraic variety $M$. The sheaves ${\mathcal{O}}^{a}=\Lambda_{{\mathcal{F}}^{a}}({\mathcal{E}}^{a})$ and ${\mathcal{O}}=\Lambda_{\mathcal{F}}({\mathcal{E}})$ rig $M$ with structures of a split algebraic and a split analytic supervariety, respectively. We call the sheaves of ${\mathbb{Z}}$-graded Lie superalgebras $\mathop{\mathrm{Der}}\nolimits{\mathcal{O}}^{a}$ and $\mathop{\mathrm{Der}}\nolimits{\mathcal{O}}$ the sheaves of vector fields on these supervarieties. {Lemma} There exists a natural injective homomorphism of sheaves of ${\mathbb{Z}}$-graded Lie superalgabras $\mathop{\mathrm{Der}}\nolimits{\mathcal{O}}^{a}\longrightarrow\mathop{\mathrm{Der}}\nolimits{\mathcal{O}}$.

There exists an injective homomorphism of ${\mathbb{Z}}$-graded Lie superalgebras of vector fields ${\mathfrak{d}}^{a}\longrightarrow{\mathfrak{d}}$, where ${\mathfrak{d}}^{a}=\Gamma(M,\mathop{\mathrm{Der}}\nolimits{\mathcal{O}}^{a})$ and ${\mathfrak{d}}=\Gamma(M,\mathop{\mathrm{Der}}\nolimits{\mathcal{O}})$. The results of [Se] imply that the homomorphism ${\mathfrak{d}}^{a}\longrightarrow{\mathfrak{d}}$ is an isomorphism for any projective variety $M$.

Now we are able to determine the action of $W_{n}$ and $DH_{n}$ on the supermanifolds constructed in § 3. Retaining the notation of § 3 let us first prove the following statement. {Lemma} Let $\gamma\in\mathop{\mathrm{Der}}\nolimits B$ be such that $\gamma V\subset\Lambda(V)$, so that

Then, $\gamma(\hat{\mathcal{O}}{}_{z}^{a})\subset\hat{\mathcal{O}}{}_{z}^{a}$ for all $z\in P(V)$.

We similarly construct an injective homomorphism

If ${\delta\in DH_{n}}$, then $\tilde{\delta}\in\mathop{\mathrm{Der}}\nolimits A$ transforms $\omega A$ into itself, and therefore determines a derivation of $A^{\prime}$ from § 3 which extends to $B^{\prime}$ and yields a derivation of ${\mathcal{O}}^{\prime a}$. {Theorem} The above-constructed homomorphisms $W_{n}\longrightarrow{\mathfrak{d}}$ and $DH_{n}\longrightarrow{\mathfrak{d}}^{\prime}$ are isomorphisms if $n\geqslant 2$ and $n\geqslant 5$, respectively.