Action of vectorial Lie superalgebras on some split supermanifolds

The"curved"super Grassmannian is the supervariety of subsupervarieties of purely odd dimension $k$ in a~supervariety of purely odd dimension $n$, unlike the"usual"super Grassmannian which is the supervariety of linear subsuperspacies of purely odd dimension $k$ in a~superspace of purely odd dimension $n$. The Lie superalgebras of all and Hamiltonian vector fields on the superpoint are realized as Lie superalgebras of derivations of the structure sheaves of certain"curved"super Grassmannians,


Introduction
The Lie superalgebra W n := Der Λ C [ξ 1 , . . ., ξ n ] consisting of all vector fields on the superpoint 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 CGr n n−1 determined by the tautological vector bundle of rank n − 1 on the complex Grassmann manifold Gr n n−1 .The Lie superalgebra H n , the subsuperalgebra of W n consisting of Hamiltonian vector fields on the superpoint C 0,n , is isomorphic to the Lie superalgebra of vector fields on a split complex supermanifold CQ n−2 associated with a vector bundle of rank n − 1 orthogonal to the tautological bundle on the quadric Q n−2 ⊂ CP 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 CGr n n−1 and CQ n−2 ; I give a new version of the proof of the above results.D. Leites told me that the supermanifolds CGr 2 Superization of a construction due to Serre 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 C, and V a second copy of the same space considered as purely even.The change of parity V −→ V will be denoted by x → x on every non-zero x ∈ V .
Construct the Koszul complex of the Z-graded algebra which can be naturally considered as a free supercommutative superalgebra.There exists a unique derivation d ∈ Der −1 A such that dx = x and dx = 0 for any x ∈ V .Obviously, d 2 = 0. Consider the Z-graded Lie superalgebra W (V ) = Der Λ(V ).Any element δ ∈ W (V ) can be uniquely extended to a derivation δ ∈ Der A such that [ δ, d] = 0, see [K].The correspondence δ → δ is a faithful linear representation of the Lie superalgebra W (V ) in A.
Let ω ∈ S 2 (V ) be a nondegenerate bilinear form.Set Then H(ω) is a Z-graded subalgebra in W (V ) called the Lie superalgebra of Hamiltonian vector fields.Set Clearly, DH(ω) is a Z-graded subalgebra of W (V ), and H(ω) is its ideal.
3 Vector bundles over CP n−1 and Q n−2 and supermanifolds Define some special supermanifolds associated with vector bundles over Q n−2 ⊂ CP n−1 and CP 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 V * are even.In V , select a basis e 1 , . . ., e n , and consider the dual bases ξ 1 , . . ., ξ n and x 1 , . . ., x n of V * and V * , respectively.In the notation of § 2 (applied to V * and V * ) we have x i = ξ i .
The elements x 1 , . . ., x n are homogeneous coordinates on P (V ); this means that the stalk F a z of the structure sheaf F a of the algebraic variety P (V ) at z ∈ P (V ) is a subring of the field C(V ) = C(x 1 , . . ., x n ) consisting of elements of the form f /g, where f, g are homogeneous polynomials of the same degree in C[x 1 , . . ., x n ] = S(V * ) and g(z) = 0.
Consider the trivial vector bundle P (V ) × V * and its subbundle E ⊂ P (V ) × V * consisting of the pairs (Cx, y), where x ∈ V \ {0} and y ∈ Ann Clearly, E is an algebraic vector bundle of rank n − 1 over P (V ) with the fiber The map Cx → Ann x identifies P (V ) with the Grassmann variety Gr n−1 (V * ), which consists of (n − 1)-dimensional subspaces in V * , and E is the tautological bundle over this Grassmann variety.
Let E a ⊂ F a ⊗ 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 Ôa = F a ⊗ Λ(V * ), and the other by its subsheaf O a = Λ(E a ).
Besides, Ô = F ⊗ Λ(V * ) and O = Λ(E) ⊂ Ô, where F is the sheaf of holomorphic functions on P (V ) and 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 ∈ Der −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(V * ) \ {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, Ôa z = F a z ⊗ Λ(V * ) ⊂ B for any z ∈ P (V ).3.1.Lemma.We have O a z = Ôa z ∩ Ker D for any z ∈ P (V ).Proof.For any x ∈ V \ {0}, consider the derivation uniquely determined by the condition d x (ξ) = ξ(x) for any ξ ∈ V * .Clearly, On the other hand, Therefore, (du)(x) = d x u(x) on a Zariski open subset of V .Hence, du = 0 if and only if u(x) ∈ Λ(E Cx ) for all x ∈ V from the domain of u.Now, define a subsupermanifold of (P (V ), O a ) whose underlying manifold is the quadric Q ⊂ P (V ).Let ω be a non-degenerate quadratic function on V ; it can be considered as an element of S 2 (V * ).Let Q be the quadric in P (V ) given by the equation ω = 0 and E = E| Q the restriction of E to Q.If we identify V * with V with the help of the nondegenerate symmetric bilinear form corresponding to ω, then E is identified with the subbundle of Q × V orthogonal to the tautological line bundle over Q.
Let F a and E a (resp.F and E ) be the sheaves of polynomial (resp.holomorphic) functions on Q and polynomial (resp.holomorphic) sections of E , respectively.Set O a := Λ(E a ) and O := Λ(E ).
Let us give a description of O a similar to the above description of O a .For this, consider the superalgebra where ) is the algebra of polynomial functions on the cone Q ⊂ V given by the equation ω = 0; the localization of 4 Several remarks on vector fields on algebraic and analytic supervarieties Let M be a nonsingular complex algebraic variety, E an algebraic vector bundle over M .Denote by F a , T a , 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, F) is the complex analytic manifold corresponding to the algebraic variety M .The sheaves O a = Λ F a (E a ) and O = Λ F (E) rig M with structures of a split algebraic and a split analytic supervariety, respectively.We call the sheaves of Z-graded Lie superalgebras Der O a and Der O the sheaves of vector fields on these supervarieties.Proof.As shown in [O], for any k ∈ Z, every γ ∈ Der k O can be identified with a pair (γ 0 , γ 1 ), where A similar statement holds also for Der O a .Now, let δ ∈ W n = W (V * ).As we have seen in § 2, δ can be uniquely extended to a derivation δ of A = S(V * ) ⊗ Λ(V * ) such that [ δ, d] = 0. Obviously, the action of δ can be uniquely extended to the localization B of A, so (see Proof of Lemma 2.1) , where h i = δξ i ∈ Λ(V ).
By Lemma 5.1 δ transforms all the algebras Ôa z , where z ∈ P (V ), into themselves.Since δ transforms Ker d into itself, then Lemma 3.1 implies that δ(O a z ) ⊂ O a z for all z ∈ P (V ).Therefore, δ determines a global derivation of O a .We have constructed a map As is easy to see, theis map is an injective homomorphism of Z-graded Lie superalgebras.By Lemma 4.1 we also have an injective homomorphism W n −→ d = Γ(P (V ), Der O).
We similarly construct an injective homomorphism  [K,Theorem 4]).A proof of these statements is contained in [S1], [S2].This proof essentially depends on Lemma 4.1.1 in [O] and actually reduces to the calculation of d 0 and d 0 and also d 0 -module d −1 and d 0 -module d −1 with the help of Bott's theorem.
n n−1 and 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.) The algebras F a z and O a z , where z ∈ Q, are embedded into B .The derivation d transforms ωA into itself, and therefore determines an odd derivation d of A and B .Lemma 3.1 implies that

4. 1 .
Lemma.There exists a natural injective homomorphism of sheaves of Z-graded Lie superalgabras Der O a −→ Der O.
DH n −→ d = Γ(Q, Der O ).If δ ∈ DH n , then δ ∈ Der A transforms ωA into itself, and therefore determines a derivation of A from § 3 which extends to B and yields a derivation of O a .5.2.Theorem.The above-constructed homomorphisms W n −→ d and DH n −→ d are isomorphisms if n 2 and n 5, respectively.Proof.By the proved above we may assume that the finite-dimensional Z-graded Lie superalgebras d and d contain W n and DH n , respectively, as subalgebras.Therefore, it suffices to prove that d and d are transitive and irreducible and d k = (W n ) k and d k = (DH n ) k for k = −1, 0 (see