Transitive irreducible Lie superalgebras of vector fields

Let $\mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf of sections of the exterior algebra of the homogeneous vector bundle $E$ over the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie group and $P$ its parabolic subgroup. Then, $\mathfrak{d}$ is transitive and irreducible whenever $E$ is defined by an irreducible $P$-module $V$ such that the highest weight of $V^*$ is dominant. Moreover, $\mathfrak{d}$ is simple; it is isomorphic to the Lie superalgebra of vector fields on the superpoint, i.e., on a $0|n$-dimensional supervariety.

(2) As is well-known, see [K*], [Sch], condition (1) is satisfied if and only if the graded Lie algebra b := gr a satisfies conditions (2).We say that a graded Lie superalgebra b is irreducible if the adjoint representation of b 0 in b −1 is irreducible.(The term irreducible is usually used to denote the image of the (Lie) algebra or superalgebra b in any irreducible representation.D.L.) Simple, and even primitive 1 , Lie (super)algebras have a filtration in which the associated graded Lie (super)algebra is both transitive and irreducible.For the classification of simple finite-dimensional Lie superalgebras over C, see [K*], [Sch].(Actually, no complete proof of the classification of simple finite-dimensional Lie superalgebras over C is known to this day; e.g., the proof of completeness of the list of examples of known deformations with odd parameters is not published.Such deformations were not even mentioned in [K*], [Sch].D.L.) Hereafter I assume that the Z-grading of every Lie superalgebra considered is compatible with parity.
In this paper, I show that such Lie superalgebras naturally appear in the study of homogeneous vector bundles over complex homogeneous flag varieties G/P , where P is a parabolic subgroup of a complex semisimple Lie group G. Namely, to each homogeneous vector bundle E −→ M there is associated a split complex supermanifold (M, Λ(E)), where E is the sheaf of holomorphic sections of E, and Λ(E) is the Grassmann algebra of E.
Here, I prove that the graded Lie superalgebra d = Γ(M, Der Λ(E)) of vector fields on this supermanifold is transitive and irreducible if E is defined by an irreducible representation ϕ of P satisfying certain natural requirements.The main point here is that the highest weight of the representation ϕ * should be dominant.This guarantees the existence of sufficiently many holomorphic sections of "the odd tangent bundle" E * over the supermanifold (M, Λ(E)).
The Bott-Borel-Weil theorem, along with other standard methods, enables one to explicitly compute the Lie superalgebra d in many cases.
In this paper, this computation is carried out in the cases where E is the cotangent bundle over M , and G is simple.

Vector fields on supermanifolds
The term "supermanifold" is meant in the sense of Berezin-Leites, in the complex-analytic situation, see [L], [MaG*], [Va*].Let us recall the definition.
1 Let a Lie algebra L contain a subalgebra L 0 which does not contain any nonzero ideal of L. Let (3) Unlike primitive Lie algebras (both finite-dimensional and Z-graded of polynomial growth) which do not differ much from the simple Lie algebras, the classification of primitive Lie Let (n, m) be a pair of non-negative integers.Let a model superspace of dimension n|m be the ringed space (C n , O), where O := Λ O [ξ] for ξ = (ξ 1 , . . ., ξ m ), and O is the sheaf of germs of holomorphic functions on C n .A topological space M equipped with a sheaf of supercommutative superalgebras O, which as a ringed space is locally isomorphic to a model superspace of dimension n|m is said to be a(n almost) complex supermanifold and O is its structure sheaf.(For details, e.g., which of almost complex supermanifolds are complex, and for phenomena indigenous not only to the "super" world -real-complex (super)manifolds, see [BGLS*].D.L.) Let us introduce parity in O by declaring all of the ξ i to be odd; so O = O0 ⊕ O1 is the sum of its homogeneous components even and odd.
Let (4) 2.1.Lemma. 1) The formulas (4) show that Der O can be naturally considered as a sheaf of filtered Lie superalgebras.
2) The sheaves gr Der O and Der gr O are naturally isomorphic.
Proof. 1) It is easy to verify that δI r ⊂ I r+p for any δ ∈ Der (p) O and r ∈ Z. (5) This implies that [Der (p) O, Der (q) O] ⊂ Der (p+q) O.
It is subject to a direct verification that the mappings δr form a derivation δ ∈ Der p gr O.
The sheaf Der (p+1) O is the kernel of the mapping δ → δ.Thus we obtain an injective sheaf homomorphism gr p Der O −→ Der p gr O.
Using local splitness of supermanifolds, it is not difficult to show that this homomorphism is, moreover, surjective.Finally, a direct verification shows that this is an isomorphism of sheaves of graded Lie superalgebras.

Split supermanifolds
Let O = Λ(F), where F is a locally free analytic sheaf of rank m on an n-dimensional complex manifold (M, O).Let T denote the tangent sheaf Der O on this manifold.
3.1.Theorem. 1) The sheaves Der p O are locally free analytic.We have 2) For p ≥ 0, there is an exact sequence of sheaves Proof. 1) It suffices to consider the case of the model supermanifold of dimension n|m.
Clearly, in this case Der O is a free sheaf of O-modules, a basis of its sections consisting of where x 1 , . . ., x n are coordinates in C n , and {ξ 1 , . . ., ξ m } is any local basis of sections of the sheaf F. Therefore, Der p O is a free sheaf of O-modules with a basis of its sections being formed by Since the sheaf O is generated by its subsheaves O and F, the derivation δ is completely determined by its restrictions The other way around, for any pair (δ 0 , δ 1 ), see eq. ( 9), which satisfies conditions (10), there exists a δ ∈ Der p O such that α(δ) = δ 0 and β(δ) = δ 1 .

Arkady Onishchik
2) For p ≥ 0, consider the sheaf homomorphism Obviously, Ker α = (Der p O) ∩ Der O O, and therefore β defines a sheaf isomorphism To compute Im α we use an isomorphism for any locally free analytic sheaf G. Let {g 1 , . . ., g q } be a basis of sections of i=1 and is the desired isomorphism ( 11).
If G = Λ p O (F), then, for any given basis {g i } q i=1 , we can take local sections Observe that for p = 0 the sequence ( 8) is of the form In particular, End F is a sheaf of ideals in Der 0 O.The sequence of sheaves leads to the exact sequence of Lie algebras where End F = Γ(M, End F) is the Lie algebra of endomorphisms of the vector bundle F .Let ε ∈ d 0 be the element corresponding to the identity automorphism of F .Clearly, ε is a grading element of d, i.e., [ε, δ] = pδ for any δ ∈ d p .
3.1.1Example.Consider the split supermanifold (M, Ω), where is the sheaf of holomorphic forms on (M, O).If F = T * , then the sheaves which appear in Theorem 3.1 coincide with Ω p T = T ⊗ O Ω p -sheaves of vector-valued holomorphic pforms, and we obtain the exact sequence which introduces maps i p and α p As was proved in [Fr] (in the C ∞ case, but this is inessential), this sequence splits.Therefore, we obtain the following theorem which I prove for completeness.d], where d : Ω −→ Ω is the exterior differential, a section of the sheaf Der 1 Ω.Then, α p (δ) = ω.Indeed, let ω = i δ i ω i , where δ i ∈ T and ω i ∈ Ω p .Then, for any ϕ ∈ O, we have

Theorem. For any p −1, we have
Therefore, the mapping ω → δ = [δ ω , d] splits the sequence (15).It is easy to see that the subsheaf of Der p Ω defined by the splitting, which is complementary to Ω p+1 T , coincides with the centralizer of d.(The first description of the centralizer of d in terms of Lie superalgebras and invariant differential operators is due to Grozman; for details, see the arXiv version of [Gro*] reproduced in this Special Volume D.L.).

Corollary. There exist isomorphisms
In particular, The latter isomorphism is described as follows: every vector field v ∈ Γ(M, T ) determines a derivation i v ∈ d −1 , called inner derivation by (or a convolution with) v. Further, we have a decomposition into a semidirect sum of Lie algebras where End T is an ideal, and the Lie subalgebra Γ(M, T ) is embedded into d 0 by means of injective homomorphism v → θ v , where θ v is the Lie derivative along the field v. Finally, Under this isomorphism the identity automorphism ε ∈ End T corresponds to the exterior The classical relations (here p(i v ) = 1, i.e., i v is odd for all vector fields v) immediately imply that d is a Lie subsuperalgebra in d.

Transitive Lie superalgebras
Let us deduce a sufficient condition for the Lie superalgebra of vector fields Γ(M, Der O) on the supermanifold (M, O) to be transitive.As before let J be the subsheaf of ideals generated by odd elements.The locally free sheaf F * = (J /J 2 ) * on the complex manifold (M, O rd ) will be called odd tangent sheaf and denoted T 1 .Let T 1 be the corresponding holomorphic vector bundle over (M, O rd ) called the odd tangent bundle.If [γ, δ] = 0 for all γ ∈ d −1 , then γ(δϕ) = 0 for all ϕ ∈ O.This follows from (16).Since δϕ ∈ Λ p (F), then under Lemma's hypothesis for p > 0, we have δϕ = 0 for all ϕ ∈ O.But then Eq. ( ) implies that γ(δs) = 0 for all γ ∈ d −1 and s ∈ F, so δs = 0 for all s ∈ F; i.e., δ = 0.
Observe that, in general, the Lie superalgebra d does not satisfy condition (2) for p = 0.Under assumptions of the Lemma we can only claim that the action of the ideal End F = End T 1 , see ( 14), on d −1 = Γ(M, T 1 ) is exact.By item 2) of Corollary 2.2, there is an injective homomorpshism gr d −→ d.
In particular, gr 4.2.Lemma.For the Lie superalgebra d := Γ(M, Der O) to be transitive the following two conditions are sufficient: a) Given any point of any fiber of the bundle T 1 , there is a section of the subspace whose image contains it.
b) The adjoint action of the Lie algebra Proof.By repeating almost verbatim the proof of Lemma 4.1 we see that the subalgebra gr d of the Lie superalgebra d satisfies conditions (2) for any p > 0. The case p = 0 is handled by condition of item b).

Homogeneous vector bundles
Let p : E −→ M be a holomorphic vector bundle over a complex manifold (M, O).A fiberwise linear biholomorphic mapping E −→ E is said to be an automorphism of the bundle E. Let A(E) be the group of all automorphisms of E. Obviously, every automorphism a of the bundle E determines an automorphism p(a) of the base M of E. We obtain an exact sequence of groups where Aut E ⊂ A(E) is the normal subgroup consisting of the automorphisms sending every fiber into itself.If M is compact, then the sequence (17) consists of complex Lie groups and their homomorphisms, see [Mo].
Observe that it is possible to describe the automorphisms of the bundle E in terms of the corresponding sheaf E. Namely, every a ∈ A(E) determines an automorphism ã of the sheaf E over the automorphism p(a) of the base M , i.e., determines an isomorphism of sheaves ã : E −→ p(a) * E. The latter isomorphism is given on local sections s by the formula ã(s)(w) = a −1 (s(p(a)(w))) for any w ∈ M .
Consider the sheaf E * = Hom O (E, O) corresponding to the dual bundle E * .The space of sections Γ(U, E * ) over any open set U ⊂ M can be identified with the subspace in Γ(p −1 (U ), O E ) consisting of the functions linear on fibers.This gives an embedding A vector field on E with a projection to M and sending E * into itself will be called an infinitesimal automorphism of the bundle E. The infinitesimal automorphisms determine a sheaf of complex Lie algebras A(E) on M .Projection to the base yields a sheaf homomorphism π : A(E) −→ T , where T is the tangent sheaf on M .If M is compact, then the Lie algebra a(E) = Γ(M, A(E)) is tangent to the Lie group A(E), and the homomorphism π : a(E) −→ Γ(M, T ) coincides with dp.
Proof.Every δ ∈ A(E) determines a pair so that conditions (10) are satisfied.Let δ ∈ Der 0 O be the derivation corresponding to the pair (δ 0 , δ 1 ), see proof of Theorem 3.1.It is easy to see that the mapping δ → δ is an injective homomorphism of sheaves of Lie algebras.
To prove surjectivity of the mapping δ → δ, consider δ ∈ (Der 0 O) w determined by a pair (δ 0 , δ 1 ) satisfying conditions (10).Let U ⊂ M be an open neighborhood of any point w ∈ M over which E admits trivialization p −1 (M ) = U × C m .Set δϕ = δ 0 ϕ, where ϕ ∈ O| U , and δ = δ 1 , where ∈ (C m ) * .These conditions completely determine an element δ ∈ A(E) w which turns into δ under this mapping .Now let (M, O) be a homogeneous space of a connected complex Lie group G, i.e., let there be given a transitive analytic G-action on M .Fix a point w 0 ∈ M , and let P be the stationary subgroup of w 0 in G.
As is well-known, M can be naturally identified with the manifold of the left cosets G/P .Let M be compact.
Recall that a holomorphic vector bundle p : E → M is homogeneous (under a G-action) if there is an analytic G-action by automorphisms of the bundle E whose projection is a given G-action on M .Equivalently, there exists an analytic homomorphism t : Associated to a homogeneous bundle E there is a linear representation ϕ : g → t(g)| Ew 0 of P on the space E w 0 .This representation completely determines the bundle E together with the G-action on E. For the role of ϕ one can take any finite-dimensional linear analytic representation of P .We write E ϕ to denote the homogeneous vector bundle over M corresponding to the representation ϕ.The corresponding locally free sheaves E ϕ are also called homogeneous.
The representation Φ of G is called induced by the representation ϕ of P .

Example.
The tangent bundle T over M is endowed with a natural G-action, and therefore is homogeneous.It corresponds to the linear representation τ : P → GL(T w 0 (M )) given by the formula τ (g) = dt(g) w 0 for any g ∈ P .
In what follows we consider the case where M is the homogeneous flag variety, i.e., where G is a connected semisimple (or reductive) complex Lie group and P is its parabolic subgroup.In this case, the space of sections of the homogeneous vector bundle and the induced representation are described by the famous Bott-Borel-Weil theorem, see [B].Let us recall it.
Let T be a maximal torus in a Borel subgroup B of G and let t and b be the respective Lie algebras.Let R be the root system with respect to T (or t); let R + be the set of positive roots corresponding to b and Π the set of simple roots and let be the set of coroots, i.e., the system of roots dual to R. The weight λ ∈ I * is called dominant if λ(h α ) 0 for all α ∈ Π. (Bott-Borel-Weil).Let P be a parabolic subgroup of G containing the opposite to B Borel group B − , and let M = G/P .Let ϕ be an irreducible analytic representation of P with highest weight λ.

Theorem
Then, Γ(M, E ϕ ) = 0 if and only if λ is dominant.In this case, the induced representation Φ in Γ(M, E ϕ ) is irreducible with highest weight λ.
This theorem is applicable only if ϕ is irreducible or completely reducible.However, one often encounters homogeneous bundles arising from representations which are not completely reducible.The following Lemma is useful for studying them.5.4.Lemma.Let M = G/P be a homogeneous flag variety, ϕ a holomorphic linear representation of P .If the induced representation Φ of G contains an irreducible representation with highest weight λ of multiplicity t, then λ is contained in the set of weights of ϕ of multiplicity t.
Proof.Let us perform induction by the length (ϕ) of the Jordan-Hoelder series of ϕ.If (ϕ) = 1, then λ is the highest weight of ϕ by Theorem 5.3.Let the Lemma be proved for representations ϕ such that (ϕ ) < (ϕ).Assuming that ϕ is reducible, consider a proper subrepresentation ϕ 1 of ϕ and the quotient representation ϕ 2 .We obtain an exact sequence of sheaves 0 which defines the exact sequence of G-modules Since every finite-dimensional analytic representation of G is completely reducible, the irreducible representation with highest weight λ is contained in Γ(M, E ϕ 1 ) and Γ(M, E ϕ 2 ) with total multiplicity t.By induction hypothesis the statement of the Lemma follows because the set of weights of ϕ is the union of the sets of weights of ϕ 1 and ϕ 2 .

Split homogeneous supervarieties
In this section we consider a particular class of split supervarieties associated with homogeneous flag varieties.Let G be a connected semisimple (or reductive) complex Lie group, P a complex Lie subgroup, and E ϕ a homogeneous vector bundle over M = G/P determined by a representation ϕ of P .Consider a split supermanifold (M, O), where O = Λ(E ϕ ).The sheaf O corresponds to the homogeneous vector bundle Λ(E ϕ ) = E Λ(ϕ) , i.e., O = E Λ(ϕ) .
The sheaf Der O is also homogeneous.Indeed, every element g ∈ G corresponds to an automorphism t(g) of O over an automorphism t(g) ∈ Aut M , see eq. ( 18).By setting we obtain the desired G-action on the locally free sheaf Der O. Also notice that the sheaves in the exact sequence (8) correspond to the homogeneous vector bundles E * ϕ ⊗ Λ p+1 (E ϕ ) and T ⊗ Λ p (E ϕ ), while the homomorphisms in this sequence are G-equivariant.
By Lemma 5.1 we can identify the Lie algebra d 0 = Γ(M, Der 0 O) with a(E * ϕ ).Since the bundle E * ϕ is also homogeneous, we have a Lie algebra homomorphism d t : g −→ d 0 .On the other hand, the G-action (19) on O determines a representation Ψ : G −→ GL(d).
Proof.This follows directly from (19): instead of g, substitute in eq. ( 19) the curve g(t) ∈ G with tangent vector u at t = 0 passing through e := g(0), and differentiate both parts with respect to t at t = 0.
A split supermanifold (M, Λ(E ϕ )) will be called homogeneous if T 1 = E * ϕ = E ϕ * has "sufficiently many" holomorphic sections.This means that every point of every fiber of this bundle is contained in the image of some section.
In what follows G is semisimple and P is its parabolic subgroup.
6.2.Lemma.Let ϕ be irreducible and λ the highest weight of representation ϕ * .The following properties are equivalent: Under these conditions the induced representation Proof.By Theorem 5.3, it remains to prove that if λ is dominant, then T 1 has sufficiently many holomorphic sections.Obviously, the restriction map r w 0 : Γ(M, E ϕ * ) → (E ϕ * ) w 0 is a P -module homomorphism.Therefore, either r w 0 is surjective or r w 0 = 0.But for any g ∈ G and s ∈ Γ(M, E ϕ * ), we have where Φ is the representation induced by ϕ * .Therefore, rw 0 = 0 implies r w = 0 for all w ∈ M , which is impossible.Thus r w is surjective for all w ∈ M .
Consider the following commutative diagram whose upper line is the exact sequence ( 14) for the bundle Let G = G 1 . . .G r be a decomposition of G into simple factors, and let Π = Π 1 • • • Π r be the corresponding decomposition of the system of simple roots.Let the highest weight λ of the representation ϕ * be dominant, and for every i assume that there exists a β ∈ Π i such that λ(h β ) > 0.
Then, the graded Lie superalgebra d = Γ(M, Der O) is transitive and irreducible.
Proof.By Lemma 6.2 and since λ is dominant, the condition of Lemma 4.1 is satisfied.Therefore, it remains to show that the adjoint representation of d 0 in d −1 = Γ(M, E ϕ * ) is irreducible and exact.The irreducibility easily follows from Lemmas 6.1 and 6.2.
Observe that the homomorphism d t is injective.Indeed, otherwise g i ⊂ Ker d t for some i.Hence, dt(g i ) = 0 which implies that G i ⊂ P .Furthermore, g i ⊂ Ker ψ by Lemma 6.1.
Let us identify g with d t(g) by means of d t.Form eq. ( 21) we see that dt = α| g and α(d 0 ) = α(g).
Consider the ideal g 0 = g ∩ End E ϕ of g.We see that g = g 0 ⊕ g, where g is also an ideal.Obviously, our assumptions imply that there is a decomposition into a semi-direct sum where End E ϕ is an ideal and g a subalgebra.Let us show that this sum is actually direct.Consider the induced representation ψ of the Lie algebra g in End E ϕ .The weights of the induced representation ϕ ⊗ ϕ * are of the form µ = ν 1 − ν 2 , where ν 1 , ν 2 are weights of ϕ.Since ϕ is irreducible, µ can be expressed in terms of the set Π P ⊂ Π corresponding to the semisimple part of P .But the roots of any subset Π i should enter the expression of any dominant weight in terms of Π with either positive or zero coefficients, see [H,§13,Exc. 8] and [R].
Since (dt)| g is injective, G i ⊂ P for all i such that g i ⊂ g.Therefore, if µ is dominant, then µ(h β ) = 0 for all β ∈ Π i and any such i.
The homogeneity of the supermanifold (M, O) proved above implies that the action of End E ϕ = End E ϕ * in d −1 is exact.The radical r of the Lie algebra d 0 is contained in End E ϕ and is non-zero since Cε ⊂ r, see § 2. Further, the Lie subalgebra ad d 0 ⊂ gl(d −1 ) is irreducible, and the dimension of its radical should be at most 1.Hence, r = Cε.
Therefore, End E ϕ = Cε ⊕ h and d 0 = End E ϕ ⊕ g are reductive Lie algebras with 1-dimensional centers.Any ideal of d 0 is contained either in End E ϕ or in g.Thanks to our assumption on λ and Theorem 5.3, the action of the Lie algebra g in d −1 is exact, and so is the d 0 -action.
Observe that the assumptions of Theorem 6.3 are satisfied, e.g., if G is simple and coincides with Aut • M , where • singles out the connected component of the unit, and ϕ is a non-trivial irreducible representation such that the highest weight of ϕ * is dominant.
On the other hand, the version of the theorem proved above, which does not presuppose that G-action on M is faithful, is useful in applications.As is well-known, the cases where the inclusions between Lie algebras α(d 0 ), (dt)(g) and Γ(M, T ) are strict do occur very seldom.
Observe that, if ϕ is irreducible and the G-action on M is locally faithful, then we have End E ϕ = Cε, see [R].
In conclusion, let us compute the Lie superalgebras d for the supermanifolds (M, Ω) of Examples 3.1.1and 5.2, where M is a flag variety.Obviously, such a supermanifold is homogeneous for any compact complex homogeneous manifold M , although the representation ϕ that determines (M, Ω) is seldom completely reducible.Proof.The equality End T = Cε is proved in [I].It can also be deduced from Theorem 5.3 by using certain facts of Lie algebra theory.
⊂ O be the subsheaf of ideals generated by subsheaf O1 and let O rd := O/I.Clearly, (M, O rd ) is a(n almost) complex-analytic manifold.The simplest supermanifolds are the split ones which are described as follows.Let (M, O) be a(n almost) complex-analytic manifold of dimension n and F a locally free analytic sheaf of rank m on M .Set O := Λ O (F).Clearly, (M, O) is a supermanifold, and O rd is naturally isomorphic to the subsheaf O ⊂ O.In what follows I will identify O rd with O in this situation and will briefly write O = Λ O (F).The structure sheaf of the split supermanifold is endowed with a Z-grading O = 0≤p≤m O p , where O p = Λ p O (F).Observe that every C ∞ of analytic supermanifold is locally split. 2 In general, using the subsheaf I 1 := I we construct the following filtration of the structure sheaf: O = I 0 ⊃ I 1 ⊃ I 2 ⊃ . . .⊃ I m ⊃ I m+1 = 0, and the associated Z-graded sheaf of superalgebras gr O = 0≤p≤m gr p O, where gr p O = I p /I p+1 .Here gr 0 O = O rd , and gr 1 O = F is a locally free sheaf of O rd -modules.Hence, it is clear that gr O = Λ O rd (F).Thus, to every supervariety (M, O) there is a corresponding split supervariety (M, gr O).Denote by Der O the sheaf of germs of derivations of the structure sheaf O. Its stalk at point w ∈ M is the Lie superalgebra Der C O w .The sections of the sheaf Der O are called vector fields on the supermanifold (M, O).The space d := Γ(M, Der O) is naturally endowed with a Lie superalgebra structure over C. If (M, O) is split, then Der O is a graded sheaf of Lie superalgebras whose homogeneous components are Der p O = {δ ∈ Der O | δ O q ⊂ O q+p for all q ∈ Z}.Therefore, d is a graded Lie superalgebra.In the general case we endow Der O with the following filtration Der (−1) O = Der O, Der (p) O = {δ ∈ Der O | δ O ⊂ I p , δI ⊂ I p+1 } for all p > 0.