On First and Second Cohomology Groups for BBW Parabolics for Classical Lie Superalgebras
aa r X i v : . [ m a t h . R T ] F e b ON FIRST AND SECOND COHOMOLOGY GROUPS FOR BBWPARABOLICS FOR CLASSICAL LIE SUPERALGEBRAS
DAVID M. GALBAN
Abstract.
Let g be a classical simple Lie superalgebra. In this paper, the author studiesthe cohomology groups for the subalgebra n + relative to the BBW parabolic subalgebrasconstructed by D. Grantcharov, N. Grantcharov, Nakano and Wu. These classical Liesuperalgebras have a triangular decomposition g = n − ⊕ f ⊕ n + where f is a detectingsubalgebra as introduced by Boe, Kujawa and Nakano. It is shown that there existsa Hochschild-Serre spectral sequence that collapses for all infinite families of classicalsimple Lie superalgebras. This enables the author to explicitly compute the first andsecond cohomologies for n + . The paper concludes with tables listing the weight spacedecompositions and dimension formulas for these cohomology groups. Introduction g a semisimple Lie algebra over C , J a subset of simple roots and p J = l J ⊕ u J the corresponding parabolic subalgebra, a famous theorem of Kostant demonstrates thatH k ( u J , L ( µ )) = M w ∈ W J , l ( w )= k L J ( w · µ ) , where L J ( w · µ ) is an irreducible finite-dimensional module corresponding to the Levi factor l J for J [UGA09]. Kostant’s theorem is piece of a larger picture where in the (parabolic)Category O J one has the isomorphism:(1) Ext n O J ( Z J ( λ ) , L ( µ )) ∼ = Hom l J ( L J ( λ ) , H n ( u J , L ( µ ))) , where Z J ( λ ) is a (parabolic) Verma module arising from inducing a finite-dimensional l J -module L J ( λ ) and L ( λ ) is an irreducible representation in O J . It is a deep theorem thatthese extension groups in (1) can be computed via Kazhdan-Lusztig polynomials [Kum02].1.2. In the case when g is a classical simple Lie superalgebra one would like to havea Kazhdan-Lusztig theory and a Kostant-type theorem in the context of a Category O theory. D. Grantcharvov, N. Grantcharov, Nakano and Wu [GGNW19] introduced thenotion of a BBW parabolic subalgebra, b , that contains the detecting subalgebra, f , earlierintroduced by Boe, Kujawa and Nakano [BKN10]. One can view that algebra f like aLevi subalgebra and b as a parabolic containing f . There exists a natural triangulardecomposition of g = n − ⊕ f ⊕ n + where b = f ⊕ n + where the Lie superalgebras n ± arenilpotent subalgebras.Recently, Lai, Nakano and Wilbert [LNW] have constructed a Category O f via thistriangular decomposition and have proved an analog to (1). Other efforts have been madein understanding a Category O for Lie superalgebras on a case-to-case basis; however,prior to [LNW] there has not been a unified treatment. A fundamental question is to The author was partially supported by NSF (RTG) grant DMS-1344994. compute H n ( n + , L ( λ )), where L ( λ ) is a finite-dimensional g -module and to determine ifthere is a Kostant-type theorem in the O f . This paper aims to provide the first calculationin this direction.1.3. Outline.
The paper is organized as follows. In Section 2 we review the definitions ofLie superalgebras, Lie superalgebra cohomology, and detecting and nilpotent subalgebras.In Section 3, a Hochschild-Serre spectral sequence is defined for each of the infinite familiesof classical Lie superalgebras and it is shown that in each case, it collapses.In Section 4, the notion of a superderivation is defined and it is shown how first coho-mology for arbitrary modules can be expressed as a quotient of the set of superderivations.We then provide a formula for H ( n , C ) and compute its dimension.In Section 5, we first interpret second cohomology as giving the set of classes of centralextensions for superalgebras. Expressions for H ( n , C ) for each classical Lie superalgebrain terms of their weight spaces are found, as well as formulas for their dimension. Finally,in Section 6 we summarize the weight spaces and dimensions of both the H and H cohomologies in a series of tables.1.4. Acknowledgements.
This paper is part of the author’s Ph.D dissertation at theUniversity of Georgia. He acknowledges his Ph.D advisor, Daniel K. Nakano, for hisguidance throughout the project. He also thanks Shun-Jen Cheng for his insights aboutthe exceptional families of simple Lie superalgebras.2.
Preliminaries
Notation.
Throughout this paper, all vector spaces, unless otherwise noted, will beover C . A superspace is a vector space V = V ¯0 ⊕ V ¯1 with a Z -grading. An element v ∈ V ¯0 is referred to as even, and an element in V ¯1 as odd. Such an element in either V ¯0 or V ¯1 is referred to as homogeneous. If v is homogeneous, we define the degree | v | of v as theelement i ∈ Z such that v ∈ V i .A Lie superalgebra is a superspace g = g ¯0 ⊕ g ¯1 equipped with a bilinear multiplication[ · , · ] satisfying the following properties:(1) [ g i , g j ] ⊆ g i + j (2) [ a, b ] = − ( − | a |·| b | [ b, a ](3) [ a, [ b, c ]] = [[ a, b ] , c ] + ( − | a |·| b | [ b, [ a, c ]] , where properties 2 and 3 hold for homogeneous elements, and the multiplication is ex-tended to all of g linearly [CW12, Definition 1.3]. A g -module M is a superspace equippedwith an action by g that is compatible with the Z grading.The notion of a universal enveloping algebra generalizes to the superalgebra case aswell. Given a superalgebra g let T ( g ) denote the tensor algebra on g . Let I denote theideal generated by elements of the form x ⊗ y − ( − | x || y | y ⊗ x − [ xy ]Let U ( g ) = T ( g ) /I and let i be the canonical embedding of g into U ( g ). Then U ( g )satisfies the universal property that if j : g → M is any linear map satisfying j ([ xy ]) = j ( x ) j ( y ) − ( − | x || y | j ( y ) j ( x )then there is a unique homomorphism φ : U ( g ) → M such that φ ◦ i = j . We let I g denotethe augmentation ideal of U ( g ). OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 3
Lie superalgebra cohomology.
We define the Lie superalgebra cohomology of g with coefficients in a module M as follows. Consider the Koszul complex whose cochaingroups are given as C n ( g , M ) = Hom(Λ ns ( g ) , M ) , where Λ ns ( g ) denotes the superexterior algebraΛ ns ( g ) := M i + j = n Λ i ( g ¯0 ) ⊗ S j ( g ¯1 ) . The differential maps d n : C n ( g , M ) → C n +1 ( g , M ), for homogeneous f , are given by theformula(2) df ( ω ∧ · · · ∧ ω n ) = n X i =0 ( − τ i ω i · f ( ω ∧ · · · ∧ b ω i ∧ · · · ∧ ω n )+ X i We now define the notion of a detectingsubalgebra, essentially an analog of the Cartan subalgebra in the classical case, follow-ing D. Grantcharov, N. Grantcharov, Nakano, and Wu [GGNW19]. We say that a Liesuperalgebra g is classical if there is a connected reductive algebraic group G ¯0 such thatLie( G ¯0 ) = g ¯0 and if the action of G ¯0 on g ¯1 differentiates to the adjoint action.If g is a classical Lie superalgebra, g ¯1 admits a stable action by G ¯0 . Following theconstruction in [BKN10, Section 8.9], fix a generic element x ∈ g ¯1 and set H = Stab G ¯0 x .We define f ¯1 = g H ¯1 and f ¯0 = [ f ¯1 , f ¯1 ] and let f = f ¯0 ⊕ f ¯1 be the detecting subalgebra .Moreover, as per [BKN10, Section 8], we can make the odd roots corresponding to f explicit and thus also those corresponding to f ¯1 and f itself. By convention, let r denotethe minimum of m and n . Let Ω denote the set of odd roots of f . Then f ¯1 = { X α ∈ Ω ( u α x α + v α x − α ) | u α , v α ∈ C } . f ¯0 can then be obtained by taking brackets.Let ǫ i and δ j be linear functionals on diagonal matrices a = diag( a , · · · , a n + m )which satisfy ǫ i ( a ) = a i DAVID M. GALBAN and δ j ( a ) = a m + j . Then for each of the classical simple Lie superalgebras, we have the following values forΩ. g Ω gl ( m | n ) { ǫ i − δ i | ≤ i ≤ r } sl ( m | n ) { ǫ i − δ i | ≤ i ≤ r } psl ( n | n ) { ǫ i − δ i | ≤ i ≤ n } osp (2 m + 1 | n ) { ǫ i − δ i | ≤ i ≤ r } osp (2 m | n ) { ǫ i − δ i | ≤ i ≤ r } D (2 , α ) { ǫ + ǫ + ǫ } G (3) { ǫ + δ } F (4) { ǫ + ǫ + ǫ + ǫ } In the case of q ( n ) we let f ¯1 be the collection of all matrices whose odd part is diagonal.Looking at the adjoint action of the maximal torus in f ¯0 on g produces a root-spacedecomposition of g , and letting n denote the space of positive roots and n − the space ofnegative ones, we obtain a triangular decomposition g = n − ⊕ f ⊕ n . We also provide atable listing the collection of root spaces corresponding to each of the n − for the classicalLie superalgebras. g Φ − ¯1 gl ( m | n ) { ǫ i − δ j , − δ i + ǫ j | i < j } sl ( m | n ) { ǫ i − δ j , − δ i + ǫ j | i < j } osp (2 m + 1 | n ) {− ǫ i + δ j , − δ i + ǫ j , − ǫ k − δ l , − δ t | i < j } osp (2 m | n ) { ǫ i − δ j , − δ i + ǫ j , − ǫ k − δ l | i < j } q ( n ) { ǫ i + ǫ j | i < j } D (2 , α ) { ( − ǫ, − ǫ, − ǫ ), ( − ǫ, − ǫ, ǫ ), ( ǫ, − ǫ, − ǫ ) } G(3) { ( − ω + ω , − ǫ ),(2 ω − ω , − ǫ ),(0 , − ǫ ),( ω − ω , − ǫ ),( − ω + ω , − ǫ ),( − ω , − ǫ ) } F(4) { ( ω − ω , − ǫ ), ( ω − ω + ω , − ǫ ), ( ω − ω , − ǫ ), ( − ω + ω , − ǫ )( − ω + ω − ω , − ǫ ), ( − ω + ω , − ǫ ), ( − ω , − ǫ ) } The Hochschild-Serre Spectral Sequence As in the case of classical Lie algebra cohomology, letting h denote an ideal of g , weconstruct an analogue of the Hochschild-Serre spectral sequence for Lie superalgebras.Consider a short exact sequence of Lie superalgebras0 → h → g → g / h → F : g / h -mod → C -mod G : g -mod → g / h -mod , which are given by F ( − ) = H ( g / h , − ) and G ( − ) = H ( h , − ). Both F and G satisfy theconditions given in [Jan03, Proposition 4.1], and so we obtain a Grothendieck spectralsequence: E p,q = R p F ( R q ( G ( − ))) , OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 5 which converges to R p + q ( F G )( − ). As F ◦ G = H ( g , − ), this simplifies to E p,q = H p ( g / h , H q ( h , − )) ⇒ H p + q ( g , − ) . Infinite families. In this section, we provide a basis for n for each of the infinitefamilies of classical simple Lie superalgebras, and define an ideal I of n . As a consequence,for each family we will obtain a short exact sequence0 → I → n → n / I → , which will give rise to a Hochschild-Serre spectral sequence E ij = H i ( n / I , H j ( I , C )) ⇒ H i + j ( n , C ) . We then show in the following section that each of these spectral sequences collapses.3.1.1. gl ( m | n ) . Let g = gl ( m | n ) where m ≥ n and let n − ⊕ f ⊕ n be its triangular decom-position. Following [CW12, Section 1.1.2] we label the rows and columns of elements of gl ( m | n ) by elements of the set { ¯1 , · · · ¯ m, , · · · n } . We let E ij denote the elementary matrixfor row i and column j . Then n is spanned by E ¯ i, ¯ j ( ǫ i − ǫ j ) 1 ≤ i < j ≤ mE i,j ( δ i − δ j ) 1 ≤ i < j ≤ nE ¯ i,j ( ǫ i − δ j ) 1 ≤ i ≤ m, ≤ j ≤ n, i < jE i, ¯ j ( δ i − ǫ j ) 1 ≤ i ≤ n, ≤ j ≤ m, i < j, where the quantity in parentheses denotes the corresponding weight under the action ofthe maximal torus.We let I ⊆ n be the subalgebra spanned by elements E ¯ i, ¯ m , E ¯ i,n , E i, ¯ m , and E i,n in thecase where m = n , and by just E ¯ i, ¯ m and E i, ¯ m when m > n , with the appropriate boundson i . Using the supercommutator identity:[ E ij , E kl ] = δ jk E il − ( − | E ij |·| E kl | δ li E jk , it is a simple computation to show that I is an ideal of n . osp (2 m + 1 | n ) . Let m ≥ n . We may view osp (2 m + 1 | n ) as being a subalgebra of gl (2 m + 1 | n ), and so we may describe its spanning set by means of the same elementarymatrices. In particular, osp (2 m + 1 | n ) will be the span of the root vectors and maximaltorus as described in [CW12, Section 1.2.4]. Restricting our view to the weight spaceslisted in the above table, let n be the subalgebra whose odd component is spanned by theelements: E k + n,i + m + E ¯ i,k ( − ǫ i + δ j ) − E i + m,k + n + E k, ¯ i ( − δ i + ǫ j ) E k + n, ¯ l − E l + m,k ( − ǫ k − δ l ) E n +1 , ¯ t + E t + m, n +1 ( δ t ) , where 1 ≤ i ≤ m and 1 ≤ k ≤ n , and whose even component is the direct sum of thenilpotent radicals of so (2 m + 1) and sp (2 n ).We let I be the subalgebra of n spanned by all root vectors with weights containing an ǫ m a δ n term. Again, it may be shown that this constitutes an ideal of n . DAVID M. GALBAN osp (2 m | n ) . The n arising from osp (2 m | n ) has a similar basis as in the osp (2 m +1 | n ) case, with an odd part given by: E k + n,i + m + E ¯ i,k ( − ǫ i + δ j ) − E i + m,k + n + E k, ¯ i ( − δ i + ǫ j ) E k + n, ¯ l − E l + m,k ( − ǫ k − δ l )and an even part given by the direct sum of the nilpotent radicals of so (2 m ) and sp (2 n ).We may define an ideal just as we did for osp (2 m + 1 | n ), letting I be the collection ofall root vectors corresponding to weights of n containing an ǫ m term.3.1.4. q ( n ) . We may view q ( n ) as the subalgebra of gl ( n | n ) spanned by the elements: e E ij := E ¯ i ¯ j + E ij ( ǫ i − ǫ j ) , E ij := E i ¯ j + E ¯ i,j ( ǫ ′ i − ǫ ′ j ) , ≤ i, j ≤ n. Then n is the subalgebra spanned by all e E ij and E ij where i < j . Let I be the subalgebraof n generated by all e E in and E in . Again, it is not too difficult to show that I is an idealof n .3.2. Collapsing.Theorem 3.2.1. For any of the infinite families of classical Lie superalgebras g , thecorresponding spectral sequence E ijr collapses on the r = 2 page.Proof. Recall that the differentials d r on the r th page of a spectral sequence have bidegree( r, − r ), sending E ijr to E i + r,j − r +1 r . Our goal is to show that for each page r ≥ 2, thedifferentials must all be 0. First, note that we may decompose all E ijr into a direct sumof weight spaces under the action of the maximal torus of f . The differentials respect thisaction, and so to show that d r is identically 0, it is sufficient to show that no weight in E ijr appears in E i + r,j − r +1 r . To demonstrate this, we split the proof up into different casesfor each classical superalgebra.(1) gl ( m | n ) Consider an arbitrary differential from the E page: d : E ij → E i +2 ,j − .The term E ij is a subquotient of Λ is ( n / I ) ∗ ⊗ Λ js ( I ) ∗ , and so any weight of E ij must also be a weight of Λ is ( n / I ) ∗ ⊗ Λ js ( I ) ∗ . As the weights of n / I are of the form ǫ k − δ l , δ k − ǫ l , ǫ k − ǫ l and δ k − δ l for 1 < k < l < n and the weights of I are ofthe form ǫ i − δ n , δ i − δ n , ǫ i − ǫ n and δ i − ǫ n for 1 < i < n , the weights of E ij allhave j summands containing either ǫ n or δ n . As the weights of E i +2 ,j − have only j − d must be the zero map.We therefore have that E ij = E ij for all i and j . However, we can apply thesame argument to the differentials on the E r page for any arbitrary r . Namely, ifthe weights in the domain of d r have j copies of ǫ m or δ n , then those in the imagehave only j − r such copies. Thus, d r must again be the 0 map. Thus for all r > E ijr = E ij , and so the spectral sequence collapses.(2) sl ( m | n ) The collection of weights corresponding to the n in sl ( m | n ) are identicalto those for gl ( m | n ). Hence, we may take the same ideal of n ⊆ sl ( m | n ) and thesame spectral sequence will collapse.(3) osp (2 m + 1 | n ) The ideal I is spanned by all weight spaces of a root containing ǫ m .Thus an arbitrary weight of E pqr must have a total of q copies of or ǫ m , whereas OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 7 those in E p + r,q +(1 − r ) r have only q + 1 − r copies. Thus any differential d r must be0, and so the spectral sequence collapses on the E page.(4) osp (2 m | n ) We defined the ideal for osp (2 m | n ) similarly to how it was definedfor osp (2 m + 1 | n ), and so the above argument follows in the same way.(5) q ( n ) As E ij is a subquotient of Λ is ( n / I ∗ ) ⊗ Λ js I ∗ , all of its weights must contain j total summands containing either copy of ǫ n , whereas E i + r,j +1 − r only contains j + 1 − r such copies, and thus an arbitrary differential d r : E ij → E i +2 ,j − mustbe 0, so the spectral sequence again collapses. (cid:3) 4. H ( n , C ) Cohomology Superderivations. It is well known that in the case of ordinary Lie algebras, H ( g , M )corresponds to derivations from g to M modulo inner derivations [HS97]. This situationgeneralizes to the Lie superalgebra case.We define a superderivation from a Lie superalgebra g to a g -module M to be a linearmap φ satisfying φ ([ xy ]) = x · φ ( y ) − ( − | x || y | y · φ ( x ) . An inner superderivation is a derivation of the form φ a ( x ) = x · a for some a ∈ M . Proposition 4.1.1. SupDer( g , M ) ∼ = Hom( I g , M ) .Proof. Let d : g → M be a superderivation. Consider the map f ′ d : T ( g ) → M given by f d ( x ⊗ · · · ⊗ x n ) = x ◦ · · · ◦ d ( x n ) and which sends T ( g ) to 0. It follows immediately that f ′ d vanishes on I and thus defines a morphism on U ( g ) which restricts to a homomorphism f d : I g → M .Conversely, given a homomorphism f : I g → M , we can extend it to a map on allof U ( g ) by setting f ( T ( g )) = 0 and letting d f = f ◦ i . It is straightforward to showthat f d f = f and d f d = d , and so the map sending f to d f is an isomorphism betweenSupDer( g , M ) and Hom( I g , M ). (cid:3) Proposition 4.1.2. H ( g , M ) ∼ = SupDer( g , M ) / InnSupDer( g , M ) . Proof. From the augmentation map, we obtain the following short exact sequence:0 → I g → U ( g ) → C → . From the corresponding long exact sequence in cohomology, we obtain thatH ( g , M ) ∼ = Coker(Hom( U ( g ) , M ) → Hom( I g , M )) ∼ = SupDer( g , M ) / Im(Hom( U ( g ) , M )) . However, if f ∈ Hom( U ( g ) , M ), and f (1) = a , then the corresponding derivation is d f ( x ) = x · a , and thus H ( g , M ) ∼ = SupDer( g , M ) / InnSupDer( g , M ) . (cid:3) In particular, when using trivial coefficients, we have the following result: Theorem 4.1.1. H ( g , C ) ∼ = ( g / [ g , g ]) ∗ . Explicit calculations. By the above theorem, to compute the first cohomology, itis sufficient to describe both n and [ n , n ]. As we have already provided bases for n inSection 3, below we do the same for [ n , n ] and give formulas for the dimensions of n , [ n , n ],and H ( n , C ). A table of corresponding weights is given in Section 6. DAVID M. GALBAN gl ( m | n ) . We have that the elementary matrices E ij that span n will be in [ n , n ]precisely when j − i ≥ 2, and so [ n , n ] will have a basis given by E ¯ i, ¯ j ≤ i, j ≤ m, j − i ≥ E i,j ≤ i, j ≤ n, j − i ≥ E ¯ i,j ≤ i ≤ m, ≤ j ≤ n, j − i ≥ E i, ¯ j ≤ i ≤ n, ≤ j ≤ m, j − i ≥ . The Lie superalgebra n has dimension (cid:0) m (cid:1) + n · ( m − n ) + 3 · (cid:0) n (cid:1) and [ n , n ] has dimension (cid:18) m − (cid:19) + 2 · (cid:18) n − (cid:19) + n · ( m − n − 1) + (cid:18) n (cid:19) , and so H ( n , C ) has dimension m − n − n − n = m + 3 n − 3. The weightsof H ( n , C ) can be found by using the information listed in the previous section and areincluded in the tables in Section 6.4.2.2. sl ( n | n ) . The weight space decomposition for n is identical to that in the gl ( n | n )case, and thus the above dimension formula and weight space decomposition hold.4.2.3. osp (2 m | n ) . The derived subalgebra [ n , n ] is spanned by the elements E j,i − E i + n,j + m E j + m,i + n − E i,j E l,k + n − E k + n,l + m E i,i + n E i,k + n + E k,i + n E i,k − E k + n,i + n E j,l + m − E l,j + m E jl − E l + m,j + m , where 1 ≤ i, k ≤ n and 1 ≤ j, l ≤ m and j − i ≥ 2. The quotient by this subalgebraconsists of root vectors solely with the corresponding weights ǫ j − δ j +1 , δ j − ǫ j +1 , ǫ m + δ n ,2 δ n , δ i − δ i +1 , and ǫ i − ǫ i +1 . As a result, H ( n , C ) has dimension2( m − 1) + 2( n − 1) + 2 = 2 m + 2 n − . osp (2 m + 1 | n ) . The only difference in terms of dimension between this and thepreceding case is the existence of a root in n not found in [ n , n ]. Thus, the dimensioncalculation may proceed in essentially the same way, yielding a dimension formula of2 m + 2 n − q ( n ) . Much like in the case of gl ( m | n ), if g = q ( n ), then [ n , n ] is spanned by thematrices: ( E i,j ≤ i, j ≤ n, j − i ≥ e E i,j ≤ i, j ≤ n, j − i ≥ OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 9 Hence, the dimension of [ n , n ] is 2 · (cid:0) n − (cid:1) = ( n − n − n is2 · (cid:0) n (cid:1) = n ( n − ( n , C ) is n ( n − − ( n − n − 1) = 2( n − . D (2 , , α ) , G (3) , and F (4) . For each of the exceptional superalgebras, we may lookat the weights given in the table from Section 2. As no two weights add up to a third, itfollows that the bracket is 0 on n ¯1 and so that n ¯1 is abelian, and so n ∼ = n / [ n , n ].5. H ( n , C ) Cohomology Central Extensions. As in the case of H ( n , C ), the classical Lie algebra interpreta-tion of equivalence classes of extensions extends to the superalgebra case. On the cochaincomplex C n ( g , M ) we set the following Z grading: C n ( g , M ) α = { f ∈ Hom(Λ ns ( g , M ) | f (Λ n ( g )) β } ⊆ M α + β , where α and β are elements of Z . As the differential map preserves this grading, thisgives rise to a Z grading on H n ( g , M ) as well.If M is a g -module, regarding M as an abelian superalgebra, we say that h is anextension of g by M if there is an exact sequence of g -modules:0 → M → h → g → , where h is a Lie superalgebra. Two such extensions are said to be equivalent if there is acommutative diagram 0 M h g M h g ϕid idϕ f . Given an even cocycle h , we define the extension E h via the short exact sequence0 → M → g ⊕ M → g → , where the product in g ⊕ M is given by[( x, m ) , ( y, n )] = ([ x, y ] , xn − ( − | m || y | ym + h ( x, y )) . Every extension will be equivalent to E h for some even cocycle h . Moreover, one canshow that two extensions E h and E h ′ are equivalent if and only if there is some even linearmap f : g → M such that df = h − h ′ , and thus the equivalence classes of extensions arein one-to-one correspondence with H ( g , M ) ¯0 [Mus12, Section 16.4].5.2. Computing H . Computing the H ( n , C ) cohomology involves a term mixing to-gether both odd and even elements, and thus requires much more care than the H case.The main idea will be to compute the dimension of these groups recursively. For simplic-ity’s sake, let us restrict our attention to g = gl ( n | n ), and let n ( n ) denote the correspondingnilpotent radical. From the collapsing of Hochschild-Serre spectral sequence, we have that:(3) H ( n ( n ) , C ) ∼ = H ( n ( n ) / I , H ( I , C )) ⊕ H ( n ( n ) / I , H ( I , C )) ⊕ H ( n ( n ) / I , H ( I , C )) , where I is the ideal described in Section 3. As I is abelian, the cohomology groupsH n ( I , C ) can be easily computed. Additionally, there is a natural isomorphism between n ( n ) / I and n ( n − directly, viewing it as the set of fixed points of H ( I , C ) under the action of n ( n − ( n ( n ) / I , H ( I , C )), which is isomorphic to H ( n ( n − , I ∗ ).5.3. Low-Dimension Examples. As an example where all of the computations are rel-atively straightforward, let us first consider the case of gl (2 | 2) where we wish to computeH ( n (2) , C ). As n (2) is abelian, all of the differentials in the cochain complex C → C → C → · · · are 0, where C i ∼ = Λ is ( n (2) ∗ ). As such, for any i , H i ( n (2) , C ) ∼ = Λ is ( n (2) ∗ ). In particular,H ( n , C ) ∼ = Λ s ( n ∗ ) ∼ = M i + j =2 Λ i ( n ¯0 ) ⊗ S j ( n ¯1 ) . Using the formulas for the dimensions of exterior and symmetric algebras on a vectorspace of dimension n , namely dim Λ i ( V ) = (cid:18) ni (cid:19) and dim S j ( V ) = (cid:18) n + j − j (cid:19) , we obtain dim H ( n , C ) = dim Λ s ( n ) = 1 · · · . Now consider the case where g = gl (3 | ( n (3) , C ). Letting n denote n (2), note that as n is abelian, n ¯0 is an ideal of n , and so we obtain a short exactsequence 0 → n ¯0 → n → n ¯1 → . This gives rise to a second Hochschild-Serre spectral sequence: E i,j = H i ( n ¯1 , H j ( n ¯0 , I ∗ ¯0 ⊗ I ∗ ¯1 )) ⇒ H i + j ( n , I ∗ ¯0 ⊗ I ∗ ¯1 ) . Again appealing to an argument with weights, the differential d sends E , to 0. As thespectral sequence is in the first quadrant, all subsequent differential must do the same.Thus, we have that H ( n , I ∗ ¯0 ⊗ I ∗ ¯1 ) ∼ = E , ⊕ E , As E , = H ( n ¯1 , H ( n ¯0 , I ∗ ¯0 ⊕ I ∗ ¯1 )) = H ( n ¯1 , C ⊕ ), we can simplify this as H ( n ¯1 ) ⊕ . As n ¯1 is abelian of dimension 2, E , must have dimension 8. On the other hand, E , ∼ =H ( n ¯1 , H ( n ¯0 , I ∗ )). However, as n ¯0 is a classical Lie algebra, by Kostant’s theorem,H ( n ¯0 , I ∗ ) ∼ = M l ( w )=1 , j ∈ J w · λ j , where w is an element of the Weyl group of n ¯0 and I ∗ = L j ∈ J L( λ j ) as a direct sum of n ¯0 modules. (Viewing I ∗ as an sl (2) × sl (2)-module shows it is isomorphic to L((1 , ⊕ L((1 , ⊕ L((0 , ⊕ L((0 , . ) As the Weyl group of n ¯0 is isomorphic to Σ × Σ , thereare 2 elements of length 1, and so E , = H ( n ¯1 , s α · I ∗ ), which has dimension 4. Thus,altogether H ( n ¯0 , I ∗ ) has dimension 8, from which an easy computation shows that thedimension of the set of fixed points under the action of n ¯1 is 4, which implies H ( n , I ∗ ) to OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 11 have a total dimension of 12. Using the argument below, we can see that H ( n , Λ s ( I ∗ ))has dimension 8 and we already know H ( n , C ) has dimension 8, so altogether, this impliesthat H ( n (3) , C ) has dimension 28. However, the argument for computing the dimensionof H ( n , I ∗ ) was only valid because n was abelian. For general gl ( n | n ) this isn’t the case,so n ¯0 is not necessarily an ideal of n .5.4. Explicit Calculations. gl ( n | n ) . Before beginning with the more general case of gl ( m | n ), we start with themore special case of gl ( n | n ). As in the general case above, we may compute H ( n , C ) bymeans of the direct sum decomposition from the spectral sequence, i.e.,H ( n , C ) ∼ = H ( n / I , Λ s ( I ∗ )) ⊕ H ( n / I , I ∗ ) ⊕ H ( n / I , C ) . The first term can be identified with the set of fixed points of Λ s ( I ∗ ) under the action of n / I , i.e., all x ∈ Λ s ( I ∗ ) such that ( n / I ) · x = 0. This set is not particularly difficult tocalculate, and we get the following result. Proposition 5.4.1. For all n , H ( n / I , Λ s ( I ∗ )) has dimension 8.Proof. Note that if a ∈ n / I and x ∈ Λ s ( I ∗ ) are weight vectors of weights λ and µ , then a · x has weight λ + µ , and so a sends distinct weight spaces to distinct weight spaces. Inparticular, if x + · · · + x n is a sum of weight vectors of distinct weights in Λ s ( I ∗ ) and a · ( x + · · · + x n ) = 0, then a · x i must equal 0 for all i . Since the standard basis for Λ s ( I ∗ )consists of root vectors all of distinct weights, it suffices to look at which basis elementsare sent to 0 by n / I . I ∗ has a basis given by E ∗ i,n , E ∗ i,n , E ∗ i,n , and E ∗ i,n , for 1 ≤ i ≤ n − 1. Based on thesupercommutator identity, if E i,j is in n / I and E ∗ k,n or E ∗ k,n is in I ∗ , E i,j · E ∗ k,n doesn’tvanish precisely when i = k . In particular, as there are no elements E i,j in n / I where i = n − n − 1, it is precisely the basis elements E ∗ n − ,n , E ∗ n − ,n , E ∗ n − ,n , and E ∗ n − ,n thatare sent to 0 for all a ∈ n / I . Any element of Λ s ( I ∗ ) that is sent to 0 is the superexteriorproduct of two such basis elements of I ∗ , and as there are two even and two odd suchbasis elements, viewing Λ s ( I ∗ ) as Λ ( I ∗ ¯0 ) ⊕ ( I ∗ ¯0 ⊗ I ∗ ¯1 ) ⊕ S ( I ∗ ¯1 ), the total dimension ofH ( n / I , Λ s ( I ∗ )) is 1 + 2 · (cid:3) Moreover, the third term may be computed recursively, using the fact that n / I isisomorphic to n from gl ( n − | n − C → C → C → · · · , where C i ∼ = Λ is ( n / I ∗ ) ⊗ I ∗ and where the differentials are as in the introduction. Thenthe middle term H ( n / I , I ∗ ) is given by the cohomology of the complex at C . Since thedifferentials preserve the action of the torus, it follows that we may break up C i into itsweight spaces. The weights of ( n / I ) ∗ are of the form α j − β k , where α and β correspondto either ǫ or δ , and k < j < n . The weights of I ∗ are of the form α ′ n − β ′ i , where i < n .All weights of C i will be sums of weights of these forms. Actually, using the fact thatthe cohomology will be a subquotient of n / I / [ n / I , n / I ] ∗ ⊗ I ∗ , we need only consider thoseweights of ( n / I ) ∗ of the form α j +1 − β j . As a shorthand, given a weight α i − β j , we let F i,j and G i,j denote the basis vector of ( n / I ) ∗ of weight α i − β j , or more explicitly: F i,j , G i,j = E ∗ ¯ j, ¯ i α = ǫ, β = ǫE ∗ j,i α = δ, β = δE ∗ j, ¯ i α = ǫ, β = δE ∗ ¯ j,i α = δ, β = ǫ. Proposition 5.4.2. The dimension for a weight space of C is at most 2.Proof. Suppose two basis vectors for C , F j +1 ,j ⊗ G n,k with weight ( α j +1 − β j ) + ( α ′ n − β ′ k )and F ′ l +1 ,l ⊗ G ′ n,m with weight ( ζ l +1 − η l ) + ( ζ ′ n − η ′ m ) actually had the same weight. Asthe ǫ i , δ j are linearly independent, any weight has a unique representation as a sum of ǫ i ’sand δ j ’s. This leads to two cases:(1) If α j +1 , βj , α ′ n , and β ′ k are all distinct, these must be, in some order, the sameweights as ζ l +1 , η l , ζ ′ n , and η ′ m . Since l + 1 < n , it follows that ζ ′ n = α ′ n and α j +1 = ζ l +1 , so j = l . Thus, either η l equals either β j or β ′ k , which leads to twopossible basis vectors of the same weight, giving a total dimension of at most 2.(2) If α j +1 = β ′ k , then ζ ′ n = α ′ n , η l = β l and ζ l +1 = η ′ m . Since this forces l + 1 to equal j + 1 and m to equal l + 1, ζ l +1 can equal only ǫ j +1 or δ j +1 , which yields at mosttwo basis vectors. (cid:3) With this in mind, we aim to determine the dimension of the image of d and kernelof d . To do this, we will determine which weights appear in both C and C and whichappear in C but not C . For the former calculation, to calculate the dimension of theimage of d , first note that since its image is in C , the differential defined in Equation 2simplifies to d f ( ω ) = ( − τ i ω · f (1) , where a function f : C → I ∗ is identified with an element of I ∗ via the map sending f to f (1). What this means is that so long as there exists an element x of nI such that x · f (1) = 0, then d does not map f to 0. If f (1) ∈ I ∗ and x ∈ n / I are nonzero weightvectors, this condition holds if the sum of the weights of f (1) and x is again a weight of I ∗ .A weight α ′ n − β ′ k of a basis vector G n,k of I ∗ may be written as a weight in C preciselywhen k < n − 1. In particular, G n,k will map to an element in the linear span of the rootvectors F ′ k +1 ,k ⊗ G ′ n,k +1 and F ′′ k +1 ,k ⊗ G ′′ n,k +1 corresponding to ( ǫ k +1 − β ′ k ) + ( α ′ n − ǫ k +1 )and ( δ k +1 − β ′ k ) + ( α ′ n − δ k +1 ), respectively. Since the differential preserves weights, and I ∗ has 4( n − − n − 2) weights of the above form, the dimension of the image of d is 4 · ( n − . To compute the dimension of the kernel, we rely heavily on the differential defined inEquation 2 and note that a generic weight will be of the form α j +1 − β j + α ′ n − β ′ i , where j < n − 1. So long as i < n − 1, this weight may be written as ( α ′ n − α ′ i +1 ) + ( α ′ i +1 − β ′ i ) + ( α j +1 − β j ), and so the differential will send the weight vector corresponding to( α j +1 − βj ) + ( α ′ n − β ′ i ) to a nonzero element of C . Thus the only weight vectors in thekernel have I ∗ component with i = n − 1. There are four basis elements of I ∗ with i = n − n − 2) basis elements of ( n / I / [ n / I , n / I ]) ∗ , so the one-dimensional weightspaces in the kernel contribute total dimension 4( n − · n − d . Besides those corresponding to weights OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 13 α ′ n − β ′ n − , which are already included in the span of the root vectors listed above, eachof these adds 1 more dimension to the kernel. As there are 4( n − 3) such elements, thisgives the kernel a total dimension of at least 20( n − − F j +1 ,j ⊗ G n,j of weight ( α j +1 − ǫ j )+( α ′ n − δ j ) and F ′ j +1 ,j ⊗ G ′ n,j of weight ( α j +1 − δ j )+ ( α ′ n − ǫ j ) be two basis vectors of the sameweight, where j < n − 1. Identify these basis elements with functions f and g from n / I to I . Using the action of the differential, we see that df will send the element F j +1 ,j ∧ H j +1 ,j of weight α j +1 − ǫ j + β j +1 − δ j to a root vector of weight α ′ n − β j +1 , where β is either ǫ or δ , depending on what α is not. However, dg will send the same element to 0. Similarly, dg will send F ′ j +1 ,j ∧ ¯ H j +1 ,j of weight α j +1 − δ j + β j +1 − ǫ j to α ′ n − β j +1 while df sends thesame element to 0. As df and dg are nonzero on different subsets of the basis elements,it follows that they must be linearly independent, and hence there is no nontrivial linearcombination of df and dg equal to 0. Since f and g span their weight space, any othernonzero element of that weight space gets mapped to a linear combination of df and dg ,and so cannot be mapped to 0 and is thus not in the kernel. Therefore, any weight of theform ( α j +1 − ǫ j ) + ( α ′ n − δ j ) does not appear in the kernel and thus the kernel must havedimension of exactly 20( n − − 4, and so the dimension of the first cohomology isdim ker d − dim Im d = 20( n − − − n − 2) = 16( n − − . Combining this with the fact that the first term in the direct sum decomposition abovehas dimension 8, we have that when n > 2, the dimension of H ( n , C ) equals8 + n X i =3 (16( i − − , which simplifies to8 + 16 n X i =3 ( i − 28) = 8 + 8( n + n ) − − n − 2) = 8 n − n + 16 . gl ( m | n ) . We now proceed to the general case of gl ( m | n ), where we assume that m > n ≥ 2. Note that in this case the ideal I is defined slightly differently from how it isin the case where m = n , leading n / I being isomorphic to the n from gl ( m − | n ). Thus,using the spectral sequence decompositionH ( n , C ) ∼ = H ( n / I , Λ s ( I ∗ )) ⊕ H ( n / I , I ∗ ) ⊕ H ( n / I , C )we can compute H( n , C ) recursively, working our way up from the n corresponding to gl ( n | n ).From here, the principals behind the computation are largely the same as in the gl ( n | n )case, where H ( n / I , Λ s ( I ∗ )) is computed by looking at the fixed points of Λ s ( I ∗ ) andH ( n / I , I ∗ ) is computed by observing how the differentials act on weights. Putting thisall together, we obtain the following formulas for the dimension of H ( n , C ) correspondingto gl ( n + ρ | n ):dim H ( n , C ) = n − n + 8 , ρ = 18 n − n + 8 , ρ = 28 n − n + 8 + 4 n ( ρ − 2) + ( ρ − +( ρ − , ρ > . q ( n ) . The calculation of the dimension of the second cohomology for q ( n ) is similarto that for gl ( n | n ). Note first that when n = 2, n is a 2-dimensional, abelian Lie super-algebra, and so the i th cohomology will be isomorphic to Λ is ( n ∗ ). Since both n ∗ ¯0 and n ∗ ¯1 have dimension 1, Λ i ( n ∗ ¯0 ) = 0 for all i > j ( n ∗ ¯1 ) has dimension 1 for all j , so Λ is ( n ∗ )is 2-dimensional for all i . For general n , we may use the same direct sum decompositionderived from the spectral sequence as in Equation 3.To compute H ( n / I , Λ s ( I ∗ )), which corresponds to fixed points of Λ s ( I ∗ ) under theaction of n / I , note that again the only weight vectors of Λ s ( I ∗ ) that will vanish underthe action of all elements n / I will be superexterior products involving maximal even rootand maximal odd root vectors, in particular, e E ∗ n − ,n and E ∗ n − ,n . Unlike in the gl ( n | n )case however, here there is only one such even root vector and one such odd root vector,so H ( n / I , Λ s ( I ∗ )) is spanned by e E ∗ n − ,n ⊗ E ∗ n − ,n and E ∗ n − ,n ⊗ E ∗ n − ,n . Thus, in the casewhere n > 2, the dimension of H ( n / I , Λ s ( I ∗ )) is equal to 2.In computing the middle term H ( n / I , I ∗ ), we may again use the fact that we candecompose the terms of the cochain complex into their weight spaces, and the differentialswill still preserve the action of the torus. Again, we may look solely at weights from( n / [ n / I , n / I ]) ∗ ⊗ I ∗ and argue as we did in the gl ( n | n ) case. Here, the kernel of d willhave dimension 4( n − 2) + 2( n − 3) and the image of d will have dimension 2( n − ( n / I , I ∗ ) a dimension of 4n-10.Combining these terms together, we have the dimension of H ( n , C ) equals2 + n X i =3 (4 i − , which simplifies to 2 n − n + 6 . osp (2 m | n ) . The same principles apply in computing the second cohomology forthe osp (2 m | n ) superalgebras. As before, we may decompose the cohomology into itsdirect sum decomposition as in Equation 3. Note that the last term is again computedrecursively, starting with the base case osp (2 | n ). In this case, n ¯0 is abelian, and so weobtain the direct sum decomposition: H ( n , C ) ∼ = H ( n ¯0 , C ) ⊕ H ( n ¯0 , n ∗ ¯1 ) ⊕ H ( n ¯0 , S ( n ∗ ¯1 )) . Since n ¯0 is the nilpotent radical of an ordinary Lie algebra, these cohomologies may be com-puted via Kostant’s theorem, which can be shown to sum up to have dimension n + n +42 .Using the fact for osp (2 m | n ), the n / I is isomorphic to the n from osp (2( m − | n ), thedimensions and weight space expressions for H ( n , C ) for m > gl ( m | n ) and q ( n ). These are listed in the tables in Section 6.5.4.5. osp (2 m +1 | n ) . We begin again with the direct sum decomposition from Equation 3.Much of the calculation is similar to that in the case of osp (2 m | n ). We begin with thebase case of osp (3 | n ) and use the recurrence from the direct sum formula to determinethe weight space decomposition for any higher osp (2 m + 1 | n ) . OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 15 D (2 , , α ) , G (3) , and F (4) . Just as in the case of H ( n , C ), the second cohomologyfor D (2 , , α ), G (3), and F (4) can be easily computed using the fact that the correspondingsubalgebras n are abelian. In particular, in each case H ( n , C ) is isomorphic to C ( n , C )in the corresponding cochain complex. A description in terms of its weight space decom-position is given in the tables below. Appendix: Tables of Weights and Dimensions In the tables below, we compile a list of all of the weights appearing in the first andsecond cohomologies for the Lie superalgebras used above, as well as their dimensions. Asa shorthand, we use the following notation. For gl ( m | n ), we let α i be the weight ǫ i +1 − ǫ i , α ′ i be the weight δ i +1 − δ i , β i be the weight δ i +1 − ǫ i , and β ′ i be the weight ǫ i +1 − δ i . Inthe case of gl ( m | n ), we assume that m > n . In the case of osp , we let µ , · · · µ m denotethe simple weights of B m or D m , and let ν , · · · , ν n be the simple weights of C n .6.1. H ( n , C ) Cohomology. H ( n , C ) Cohomology (Classical Cases) Lie Superalgebra CorrespondingEven Weights CorrespondingOdd Weights gl ( n | n ) α i , 1 ≤ i ≤ n − α ′ j , 1 ≤ j ≤ n − β i , 1 ≤ i ≤ n − β ′ j , 1 ≤ j ≤ n − gl ( m | n ) α i , 1 ≤ i ≤ m − α ′ j , 1 ≤ j ≤ n − β i , 1 ≤ i ≤ n − β ′ j , 1 ≤ j ≤ n osp (2 m | n ) − µ i , 1 ≤ i ≤ m − ν i , 1 ≤ i ≤ n ǫ i +1 − δ i , 1 ≤ i ≤ rδ i +1 − ǫ i , 1 ≤ i ≤ r osp (2 m + 1 | n ) − µ i , 1 ≤ i ≤ m − ν i , 1 ≤ i ≤ n ǫ i +1 − δ i , 1 ≤ i ≤ rδ i +1 − ǫ i , 1 ≤ i ≤ r q ( n ) ǫ i +1 − ǫ i ,1 ≤ i ≤ n − δ i +1 − δ i ,1 ≤ i ≤ n − ( n , C ) Cohomology (Exceptional Cases) Lie Superalgebra CorrespondingEven Weights CorrespondingOdd Weights D (2 , , α ) − µ , − µ , − µ ( − ǫ, − ǫ, − ǫ ),( − ǫ, − ǫ, ǫ ),( ǫ, − ǫ, − ǫ ) G (3) − µ , − α , − β ( − ω + ω , − ǫ ),(2 ω − ω , − ǫ ),(0 , − ǫ ),( ω − ω , − ǫ ),( − ω + ω , − ǫ ),( − ω , − ǫ ) F (4) − µ , − ν , − ν , − ν ( ω − ω , − ǫ ),( ω − ω + ω , − ǫ ),( ω − ω , − ǫ ),( − ω + ω , − ǫ ),( − ω + ω − ω , − ǫ ),( − ω + ω , − ǫ ),( − ω , − ǫ ) OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 17 H ( n , C ) Cohomology Dimensions (Classical Cases) Lie Superalgebra Even Odd Total gl ( n | n ) 2( n − n − 1) 4( n − gl ( m | n ) m − n − 1, 2 n − m + 3 n − osp (2 m | n ) m + n − m + n − m + 2 n − osp (2 m + 1 | n ) m + n r m + n + 2 r q ( n ) n − n − n − ( n , C ) Cohomology Dimensions (Exceptional Cases) Lie Superalgebra Even Odd Total D (2 , , α ) 3 3 6 G (3) 3 6 9 F (4) 4 7 11 ( n , C ) Cohomology. Note that every weight in the H ( n , C ) corresponds to thesum of two roots of the Lie superalgebra. Below, we classify the weights by whether theyare the sum of two even roots, two odd roots, or of an even root and and odd root.H ( n , C ) Cohomology (Classical Cases) Lie Superalgebra Even+Even Weights Odd+Odd Weights Odd+Even Weights gl ( n | n ) α i + α j ,1 ≤ i < j ≤ n − α ′ i + α ′ j ,1 ≤ i < j ≤ n − α i + β j , α i + β ′ j ,1 ≤ i < i +1 < j ≤ n − α ′ i + β j , α ′ + β ′ j ,1 ≤ i < i +1 < j ≤ n − α i ± β j ,1 ≤ i < i + 1 < j ≤ r − α ′ i ± β j ,1 ≤ i < i + 1 < j ≤ r − ± β i ± β j ,1 ≤ i < j ≤ r gl ( m | n ) α i + α j ,1 ≤ i < i + 1 < j ≤ m − α ′ i + α ′ j ,1 ≤ i < i + 1 < j ≤ n − α i ± β j ,1 ≤ i < i + 1 < j ≤ r − α ′ i ± β j ,1 ≤ i < i + 1 < j ≤ r − ± β i ± β j ,1 ≤ i < j ≤ r osp (2 m | n ) µ i + µ j ,1 ≤ i < i + 1 < j ≤ m − ν ′ i + ν ′ j ,1 ≤ i < i + 1 < j ≤ n − µ i ± δ j +1 − ǫ j ,1 ≤ i < i + 1 < j ≤ r − ν i ± δ j +1 − ǫ j ,1 ≤ i < i + 1 < j ≤ r − ± ( δ i +1 − ǫ i ) ± ( δ j +1 − ǫ j ),1 ≤ i < j ≤ r osp (2 m + 1 | n ) µ i + µ j ,1 ≤ i < i + 1 < j ≤ m − ν ′ i + ν ′ j ,1 ≤ i < i + 1 < j ≤ n − µ i ± δ j +1 − ǫ j ,1 ≤ i < i + 1 < j ≤ r − ν i ± δ j +1 − ǫ j ,1 ≤ i < i + 1 < j ≤ r − ± ( δ i +1 − ǫ i ) ± ( δ j +1 − ǫ j ),1 ≤ i < j ≤ r q ( n ) α i + α j ,1 ≤ i < i + 1 < j ≤ n − α i ± β j ,1 ≤ i < i + 1 < j ≤ n − ± β i ± β j ,1 ≤ i < j ≤ n − OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 19 H ( n , C ) Cohomology (Exceptional Cases) Lie Superalgebra Even+Even Weights Odd+Odd Weights Odd+Even Weights D (2 , , α ) Sums of any distincttwo following weights: − µ , − µ , − µ Sums of one weightfrom left column withone from right column Sums of any two ofthe following weights:( − ǫ, − ǫ, − ǫ ),( − ǫ, − ǫ, ǫ ),( ǫ, − ǫ, − ǫ ) G (3) Sums of any distincttwo following weights: − µ , − α , − β Sums of one weightfrom left column withone from right column Sums of any two ofthe following weights:( − ω + ω , − ǫ ),(2 ω − ω , − ǫ ),(0 , − ǫ ),( ω − ω , − ǫ ),( − ω + ω , − ǫ ),( − ω , − ǫ ) F (4) Sums of any distincttwo following weights: − µ , − ν , − ν , − ν Sums of one weightfrom left column withone from right column Sums of any two ofthe following weights:( ω − ω , − ǫ ),( ω − ω + ω , − ǫ ),( ω − ω , − ǫ ),( − ω + ω , − ǫ ),( − ω + ω − ω , − ǫ ),( − ω + ω , − ǫ ),( − ω , − ǫ ) H ( n , C ) Cohomology Dimensions (Classical Cases) Lie Superalgebra Even+Even Odd+Odd Odd+Even Total gl ( n | n ) 4 n − n + 8 4 n − n + 8 2( r + r ) 8 n − n + 16 gl ( m | n ) (( n − + ( n − m − +( m − 1) ( n + m − r ) 2( r + r ) (( n − + ( n − 1) + ( m − +( m − 1) + ( n + m − r ) + 2( r + r ) osp (2 m | n ) (( n − + ( n − m − +( m − 1) ( n + m − r ) 2( r + r ) (( n − + ( n − 1) + ( m − +( m − 1) + ( n + m − r ) + 2( r + r ) osp (2 m + 1 | n ) (( n − + ( n − m − +( m − 1) ( n + m − r ) 2( r + r ) (( n − + ( n − 1) + ( m − +( m − 1) + ( n + m − r ) + 2( r + r ) q ( n ) (( n − +( n − n − (( n − + ( n − n − + ( n − ( n , C ) Cohomology Dimensions (Exceptional Cases) Lie Superalgebra Even+Even Odd+Odd Odd+Even Total D (2 , , α ) 3 9 6 18 G (3) 3 18 21 42 F (4) 6 28 28 62 OHOMOLOGY GROUPS FOR BBW PARABOLICS FOR LIE SUPERALGEBRAS 21 References [BKN10] Brian D. 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