Cohomology of algebraic structures: from Lie algebra to vertex algebra cohomology Victor Kac MIT 1 / 37
In any cohomology theory the notion of a vector superspace and a superalgebra are indispensable. A vector superspace is a vector space V with a decomposition V = V ¯ 0 ⊕ V ¯ 1 . For v ∈ V α , α ∈ Z / 2 Z = { 0 , 1 } , one calls p ( v ) = α the parity of v. A superalgebra is a Z / 2 Z -graded algebra: V α V β ⊂ V α + β . Basic Example: End V = (End V ) ¯ 0 ⊕ (End V ) ¯ 1 , where (End V ) ¯ 0 (resp. (End V ) ¯ 1 ) consists of parity preserving (resp. reversing) endomorphisms. 2 / 37
The Lie bracket on a superalgebra is [ a, b ] = ab − ( − 1) p ( a ) p ( b ) ba. (1) If a superalgebra is associative, then the Lie bracket (1) on it defines a Lie superalgebra structure. Axioms of a Lie superalgebra [ a, b ] = − ( − 1) p ( a ) p ( b ) [ b, a ] (skew-commutivity) [ a, [ b, c ]] = [[ a, b ] , c ] + ( − 1) p ( a ) p ( b ) [ b, [ a, c ]] (Jacobi identity) Recall Koszul rule: sign changes iff odd passes odd Basic example: End V with bracket (1) is the general linear Lie superalgebra Another example: if V carries a structure of a superalgebra, then Der V = { D ∈ End V | D ( ab ) = D ( a ) b + ( − 1) p ( D ) p ( a ) aD ( b ) } is the Lie superalgebra of derivations of the superalgebra V. 3 / 37
An important special case ( W for Witt): W ( V ) = Der S ( V ) , the Lie superalgebra of derivations of the (commutative) superalgebra of polynomial functions on V ∗ (= the Lie superalgebra of polynomial vector fields on V ∗ ). The Lie superalgebra W ( V ) carries a natural Z -grading, coming from S ( V ) : � W j ( V ) , W ( V ) = j ≥− 1 where W − 1 ( V ) = V, W 0 ( V ) = End V, W j ( V ) = Hom( S j +1 ( V ) , V ) . 4 / 37
Explicit formula for the bracket on W ( V ) : [ X, Y ] = X � Y − ( − 1) p ( X ) p ( Y ) Y � X, (2) where X ∈ W n ( V ) , Y ∈ W m ( V ) , and ( X � Y )( v 0 ⊗ . . . ⊗ v m + n ) = � ǫ v ( i 0 , . . . , i m + n ) X ( Y ( v i 0 ⊗ . . . ⊗ v i m ) ⊗ v i m +1 ⊗ . . . ⊗ v i m + n ) . i 0 <...<i m i m +1 <...<i m + n (3) The summation is over the shuffles in S m + n +1 , and ǫ v = ( − 1) N , where N = # of interchanges of indices of odd v i ’s. 5 / 37
Since W 1 ( V ) = Hom( S 2 V, V ) , even elements of the vector superspace W 1 ( V ) correspond bijectively to commutative superalgebra structures (i.e. ab = ( − 1) p ( a ) p ( b ) ba ) on V. But if we want skew-commutative, we need to reverse the parity of V : consider the vector superspace Π V with the even part V ¯ 1 and the odd part V ¯ 0 , and consider the Lie superalgebra W (Π V ) . The odd elements X ∈ W 1 (Π V ) are in bijective correspondence with skew-commutative superalgebra structures on V : [ a, b ] = ( − 1) p ( a ) X ( a ⊗ b ) , a, b ∈ V (4) 6 / 37
A remarkable fact is that (4) satisfies the Jacobi identity (hence defines a Lie superalgebra on V ) if and only if [ X, X ] = 0 . (5) Since, by Jacobi identity, ad[ X, X ] = 2(ad X ) 2 , we see that, given a Lie superalgebra structure on V, with the bracket, defined by X ∈ W 1 (Π V ) , we obtain a cohomology complex � ( C · = C j , ad X ) , where C j = W j − 1 (Π V ) . j ≥ 0 This is the Chevalley-Eilenberg Lie (super)algebra cohomology complex with coefficients in the adjoint representation. 7 / 37
More generally, given a module M over the Lie superalgebra V, one considers, instead of V, the Lie superalgebra V ⋉ M with M an abelian ideal, and by a simple reduction procedure construct the Chevalley-Eilenberg complex of V with coefficients in M. 8 / 37
Basic idea. Given an algebraic structure A on a vector (super)space V, construct a Z -graded Lie superalgebra � W j W A (Π V ) = A (Π V, ) j ≥− 1 such that an odd element X ∈ W 1 A (Π V ) , satisfying [ X, X ] = 0 , defines the algebraic structure A on V. Then we obtain a cohomology complex for the structure A : � C j A , ad X ) , where C j A = W j − 1 ( C · A = (Π V ) . A j ≥ 0 For a V -module M one uses a reduction procedure as in the Lie (super)algebra case. 9 / 37
The above construction of the Lie superalgebra W ( V ) is easy to reformulate in terms of the linear operad P = Hom( V ) , where P ( j ) = Hom( V ⊗ j , V ) , j ≥ 0 , so that W j P = P ( j + 1) S j +1 . A straightforward generalization produces a cohomology theory for any linear symmetric (super)operad P. The art is how to construct the linear operad P, which produces a cohomology theory of a given algebraic structure A . 10 / 37
The first example beyond the Lie (super)algebra is the Lie conformal (super)algebra (LCA). Recall that an LCA structure on a vector superspace V with an even endomorphism ∂ is defined by the λ -bracket ( λ indeterminate, p ( λ ) = ¯ 0) [ . λ . ] : V ⊗ 2 → V [ λ ] , a ⊗ b �→ [ a λ b ] , satisfying the following axioms: (sesquilinearity) [ ∂a λ b ] = − λ [ a λ b ] , [ a λ ∂b ] = ( ∂ + λ )[ a λ b ] , [ b λ a ] = − ( − 1) p ( a ) p ( b ) [ a − λ − ∂ b ] . (skew-commutivity) [ a λ [ b µ c ]] = [[ a λ b ] λ + µ c ] + ( − 1) p ( a ) p ( b ) [ b µ [ a λ c ]] . (Jacobi identity) Explanation: writing [ a µ b ] = � n ≥ 0 µ n c n , [ a − λ − ∂ b ] is � n ≥ 0 ( − λ − ∂ ) n c n . 11 / 37
In order to construct the corresponding Z -graded Lie superalgebra W LCA ( V ) , let V n = V [ λ 1 , . . . , λ n ] / ( ∂ + λ 1 + . . . + λ n ) V [ λ 1 , . . . , λ n ] . (6) LCA ( V ) consists of linear maps Y λ 0 ,...,λ n : V ⊗ ( n +1) → V n +1 , Then W n which are invariant w.r. to the simultaneous permutation of factors in V ⊗ ( n +1) and of the λ i ’s, and which satisfy the sesquilinearity properties (0 ≤ j ≤ n ) Y λ 0 ,...,λ n ( v 0 ⊗ . . . ⊗ ∂v j ⊗ . . . ⊗ v n ) = − λ j Y λ 0 ,...,λ n ( v 0 ⊗ . . . ⊗ v n ) . The box product on W LCA ( V ) = � W j LCA ( V ) is defined by a j ≥− 1 formula, similar to (3), but “decorated” by λ i ’s: 12 / 37
( X � Y ) λ 0 ,...,λ m + n ( v 0 , . . . , v m + n ) � = ǫ v ( i 0 , . . . , i m + n ) X λ i 0 + ... + λ in ,λ in +1 ,...,λ im + n (7) ( Y λ i 0 ,...,λ im ( v i 0 ⊗ . . . ⊗ v i m ) ⊗ v i m +1 ⊗ . . . ⊗ v i m + n ) . Here, as in (3), the summation is over the shuffles in S m + n +1 , and ǫ v is the same. Then formula (2) defines a structure of Z -graded Lie superalgebra on W LCA ( V ) . 13 / 37
As in the Lie (super)algebra case, we consider the Z -graded Lie superalgebra W LCA (Π V ) . Then W − 1 LCA (Π V ) = Π( V/∂V ) , W 0 LCA (Π V ) = End ∂ V, and odd elements X ∈ W 1 LCA (Π V ) correspond bijectively to the linear maps [ . λ . ] : V ⊗ 2 → V [ λ ] , satisfying sesquilinearity and skew-commutativity axioms of LCA. Explicitly, this bijection is given by (cf. with (4)): [ a λ b ] = ( − 1) p ( a ) X λ, − λ − ∂ ( a ⊗ b ) . (8) As in the Lie superalgebra case, [ X, X ] = 0 iff (8) satisfies the Jacobi identity. 14 / 37
We obtain an LCA cohomology complex for the adjoint module: � C j LCA , ad X ) , where C j LCA = W j − 1 ( C LCA = LCA (Π V ) . j ≥ 0 This complex, for any V -module M, was constructed by Bakalov-VK-Voronov in [BKV99], where also its basic properties were studied and cohomology of main examples was computed. Basic examples of LCA. All simple finitely generated over C [ ∂ ] Lie conformal algebras were classified by [D’Andrea-VK 98]: ◮ Virasoro LCA : Vir = C [ ∂ ] L, [ L λ L ] = ( ∂ + 2 λ ) L, ◮ Affine LCA : Cur g = C [ ∂ ] g , [ a λ b ] = [ a, b ] , where g is a simple Lie algebra. 15 / 37
Theorem 1 [BKV99] (a) dim H n LCA (Vir , F ) = 1 for n = 0 , 2 , 3; = 0 otherwise. (b) H n LCA (Cur g , F ) ≡ H n ( g , F ) ⊕ H n +1 ( g , F ) . Theorem 2 [BKV99] (a) H LCA (Vir , Vir) = 0 (b) H n LCA (Cur g , Cur g ) = H n − 1 ( g , F ) . Corollary Since H i LCA ( V, V ) = Casimirs for i = 0, =derivations modulo inner derivations for i = 1, and = 1st order deformations for i = 2, we obtain that all Casimirs of Vir and Cur g are trivial, all derivations are inner, and all their 1-st order deformation are the obvious ones. Theorem 3 [BKV99] (cf. [Ritt50] for n = 2) For the Vir-module M ∆ = F [ ∂ ] v, L λ v = ( ∂ + ∆ λ ) v, ∆ ∈ F , on has: � 2 if n = r + 1 , and r ∈ Z + dim H n LCA (Vir , M 1 − 3 r 2 ± r ) = 1 if n = r, r + 2 , and r ∈ Z + , 2 and H n LCA (Vir , M ∆ ) = 0 otherwise 16 / 37
The main tool for computing the LCA cohomology is the basic complex � ( � � C n C LCA = LCA , d X ) , n ≥ 0 obtained from from the definition of C LCA , by replacing in the definition of W n LCA ( V ) the space V n , defined by (6), by V [ λ 1 , . . . , λ n ] . One gets thereby an LCA � W LCA ( V ) with canonically defined representation of W LCA ( V ) on it, giving, in particular the action d X of X, and an exact sequence of maps: π 0 → ∂ � W LCA ( V ) → � W LCA ( V ) → W LCA ( V ) . In good situations, e.g. if V is free as an F [ ∂ ]-module, the map π is surjective. Hence this short exact sequence induces a cohomology long exact sequence. 17 / 37
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