On the semi-regular module and vertex operator algebras
aa r X i v : . [ m a t h . R T ] D ec ON THE SEMI-REGULAR MODULE AND VERTEX OPERATORALGEBRAS
MINXIAN ZHU Introduction
The aim of this paper is to give a proof of a conjecture stated in a previous paperby the author ([Z1]).Let g be a simple complex Lie algebra, ˆ g be the affine Lie algebra and h ∨ be thedual Coxeter number of g . Let A g ,k be the vertex algebroid associated to g and acomplex number k , according to [GMS1], we can construct a vertex algebra U A g ,k ,called the enveloping algebra of A g ,k . Set V = U A g ,k . It is shown in [AG] and [GMS2]that not only V is a ˆ g -representation of level k , it is also a ˆ g -representation of thedual level ¯ k = − h ∨ − k . Moreover the two copies of ˆ g -actions commute with eachother, i.e. V is a ˆ g k ⊕ ˆ g ¯ k -representation.When k / ∈ Q , the vertex operator algebra V decomposes into ⊕ λ ∈ P + V λ,k ⊗ V λ ∗ , ¯ k as a ˆ g k ⊕ ˆ g ¯ k -module (see [FS], [Z1]). Here P + is the set of dominant integral weightsof g , V λ,k is the Weyl module induced from V λ , the irreducible representation of g with highest weight λ , in level k , and V λ ∗ , ¯ k is induced from V ∗ λ in the dual level¯ k . In fact the vertex operators can be constructed using intertwining operators andKnizhnik-Zamolodchikov equations (see [Z1]).In the case where k ∈ Q , the ˆ g k ⊕ ˆ g ¯ k -module structure of V is much more com-plicated. In the present paper, we prove a result about the existence of canonicalfiltrations of V conjectured at the end of [Z1]. More precisely we will prove thefollowing. Theorem 1.
Let k ∈ Q , k > − h ∨ . The vertex operator algebra V admits an in-creasing (resp. a decreasing) filtration of ˆ g k ⊕ ˆ g ¯ k -submodules with factors isomorphicto V λ,k ⊗ V cλ, ¯ k ( resp. V cλ,k ⊗ V λ, ¯ k ) , λ ∈ P + , where V cλ, ¯ k is the contragredient module of V λ, ¯ k defined by the anti-involution: x ( n ) x ( − n ) , c c of ˆ g . We need two ingredients to prove the theorem: one is the semi-regular module; theother is the regular representation of the corresponding quantum group at a root ofunity.The standard semi-regular module was first introduced by A. Voronov in [V] to treatthe semi-infinite cohomology of infinite dimensional Lie algebras as a two-sided derivedfunctor of a functor that is neither left nor right exact. It was also studied rigorouslyby S. M. Arkhipov. He defined the associative algebra semi-infinite cohomology in the derived categories’ setting (see [A1]), and discovered a deep semi-infinite dualitywhich generalizes the classical bar duality of graded associative algebras (see [A2]).The semi-regular module S γ associated to a semi-infinite structure γ of ˆ g (see [V]) isthe semi-infinite analogue of the universal enveloping algebra U of ˆ g . In particular S γ is a U -bimodule, and the tensor product S γ ⊗ U V becomes a ˆ g − ¯ k ⊕ ˆ g ¯ k -representation.We will show in Section 3 that S γ ⊗ U V can be embedded into U ∗ as a bisubmodule.In fact it is spanned by the matrix coefficients of modules from the category O ¯ k + h ∨ ,defined and studied by Kazhdan and Lusztig in [KL1-4] for ¯ k < − h ∨ .In the series of papers [KL1-4], Kazhdan and Lusztig defined a structure of braidedcategory on O ¯ k + h ∨ , and constructed an equivalence between the tensor category O ¯ k + h ∨ and the category of finite dimensional integrable representations of the quan-tum group with parameter e iπ/ (¯ k + h ∨ ) (in the simply-laced case). It motivated theauthor to study the structure of regular representations of the quantum group atroots of unity (see [Z2]).One of the main results in [Z2] is that the quantum function algebra admits anincreasing filtration of (bi)submodules such that the subquotients are isomorphic tothe tensor products of the dual of Weyl modules W ∗− ω λ ⊗ W ∗ λ ( ω being the longestelement in the Weyl group). Translating this to the affine Lie algebra, it meansthat S γ ⊗ U V admits an increasing filtration of ˆ g − ¯ k ⊕ ˆ g ¯ k -submodules with factorsisomorphic to V ∗− ω λ, ¯ k ⊗ V cλ, ¯ k . Applying the functor H om U ( S γ , − ) (see [S, Theorem2.1]) to this filtration of S γ ⊗ U V , we obtain an increasing filtration of ˆ g k ⊕ ˆ g ¯ k -submodules of the vertex operator algebra V with factors described in Theorem 1. Thecorresponding decreasing filtration is obtained by using the non-degenerate bilinearform on V constructed in [Z1].The paper is organized as follows: In Section 2, we follow [S] to recall the definitionof semi-regular module S γ and the two functors defined with it. In Section 3, weembed S γ ⊗ U V into the dual of U as a (bi)submodule. In Section 4, we prove themain theorem about the filtrations of the vertex operator algebra V using results of[Z2]. 2. Semi-regular module S γ and equivalence of categories The semi-regular module of a graded Lie algebra with a semi-infinite structure wasfirst introduced by A. Voronov in [V], where it was called the “standard semijectivemodule”. It replaces the universal enveloping algebra (and its dual) in the semi-infinite theory, and like the universal enveloping algebra, it possesses left and right(semi)regular representations. Voronov used semijective complexes and resolutions todefine the semi-infinite cohomology of infinite dimensional Lie algebras as a two-sidedderived functor of a functor that is intermediate between the functors of invariantsand coinvariants.In [A2], S. M. Arkhipov generalized the classical bar duality of graded associa-tive algebras to give an alternative construction of the semi-infinite cohomology ofassociative algebras. Given a graded associative algebra A with a triangular decom-position, he introduced the endomorphism algebra A ♯ of a semi-regular A -module S A (see [A1]). In the case where A is the universal enveloping algebra of a graded Liealgebra, the algebra A ♯ is also a universal enveloping algebra of a Lie algebra whichdiffers from the previous one by a 1-dimensional central extension (determined by N THE SEMI-REGULAR MODULE AND VERTEX OPERATOR ALGEBRAS 3 the critical 2-cocycle). In the affine Lie algebra case, he proved that the categoryof all ˆ g -modules with a Weyl filtration in level k is contravariantly equivalent to theanalogous category in the dual level ¯ k . This equivalence was obtained directly in [S],where W. Soergel used it to find characters of tilting modules of affine Lie algebrasand quantum groups.Let us recall the definition of the semi-regular module from [S, Theorem 1.3].Let g be a simple complex Lie algebra. Let ˆ g = g ⊗ C [ t, t − ] ⊕ C c be the affine Liealgebra, where the commutator relations are given by[ x ( m ) , y ( n )] = [ x, y ]( m + n ) + mδ m + n, ( x, y ) c. Here x ( n ) = x ⊗ t n for x ∈ g , ( , ) is the normalized Killing form on g and c is thecenter. Define a Z -grading on ˆ g by deg x ( n ) = n and deg c = 0.Set ˆ g > = g ⊗ t C [ t ], ˆ g < = g ⊗ t − C [ t − ], ˆ g = g ⊕ C c and ˆ g ≥ = ˆ g > ⊕ ˆ g . Denote theenveloping algebras of ˆ g , ˆ g ≥ , ˆ g < by U, B, N . Obviously
U, B, N inherit Z -gradingsfrom the corresponding Lie algebras.Define a character γ : ˆ g = g ⊕ C c → C ; γ | g = 0 , γ ( c ) = 2 h ∨ , where h ∨ is the dual Coxeter number of g . It is easy to check that γ is a semi-infinitecharacter for ˆ g (see [S, Definition 1.1]).For any two Z -graded vector spaces M, M ′ , define the Z -graded vector space H om C ( M, M ′ ) with homogeneous components H om C ( M, M ′ ) j = { f ∈ Hom C ( M, M ′ ) | f ( M i ) ⊂ M ′ i + j } . The graded dual N ⊛ = ⊕ i N ∗ i of N is an N -bimodule via the prescriptions ( nf )( n ) = f ( n n ) and ( f n )( n ) = f ( nn ) for any n, n ∈ N , f ∈ N ⊛ . We have N ⊛ = H om C ( N, C ), if we equip C with the Z -grading C = C .Consider the following sequence of isomorphisms of ( Z -graded) vector spaces: H om B ( U, C γ ⊗ C B ) ˜ → H om C ( N, B ) ˜ ← N ⊛ ⊗ C B ˜ → N ⊛ ⊗ N U, here C γ is the one-dimensional representation of ˆ g ≥ defined by the character γ :ˆ g → C and the surjection ˆ g ≥ ։ ˆ g , and C γ ⊗ C B is the tensor product of thesetwo representations as a left ˆ g ≥ -module. In the leftmost term, U is considered a B -module via left multiplication of B on U , and H om B ( U, C γ ⊗ C B ) is made into a(left) U -module via the right multiplication of U onto itself. The first isomorphism isdefined as the restriction to N using the identification C γ ⊗ C B ˜ → B ; 1 ⊗ b b .As a vector space, the semi-regular module S γ = N ⊛ ⊗ C B. It is also a U -bimodule: the left (resp. right) U -action on S γ is defined via the firsttwo (resp. last) isomorphisms. The semi-infinite character γ ensures that these twoactions commute. Lemma 2.1. c · s = s · c + 2 h ∨ s for any s ∈ S γ , where c · s and s · c stand for the leftand right actions of c on s ∈ S γ .Proof. Easily verified. (cid:3)
MINXIAN ZHU
Proposition 2.2. [S, Theorem 1.3] The map ι : N ⊛ ֒ → S γ ; f f ⊗ is an inclusionof N -bimodules. The maps U ⊗ N N ⊛ → S γ ; u ⊗ f u · ι ( f ) and N ⊛ ⊗ N U → S γ ; f ⊗ u ι ( f ) · u are bijections. Remark 2.3.
The sequence of isomorphisms S γ = U ⊗ N N ⊛ ∼ = B ⊗ C N ⊛ ∼ = H om C ( N, B ) ˜ → H om B − right ( U, C − γ ⊗ B )induces a right U -map from S γ to H om B − right ( U, C − γ ⊗ B ). The right U -modulestructure of the latter is given by the left multiplication of U on the first argumentin H om.Let P + be the dominant integral weights of g and λ ∈ P + . Denote by V λ,k =Ind ˆ g ˆ g ≥ V λ the Weyl module induced from the finite dimensional irreducible represen-tation of g with highest weight λ in level k . Let V ∗ λ,k be the graded dual of V λ,k , onwhich ˆ g acts by Xf ( v ) = − f ( Xv ) for any X ∈ ˆ g , f ∈ V ∗ λ,k and v ∈ V λ,k .Let M (resp. K ) denote the category of all Z -graded representations of ˆ g , whichare over N isomorphic to finite direct sums of may-be grading shifted copies of N (resp. N ⊛ ). In fact M (resp. K ) consists precisely of those Z -graded ˆ g -modules,which admit a finite filtration with factors isomorphic to Weyl modules (resp. thedual of Weyl modules) (see [S, Remarks 2.4]). Proposition 2.4. [S, Theorem 2.1] The functor S γ ⊗ U − : M → K defines anequivalence of categories with inverse H om U ( S γ , − ) , such that short exact sequencescorrespond to short exact sequences.Proof. Note that S γ ⊗ U − ∼ = N ⊛ ⊗ N − and H om U ( S γ , − ) ∼ = H om N ( N ⊛ , − ) by Propo-sition 2.2. (cid:3) Proposition 2.5.
Let E be a Z -graded B -module bounded from below, the functor S γ ⊗ U − maps U ⊗ B E to H om B ( U, C γ ⊗ E ) .Proof. Similar to the construction of the semi-regular module S γ , consider the follow-ing sequence of isomorphisms of Z -graded vector spaces: S γ ⊗ U ( U ⊗ B E ) ∼ = N ⊛ ⊗ C E ˜ → H om C ( N, E ) ˜ ← H om B ( U, C γ ⊗ E ) . It is straightforward to check that, under these isomorphisms, the (left) U -modulestructure of S γ ⊗ U ( U ⊗ B E ) agrees with that of H om B ( U, C γ ⊗ E ). (cid:3) Remark 2.6.
In general for any Z -graded B -module E ′ , the inclusion S γ ⊗ U ( U ⊗ B E ′ ) ∼ = N ⊛ ⊗ C E ′ ֒ → H om B ( U, C γ ⊗ E ′ ) is a U -map. Proposition 2.7.
Let F be a Z -graded B -module bounded from above, then the func-tor H om U ( S γ , − ) maps H om B ( U, F ) to U ⊗ B ( C − γ ⊗ F ) .Proof. The isomorphism of vector spaces U ⊗ B ( C − γ ⊗ F ) ˜ → H om U ( S γ , H om B ( U, F )),induced from H om U ( S γ , H om B ( U, F )) ∼ = H om N ( N ⊛ , H om C ( N, F )) ∼ = H om C ( N ⊛ , F ) ∼ = N ⊗ C F ∼ = U ⊗ B ( C − γ ⊗ F ) , agrees with the composition of (left) U -maps U ⊗ B ( C − γ ⊗ F ) → H om U ( S γ , S γ ⊗ U ( U ⊗ B ( C − γ ⊗ F ))) → H om U ( S γ , H om B ( U, F )) , hence it is a U -isomorphism. (cid:3) N THE SEMI-REGULAR MODULE AND VERTEX OPERATOR ALGEBRAS 5
In particular S γ ⊗ U − transforms Weyl modules to the dual of Weyl modules, and H om U ( S γ , − ) transforms the latter to the former (both with a level shift). Corollary 2.8. S γ ⊗ U V λ,k ∼ = V ∗ λ ∗ , ¯ k , and H om U ( S γ , V ∗ λ, ¯ k ) ∼ = V λ ∗ ,k , here λ ∗ denotesthe highest weight of V ∗ λ .Proof. Note that U ⊗ B V λ = V λ,k and H om B ( U, C γ ⊗ V λ ) ∼ = V ∗ λ ∗ , ¯ k if c acts on V λ asscalar multiplication by k . (cid:3) Realization of S γ ⊗ U V inside U ∗ Fix a complex number k , and let V = U A g ,k be the vertex operator algebra asso-ciated to the vertex algebroid A g ,k (see [AG], [GMS1, 2], [Z1]). Note that in [Z1], weused V to denote the vertex operator algebra for generic values of k / ∈ Q , but here weadopt this notation with no restriction on k .The vertex operator algebra V admits two commuting actions of ˆ g in dual levels k, ¯ k = − h ∨ − k . It follows from Lemma 2.1 that S γ ⊗ U V , using the ˆ g k -modulestructure of V , becomes a ˆ g − ¯ k ⊕ ˆ g ¯ k -representation. Define U (ˆ g , k ) = U (ˆ g ) / ( c − k ) U (ˆ g ).Our goal is to construct an embedding of U -bimodulesΦ : S γ ⊗ U V ֒ → U (ˆ g , ¯ k ) ∗ . Let B = ⊕ i ≤ B i (denoted by “ B ” with opposite grading in [Z1]) be the commutativevertex subalgebra of V generated by A , where A is the commutative algebra of regularfunctions on an affine connected algebraic group G with Lie algebra g . Recall that B is closed under the actions of U (ˆ g ≥ , k ) and U (ˆ g ≥ , ¯ k ). As a ˆ g k -module, we have V ∼ = U ⊗ B B ∼ = N ⊗ C B (see e.g. [Z1, Proposition 3.16]). Since S γ ∼ = N ⊛ ⊗ N U as aright U -module, we have S γ ⊗ U V ∼ = N ⊛ ⊗ N V ∼ = N ⊛ ⊗ C B . Define a functional ǫ : B → C as follows: ǫ | B ≤ = 0 and its restriction to B = A isthe evaluation of functions at identity.Multiplication induces isomorphism of vector spaces: N ⊗ C B ∼ = U , hence any u ∈ U can be written as u = u < u ≥ with u < ∈ N and u ≥ ∈ B .Let ¯ : U → U ; u → u be the anti-involution of U determined by − Id : ˆ g → ˆ g .Define a map Φ : S ⊗ U V → U ∗ as follows: for any f ∈ N ⊛ , b ∈ B ,Φ( f ⊗ b )( u < u ≥ ) = f ( u < ) ǫ ( u r ≥ · b ) , here u r ≥ · b means the U (ˆ g ≥ , ¯ k )-action on B . In fact Φ( f ⊗ b ) ∈ U (ˆ g , ¯ k ) ∗ .The dual space U ∗ is a U -bimodule via the recipes ( u · g )( u ) = g ( u u ) and ( g · u )( u ) = g ( uu ) for any u, u ∈ U , g ∈ U ∗ . Theorem 3.1.
For any u ∈ U and f ⊗ b ∈ N ⊛ ⊗ C B ( ∼ = S γ ⊗ U V ), we have Φ( u l · ( f ⊗ b )) = (Φ( f ⊗ b )) · ¯ u, Φ( u r · ( f ⊗ b )) = u · (Φ( f ⊗ b )) , here u l · ( f ⊗ b ) , u r · ( f ⊗ b ) stand for the ˆ g − ¯ k - and ˆ g ¯ k -actions on S γ ⊗ U V respectively. MINXIAN ZHU
To prove the theorem, we need some preparations. First, letΘ : S ⊗ U V → H om B ( U, C γ ⊗ B )be the (left) U -map described in Remark 2.6 (taking E ′ = B ). Note that we regard V , B as non-positively graded, i.e. taking the opposite of the grading defined by theconformal weights of the vertex operator algebra V . Here B is regarded as a left B -module via the U (ˆ g ≥ , k )-action on B , and Θ is a U (ˆ g , − ¯ k )-map.Following [GMS2, Z1], let τ i be an orthonormal basis of g with respect to thenormalized Killing form ( , ). Let C ijk be the structure constants determined by[ τ i , τ j ] = C ijk τ k . We identify g with the tangent space to the identity of G . Let τ Li (resp. τ Ri ) be the left (resp. right) invariant vector fields valued τ i (resp. − τ i )at the identity, there exist regular functions a ij ∈ A such that τ Ri = a ij τ Lj and ǫ ( a ij ) = − δ ij . Lemma 3.2.
Let β : B → B be the automorphism which restricts to ˆ g ≥ as X γ ( X ) + X , then for any u ≥ ∈ B and b ∈ B , we have ǫ ( β ( u ≥ ) l · b ) = ǫ ( u ≥ r · b ) , here β ( u ≥ ) l · b , u ≥ r · b denote the U (ˆ g ≥ , k ) - and U (ˆ g ≥ , ¯ k ) -actions on B .Proof. By [Z1, Lemma 3.14 (10)], we have τ j ( n ) l · b = P i P p ≥ a ij ( − − p ) τ i ( n + p ) r · b for any n ≥ b ∈ B . Since ǫ | B ≥ = 0, we have ǫ ( τ j ( n ) l · b ) = P i ǫ ( a ij ( − τ i ( n ) r · b ) = P i ( − δ ij ) ǫ ( τ i ( n ) r · b ) = − ǫ ( τ j ( n ) r · b ). Since the U (ˆ g ≥ , k )- and U (ˆ g ≥ , ¯ k )-actions on B commute, for any u ≥ = τ j ( n ) · · · τ j q ( n q ), we have ǫ ( β ( u ≥ ) l · b ) = ǫ ( u l ≥ · b ) = ǫ ( − τ j ( n ) r · ( τ j ( n ) · · · ) l · b ) = ǫ (( τ j ( n ) · · · ) l · ( − τ j ( n )) r · b ) = ǫ (( τ j ( n ) · · · ) l · ( − τ j ( n )) r · ( − τ j ( n )) r · b ) = · · · = ǫ (( − τ j q ( n q )) r · · · · ( − τ j ( n )) r · b ) = ǫ ( u ≥ r · b ) . We also have ǫ ( β ( c ) l · b ) = ǫ (( c + 2 h ∨ ) l · b ) = ǫ (( k + 2 h ∨ ) b ) = ǫ ( − ¯ kb ) = ǫ (¯ c r · b ), hencethe lemma is proved. (cid:3) Proposition 3.3.
For any f ⊗ b ∈ N ⊛ ⊗ C B , we have Φ( f ⊗ b ) = ǫ Θ( f ⊗ b )¯ .Proof. By the definition of Θ, for any u = u < u ≥ ∈ U , we have Θ( f ⊗ b )(¯ u ) = Θ( f ⊗ b )( u ≥ u < ) = f ( u < ) β ( u ≥ ) l · b . Then it follows from Lemma 3.2 that ǫ Θ( f ⊗ b )(¯ u ) = f ( u < ) ǫ ( u r ≥ · b ) = Φ( f ⊗ b )( u ). (cid:3) Corollary 3.4.
For any u ∈ U and f ⊗ b ∈ N ⊛ ⊗ C B , we have Φ( u l · ( f ⊗ b )) =(Φ( f ⊗ b )) · ¯ u .Proof. Since Θ is a (left) U -map, by Proposition 3.3, we haveΦ( u l · ( f ⊗ b )) = ǫ Θ( u l · ( f ⊗ b ))¯= ǫ ( u · Θ( f ⊗ b ))¯= ǫ Θ( f ⊗ b ) r u ¯ = ǫ Θ( f ⊗ b )¯ l ¯ u = Φ( f ⊗ b ) l ¯ u = (Φ( f ⊗ b )) · ¯ u, where r u , l ¯ u : U → U denote the right and left multiplications by u and ¯ u respectively.Hence we proved one half of Theorem 3.1. (cid:3) Next we prove the other half of Theorem 3.1, which is to show thatΦ( u r · ( f ⊗ b )) = u · (Φ( f ⊗ b )) . If u = u ≥ ∈ B , then u ≥ r · ( f ⊗ b ) = f ⊗ u ≥ r · b . Hence Φ( f ⊗ ( u ≥ r · b ))( u ′ < u ′≥ ) = f ( u ′ < ) ǫ ( u ′≥ r · u ≥ r · b ) = Φ( f ⊗ b )( u ′ < u ′≥ u ≥ ) = u ≥ · (Φ( f ⊗ b ))( u ′ < u ′≥ ), whichmeans that Φ( u ≥ r · ( f ⊗ b )) = u ≥ · (Φ( f ⊗ b )). N THE SEMI-REGULAR MODULE AND VERTEX OPERATOR ALGEBRAS 7
To prove it holds for u = u < ∈ N as well, it suffices to show that Φ( τ i ( − r · ( f ⊗ b )) = τ i ( − · (Φ( f ⊗ b )) since ˆ g < is generated by ˆ g − .Recall that although B is only closed under the action of U (ˆ g ≥ , ¯ k ), it can beequipped with a ˆ g ¯ k -module structure ˜ ρ : U → End( B ) such that ˜ ρ ( u ≥ ) b = u ≥ r · b forany u ≥ ∈ B and b ∈ B (see [Z1, Lemma 3.29, Remark 3.30]). In addition, we have τ i ( − r · ( f ⊗ b ) = X j f · τ j ( − ⊗ ( a ij b ) + f ⊗ ˜ ρ ( τ i ( − b (see [Z1, Lemma 3.14 (9)]). Hence for any u < ∈ N , u ∈ U (ˆ g ) and u > ∈ U (ˆ g > ),we have Φ( τ i ( − r · ( f ⊗ b ))( u < u u > )= X j f ( τ j ( − u < ) ǫ ( u r · u > r · ( a ij b )) + f ( u < ) ǫ ( u r · u > r · ˜ ρ ( τ i ( − b )= X j f ( τ j ( − u < ) ǫ ( u r · a ij ( − u > r · b ) + f ( u < ) ǫ ( u r · [ u > , τ i ( − r · b ) . The last equality is because [ u > r , a ij ( − ] | B = 0 (see [Z1, Lemma 3.14 (4)]), and[ u > , τ i ( − ∈ B , ǫ | B ≥ = 0.On the other hand, we have τ i ( − · (Φ( f ⊗ b ))( u < u u > ) = Φ( f ⊗ b )( u < u u > τ i ( − f ⊗ b )( u < u [ u > , τ i ( − u < [ u , τ i ( − u > + u < τ i ( − u ≥ )= f ( u < ) ǫ ( u r · [ u > , τ i ( − r · b ) + X s f ( u < τ s ( − ǫ ( F i,s ( u ) r · u > r · b )+ f ( u < τ i ( − ǫ ( u ≥ r · b )where F i,s : U (ˆ g ) → U (ˆ g ) are maps such that [ u , τ i ( − P s τ s ( − F i,s ( u ) forany u ∈ U (ˆ g ).Since τ Rk ( a ij ) = C kip a pj , we have [ τ k (0) r , a ij ( − ] = C kip a pj ( − (see [Z1, Lemma 3.14(4)]). Compare it with the commutator [ τ k (0) , τ i ( − C kip τ p ( − u r , a ij ( − ] = P s a sj ( − F i,s ( u ) r . Hence we have X j f ( τ j ( − u < ) ǫ ( u r · a ij ( − u > r · b )= X j f ( τ j ( − u < ) ǫ ( X s a sj ( − F i,s ( u ) r · u > r · b ) + X j f ( τ j ( − u < ) ǫ ( a ij ( − u ≥ r · b )= X j f ( τ j ( − u < ) ǫ ( − F i,j ( u ) r · u > r · b ) + f ( τ i ( − u < ) ǫ ( − u ≥ r · b )= X j f ( u < τ j ( − ǫ ( F i,j ( u ) r · u > r · b ) + f ( u < τ i ( − ǫ ( u ≥ r · b ) , which proves that Φ( τ i ( − r · ( f ⊗ b )) = τ i ( − · (Φ( f ⊗ b )). The proof of Theorem3.1 is now complete. MINXIAN ZHU
Remark 3.5.
Following the notations in [Z1], let { e ω i } be right invariant 1-forms dualto { τ Ri } , and let f B be the linear span of elements of the form ∂ ( j ) e ω i · · · ∂ ( j n ) e ω i n ,then B = A ⊗ f B . There is a non-degenerate pairing between U (ˆ g > ) and f B , definedby ( u > , ˜ b ) = ǫ ( u > r · ˜ b ), via which f B can be identified with U (ˆ g > ) ⊛ , the graded dualof U (ˆ g > ). The regular functions A can be identified with the Hopf dual U ( g ) ∗ Hopf ,which is a subalgebra of U ( g ) ∗ defined by U ( g ) ∗ Hopf = { φ ∈ U ( g ) ∗ | Ker φ contains a two-sided ideal J ⊂ U ( g )of finite codimension } . It is not hard to see that ǫ ( u r · u r> · a ˜ b ) = ǫ ( u r · a ) ǫ ( u r> · ˜ b ) for any u ∈ U ( g ) , u > ∈ U (ˆ g > ) , a ∈ A and ˜ b ∈ f B . Hence S ⊗ U V ∼ = N ⊛ ⊗ B ∼ = N ⊛ ⊗ A ⊗ f B ∼ = U (ˆ g < ) ⊛ ⊗ U ( g ) ∗ Hopf ⊗ U (ˆ g > ) ⊛ ⊂ U (ˆ g , ¯ k ) ∗ , and Φ is injective.4. filtrations of the vertex operator algebra V Fix k ∈ Q , k > − h ∨ ; set κ = k + h ∨ >
0. Let O − κ be the full subcategory of thecategory of ˆ g ¯ k -modules defined by Kazhdan and Lusztig in [KL1-4]. They constructeda tensor structure on O − κ , and established an equivalence of tensor categories between O − κ and the category of finite-dimensional integrable representations of the quantumgroup with quantum parameter q = e − iπ/ κ (in the simply-laced case).Let V λ, ¯ k = Ind ˆ g ˆ g ≥ V λ be a Weyl module, denote the irreducible quotient of V λ, ¯ k by L λ, ¯ k . Definition 4.1. [KL1, Definition 2.15] O − κ is the full subcategory of ˆ g ¯ k -modules,which admits a finite composition series with factors of the form L λ, ¯ k for various λ ∈ P + .Let us recall some basic facts about O − κ . The Z > -grading on ˆ g > induces an N -grading on the enveloping algebra: U (ˆ g > ) = L n ≥ U (ˆ g > ) n . For any V ∈ O − κ , v ∈ V , there exists an n ∈ N such that U (ˆ g > ) n · v = 0.A module N over g ⊗ C [ t ] is said to be a nil-module if dim C N < ∞ and there existsa n ≥ U (ˆ g > ) n N = 0. Extend N to a ˆ g ≥ -module by defining the actionof c to be multiplication by ¯ k , and let N ¯ k = Ind ˆ g ˆ g ≥ N be the induced module. We saythat N ¯ k is a generalized Weyl module. Proposition 4.2. [KL1, Theorem 2.22] A ˆ g ¯ k -module V is in O − κ if and only if V is a quotient of a generalized Weyl module. Given V ∈ O − κ , let ¯ L : V → V be the Sugawara operator defined by ¯ L v = − κ P j> P i τ i ( − j ) τ i ( j ) v − κ P i τ i (0) τ i (0) v , where { τ i } is an orthonormal basis of g with respect to the normalized Killing form. Note that this operator is well definedand locally finite. Let V z be the generalized eigenspace of ¯ L with eigenvalue − z ∈ C ,we have V = L z ∈ C V z with dim V z < ∞ . In fact there exist z , · · · , z m ∈ Q such that { z | V z = 0 } ⊂ { z − N } ∪ · · · ∪ { z m − N } , and V becomes a Q -graded ˆ g ¯ k -representation,i.e. x ( n ) V z ⊂ V z + n for any x ( n ) ∈ ˆ g (see [KL1, Lemma 2.20, Proposition 2.21]). In N THE SEMI-REGULAR MODULE AND VERTEX OPERATOR ALGEBRAS 9 case V = V λ, ¯ k is a Weyl module, ¯ L acts on V λ, ¯ k semisimply. More specifically, wehave ¯ L | U (ˆ g < ) − n ⊗ V λ = − h λ,λ +2 ρ i κ + n , where ρ is the half sum of positive roots.Define the dual representation of V as follows: as a vector space V ∗ = L z ( V z ) ∗ ; theˆ g -action is given by Xf ( v ) = f ( − Xv ) for any X ∈ ˆ g , f ∈ V ∗ , v ∈ V . In particular V ∗ is a ˆ g − ¯ k -module and locally U (ˆ g < )-finite. In order for V ∗ to be a graded ˆ g -moduleas well, set ( V ∗ ) z = ( V − z ) ∗ , or equivalently set ( V ∗ ) z to be the generalized ( − z )-eigenspace of the operator L ′ = κ P j> P i τ i ( j ) τ i ( − j ) + κ P i τ i (0) τ i (0) which actson V ∗ .The contragredient dual V c is isomorphic to V ∗ as a vector space, but insteadof using − Id : ˆ g → ˆ g , we use the anti-involution x ( n )
7→ − x ( − n ) , c c to definethe ˆ g -action on V c . Unlike V ∗ , the contragredient module V c is a ˆ g ¯ k -representation,locally U (ˆ g > )-finite, and in fact belongs to O − κ .Given V ∈ O − κ , define a map φ V : V ∗ ⊗ V → U (ˆ g , ¯ k ) ∗ ; φ V ( f ⊗ v )( u ) = f ( u · v ) forany f ∈ V ∗ , v ∈ V, u ∈ U (ˆ g ). It is easy to see that φ V is a ˆ g − ¯ k ⊕ ˆ g ¯ k -map, where theˆ g − ¯ k ⊕ ˆ g ¯ k -module structure of U (ˆ g , ¯ k ) ∗ is given by ( X, · g = − g · X and (0 , X ) · g = X · g for any X ∈ ˆ g , g ∈ U (ˆ g , ¯ k ) ∗ . Denote the image of φ V by M ( V ), which is called thematrix coefficients of V .Recall the ˆ g − ¯ k ⊕ ˆ g ¯ k -map Φ : S γ ⊗ U V → U (ˆ g , ¯ k ) ∗ defined in Section 3. As pointedout in Remark 3.5, the map Φ is injective and its image, which we denote by M O − κ ,is isomorphic to U (ˆ g < ) ⊛ ⊗ U ( g ) ∗ Hopf ⊗ U (ˆ g > ) ⊛ . Here U (ˆ g < ) ⊛ = L n ≤ ( U (ˆ g < ) n ) ∗ , U (ˆ g > ) ⊛ = L n ≥ ( U (ˆ g < ) n ) ∗ are graded duals. Proposition 4.3. M O − κ consists of matrix coefficients of modules from the category O − κ , i.e. M O − κ = P V ∈O − κ M ( V ) .Proof. Let V = L z V z ∈ O − κ , v ∈ V and f ∈ V ∗ , for any u = u < u u > ∈ U = U (ˆ g ),we have φ V ( f ⊗ v )( u ) = h f, u < u u > · v i = h u < · f, u · u > · v i . Since V ∈ O − κ ,there exist n , n ∈ N such that U (ˆ g < ) − n · f = U (ˆ g > ) n · v = 0. Moreover each V z is finite-dimensional and semisimple as a g -module, therefore it is not hard to seethat φ V ( f ⊗ v ) ∈ U (ˆ g < ) ⊛ ⊗ U ( g ) ∗ Hopf ⊗ U (ˆ g > ) ⊛ , i.e. M ( V ) ⊂ M O − κ .On the other hand, let g ∈ M O − κ , there exists an n ∈ N such that U (ˆ g > ) n · g = 0.Since each U (ˆ g > ) n ′ is finite-dimensional and g acts on M O − κ locally finitely, theˆ g ≥ -submodule generated by g is a nil-module. Hence the ˆ g -submodule W = U (ˆ g ) · g generated by g is a quotient of a generalized Weyl module, hence it belongs to O − κ .Let δ be the functional on U ∗ defined by δ ( g ′ ) = g ′ (1), then δ ∈ W ∗ and g = φ W ( δ ⊗ g ) ⊂ M ( W ). (cid:3) Define two operators ¯ L , L ′ that act on M O − κ as follows: for any g ∈ M O − κ , set¯ L g = − κ P j> P i τ i ( − j ) · τ i ( j ) · g − κ P i τ i (0) · τ i (0) · g and L ′ g = κ P j> P i g · τ i ( − j ) · τ i ( j ) + κ P i g · τ i (0) · τ i (0). Let M O − κ z ′ ,z be the subspace consisting of all g ∈ M O − κ such that g is in the kernel of some power of ¯ L + z Id and the kernel ofsome power of L ′ + z ′ Id. Then M O − κ = L z,z ′ M O − κ z ′ ,z , and φ V (( V ∗ ) z ′ ⊗ V z ) ⊂ M O − κ z ′ ,z for any V ∈ O − κ . Moreover M O − κ z ′ ,z · x ( n ) ⊂ M O − κ z ′ + n,z and x ( n ) · M O − κ z ′ ,z ⊂ M O − κ z ′ ,z + n forany x ( n ) ∈ ˆ g . Define a Z -grading on M O − κ : for any g ∈ ( U (ˆ g < ) n ) ∗ , a ∈ U ( g ) ∗ Hopf , g ∈ ( U (ˆ g > ) n ′ ) ∗ , define deg g ⊗ a ⊗ g = − n − n ′ ; set M O − κ n = { g | deg g = n } . It isnot difficult to see that M O − κ n = L z + z ′ = n M O − κ z ′ ,z . Following [KL1, 3.3], define a partial order on P + as follows: λ ≤ µ if either λ = µ or h λ, λ + 2 ρ i < h µ, µ + 2 ρ i . Let O s − κ be the full subcategory of O − κ whose objectsare the V in O − κ such that the composition factors of V are of the form L λ, ¯ k forsome λ in the finite set F s = { λ ∈ P + |h λ, λ + 2 ρ i ≤ s } .We say that a module V ∈ O − κ is tilting if both V and V c have a Weyl filtration.For any λ ∈ P + , there exists an indecomposable tilting module T λ, ¯ k such that V λ, ¯ k ֒ → T λ, ¯ k , and any other Weyl modules V µ, ¯ k entering the Weyl filtration of T λ, ¯ k satisfy µ < λ (see [KL4, Proposition 27.2]). Lemma 4.4.
Let
V, V ′ ∈ O − κ . (1) If V has a (finite) Weyl filtration with factors isomorphic to V λ i , ¯ k for various λ i ∈ P + , then M ( V ) ⊂ P i M ( T λ i , ¯ k ) . (2) If V ′ has a (finite) filtration with factors isomorphic to V cµ i , ¯ k for various µ i ∈ P + , then M ( V ′ ) ⊂ P i M ( T cµ i , ¯ k ) .Proof. The proof is exactly the same as that of [Z2, Lemma 3.2]: we can construct aninjection
V ֒ → L i T λ i , ¯ k , and a surjection L i T cµ i , ¯ k ։ V ′ , since Ext O − κ ( V λ, ¯ k , V cµ, ¯ k ) = 0(see [KL4, Proposition 27.1]). (cid:3) Corollary 4.5. M O − κ consists of the matrix coefficients of tilting modules from O − κ ,i.e. M O − κ = P V ∈O − κ ,V tilting M ( V ) .Proof. For any V ∈ O − κ , choose s such that V ∈ O s − κ . By [KL1, Proposition 3.9],there exists a P , projective in O s − κ and having a (finite) Weyl filtration, such that V a quotient of P . Hence by Lemma 4.4 (1), we have M ( V ) ⊂ M ( P ) ⊂ P i M ( T λ i , ¯ k ) forsome λ i ∈ F s . (cid:3) Proposition 4.6.
Order the dominant weights in such a way P + = { ν , · · · , ν i , · · · } that ν i < ν j implies i < j . Set M O − κ ,i = P j ≤ i M ( T ν j , ¯ k ) , then M O − κ , ⊂ · · · ⊂ M O − κ ,i − ⊂ M O − κ ,i ⊂ · · · is an increasing filtration of ˆ g − ¯ k ⊕ ˆ g ¯ k -submodules of M O − κ with factors M O − κ ,i / M O − κ ,i − isomorphic to V ∗ ν i , ¯ k ⊗ V c − ω ν i , ¯ k , where ω is the longestelement in the Weyl group.Proof. The proof is the same as that of [Z2, Theorem 3.3], using Lemma 4.4. (cid:3)
Remark 4.7.
The category O − κ is a direct sum of subcategories correspondingto the orbits of the shifted action of affine Weyl group on the weight lattice (see[KL4, Lemma 27.7]). Hence we can decompose M O − κ , as a ˆ g − ¯ k ⊕ ˆ g ¯ k -module, intosummands corresponding to the orbits as well. Some summands are semisimple (see[KL4, Proposition 27.4], [Z2, Proposition 3.1]), but all have an increasing filtration ofthe above type. Proposition 4.8.
The vertex operator algebra V is isomorphic to H om U ( S γ , M O − κ ) as a ˆ g k ⊕ ˆ g ¯ k -module.Proof. Recall that M O − κ ∼ = S γ ⊗ U V = N ⊛ ⊗ B . Hence H om U ( S γ , M O − κ ) ∼ = H om N ( N ⊛ , N ⊛ ⊗ B ) ∼ = H om C ( N ⊛ , B ) ∼ = N ⊗ B ∼ = V , the second to last isomorphism is because B isnon-positively graded while N ⊛ is non-negatively graded. Moreover the induced iso-morphism V → H om U ( S γ , M O − κ ) ∼ = H om U ( S γ , S γ ⊗ U V ) is a ˆ g ⊕ ˆ g -map. (cid:3) Lemma 4.9.
For any b ∈ B , there exists an i such that N ⊛ ⊗ b ⊂ M O − κ ,i . N THE SEMI-REGULAR MODULE AND VERTEX OPERATOR ALGEBRAS 11
Proof.
For any f ∈ N ⊛ and u ≥ ∈ U (ˆ g ≥ ), we have u ≥ · ( f ⊗ b ) = f ⊗ ( u r ≥ · b ). Let N be the U (ˆ g ≥ , ¯ k )-submodule of B generated by b , then N is a nil-module and theˆ g ¯ k -submodule U (ˆ g , ¯ k ) · ( f ⊗ b ) generated by f ⊗ b is a quotient of the generalized Weylmodule N ¯ k . Hence f ⊗ b ∈ M ( N ¯ k ) for any f ∈ N ⊛ , hence there exists an i such that N ⊛ ⊗ b ⊂ M O − κ ,i . (cid:3) Theorem 4.10.
Set Σ i = H om U ( S γ , M O − κ ,i ) , then V = S i Σ i and Σ ⊂ · · · ⊂ Σ i − ⊂ Σ i ⊂ · · · is an increasing filtration of ˆ g k ⊕ ˆ g ¯ k -submodules of V with factors Σ i / Σ i − isomorphic to V − ω ν i ,k ⊗ V c − ω ν i , ¯ k .Proof. For any u < ⊗ b ∈ N ⊗ B ∼ = V , let N ′ ⊂ B be the U (ˆ g ≥ , k )-submodule generatedby b , then N ′ is finite-dimensional. For any s ∈ S γ we have p ( s ⊗ ( u < ⊗ b )) ∈ N ⊛ ⊗ N ′ , where p : S γ ⊗ V → S γ ⊗ U V is the canonical projection. By Lemma4.9, there exists an i such that p ( s ⊗ ( u < ⊗ b )) ∈ M O − κ ,i for any s ∈ S γ , hence u < ⊗ b ∈ H om U ( S γ , M O − κ ,i ) = Σ i . This proves that V = S i Σ i .Note that M O − κ ,i = L z ′ ,z M O − κ ,iz ′ ,z with dim M O − κ ,iz ′ ,z < ∞ . Fix z , the exact se-quence of ˆ g − ¯ k ⊕ ˆ g ¯ k -modules 0 → M O − κ ,i − → M O − κ ,i → V ∗ ν i , ¯ k ⊗ V c − ω ν i , ¯ k → g − ¯ k -modules 0 → L z ′ M O − κ ,i − z ′ ,z → L z ′ M O − κ ,iz ′ ,z → V ∗ ν i , ¯ k ⊗ ( V c − ω ν i , ¯ k ) z →
0. Since V ∗ ν i , ¯ k ⊗ ( V c − ω ν i , ¯ k ) z is isomorphic to a finite direct sumof grading-shifted copies of N ⊛ over N , by induction on i , so does L z ′ M O − κ ,iz ′ ,z foreach i , and the two exact sequences split over N , which means that there exists agrading preserving N -map V ∗ ν i , ¯ k ⊗ V c − ω ν i , ¯ k → M O − κ ,i so that its composition with theprojection is identity on the former. Therefore the sequence of ˆ g k ⊕ ˆ g ¯ k -modules 0 →H om U ( S γ , M O − κ ,i − ) → H om U ( S γ , M O − κ ,i ) → H om U ( S γ , V ∗ ν i , ¯ k ⊗ V c − ω ν i , ¯ k ) → H om U ( S γ , − ) ∼ = H om N ( N ⊛ , − ). Hence we have Σ i / Σ i − ∼ = H om U ( S γ , V ∗ ν i , ¯ k ⊗ V c − ω ν i , ¯ k ), which is isomorphic to V − ω ν i ,k ⊗ V c − ω ν i , ¯ k by Proposition 2.7, Lemma 2.8and the fact that the grading on V c − ω ν i , ¯ k is bounded from above. (cid:3) Remark 4.11.
The decomposition of M O − κ discussed in Remark 4.7 leads to adecomposition of V , as a ˆ g k ⊕ ˆ g ¯ k -module, into summands corresponding to the orbitsof the affine Weyl group on the weight lattice. Again some summands are semisimple,but each has an increasing filtration of the above type. Corollary 4.12.
The vertex operator algebra V admits a decreasing filtration of ˆ g k ⊕ ˆ g ¯ k -submodules V ⊃ Ξ ⊃ · · · ⊃ Ξ i − ⊃ Ξ i ⊃ · · · with factors Ξ i − / Ξ i isomorphic to V c − ω ν i ,k ⊗ V − ω ν i , ¯ k , and T i Ξ i = 0 .Proof. Let L , ¯ L : V → V be the Sugawara operators associated to the ˆ g k - and ˆ g ¯ k -actions on V respectively, i.e. L = κ P j> P i τ i ( − j ) τ i ( j ) + κ P i τ i (0) τ i (0), and¯ L = − κ P j> P i ¯ τ i ( − j ) ¯ τ i ( j ) − κ P i ¯ τ i (0) ¯ τ i (0). Now we regard the vertex operatoralgebra V = L n ≥ V n as non-negatively graded, then the sum L = L + ¯ L is thegradation operator, i.e. L | V n = n Id (see [Z1, Proposition 3.20, 3.24]).Let V z ,z be the subspace consisting of v ∈ V such that v is killed by some powerof L − z Id and some power of ¯ L − z Id. It follows from Theorem 4.10 that V = L z ,z V z ,z with dim V z ,z < ∞ .Recall the symmetric non-degenerate bilinear form h , i : V × V → C constructedin [Z1, Proposition 3.28]. It is shown to be compatible with the vertex operator algebra structure of V , in particular we have h x ( n ) · , ·i = h· , − x ( − n ) ·i and h ¯ y ( n ) · , ·i = h· , − ¯ y ( − n ) ·i for any x ( n ) ∈ ˆ g k , ¯ y ( n ) ∈ ˆ g ¯ k . It implies that h L · , ·i = h· , L ·i and h ¯ L · , ·i = h· , ¯ L ·i . Hence h , i| V z ,z × V z ′ ,z ′ = 0 except when z = z ′ and z = z ′ , inwhich case the pairing is non-degenerate.Let V c = L z ,z V ∗ z ,z be the contragredient dual of V , where the ˆ g k - and ˆ g ¯ k -actions on V c are both defined by the anti-involution x ( n )
7→ − x ( − n ); c c of ˆ g .Then we have V ∼ = V c because of the bilinear form h , i .Set Ξ i = { v ∈ V | h v, Σ i i = 0 } , then Ξ i is a ˆ g k ⊕ ˆ g ¯ k -submodule of V . Moreover wehave Ξ i ⊂ Ξ i − , and T i Ξ i = 0 because S i Σ i = V and h , i is non-degenerate. In factΞ i − / Ξ i ∼ = (Σ i / Σ i − ) c ∼ = ( V − ω ν i ,k ⊗ V c − ω ν i , ¯ k ) c ∼ = V c − ω ν i ,k ⊗ V − ω ν i , ¯ k . (cid:3) References [A1] S. Arkhipov, Semi-infinite cohomology of quantum groups, Comm. Math. Phys. 188 (1997),no. 2, 379-405.[A2] S.M.Arkhipov, Semi-infinite cohomology of associative algebras and bar duality, Int. Math.Res. Not. 1997, no. 17, 833-863.[AG] S. Arkhipov, D. Gaitsgory, Differential operators on the loop group via chiral algebras, Int.Math. Res. Not. 2002, no. 4, 165-210.[FS] I. Frenkel, K. Styrkas, Modified regular representations of affine and Virasoro algebras, VOAstructure and semi-infinite cohomology, Adv. Math. 206 (2006), 57-111.[GMS1] V. Gorbounov, F. Malikov, V. Schechtman, Gerbes of chiral differential operators, II, Vertexalgebroids, Invent. Math. 155 (2004), no. 3, 605-680.[GMS2] V. Gorbounov, F. Malikov, V. Schechtman, On chiral differential operators over homogeneousspaces, Int. J. Math. Math. Sci. 26 (2001), no. 2, 83-106.[KL1] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras I, J. Amer. Math.Soc. 6 (1993), 905-947.[KL2] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras II, J. Amer. Math.Soc. 6 (1993), 949-1011.[KL3] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras III, J. Amer. Math.Soc. 7 (1994), 335-381.[KL4] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras IV, J. Amer. Math.Soc. 7 (1994), 383-453.[S] W. Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent.Theory, 2 (1998), 432-448.[V] A. Voronov, Semi-infinite homological algebra, Invent. Math. 113 (1993), 103-146.[Z1] M. Zhu, Vertex operator algebras associated to modified regular representations of affine Liealgebras, math.QA/0611517.[Z2] M. Zhu, Regular representations of the quantum groups at roots of unity, preprint, 2007.
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