MODIFIED REGULAR REPRESENTATIONS

OF AFFINE AND VIRASORO ALGEBRAS, VOA STRUCTURE AND SEMI-INFINITE COHOMOLOGY.

IGOR B. FRENKEL AND KONSTANTIN STYRKAS

Abstract. We find a counterpart of the classical fact that the regular representation ℜ(G) of a simple complex group G is spanned by the matrix elements of all irreducible representations of G. Namely, the algebra of functions on the big cell G_{0} ⊂ G of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category O of representations of the Lie algebra 𝔤 of G, and has the structure of a 𝔤 ⊕ 𝔤-module.

The standard regular representation ℜ(Ĝ) of the affine group Ĝ has two commuting actions of the Lie algebra 𝔤̂ with total central charge 0, and carries the structure of a conformal field theory. The modified versions ℜ′(Ĝ) and ℜ′(Ĝ_{0}), originating from the loop version of the Bruhat decomposition, have two commuting 𝔤̂-actions with central charges shifted by the dual Coxeter number, and acquire vertex operator algebra structures derived from their Fock space realizations, given explicitly for the case G = SL(2, ℂ).

The quantum Drinfeld-Sokolov reduction transforms the representations of the affine Lie algebras into their W-algebra counterparts, and can be used to produce analogues of the modified regular representations. When 𝔤 = 𝔰𝔩(2, ℂ) the corresponding W-algebra is the Virasoro algebra. We describe the Virasoro analogues of the modified regular representations, which are vertex operator algebras with the total central charge equal to 26.

The special values of the total central charges in the affine and Virasoro cases lead to the non-trivial semi-infinite cohomology with coefficients in the modified regular representations. The inherited vertex algebra structure on this cohomology degenerates into a supercommutative associative superalgebra. We describe these superalgebras in the case when the central charge is generic, and identify the 0th cohomology with the Grothendieck ring of finitedimensional G-modules. We conjecture that for the integral values of the central charge the 0th semi-infinite cohomology coincides with the Verlinde algebra and its counterpart associated with the big projective modules.

0. Introduction.

The study of the regular representation of a simple complex Lie group G is at the foundation of representation theory of G. Realized as the space of regular functions on G, the regular representation ℜ(G) carries two compatible structures of a G-bimodule and of a commutative associative algebra. An algebro-geometric version of the Peter-Weyl theorem establishes the decomposition of ℜ(G) into a direct sum of subspaces, spanned by matrix elements of all irreducible finite-dimensional representations V_{λ} of G, indexed by integral dominant highest weights λ ∈ P^{+}. In other words, we have an isomorphism of G-bimodules

ℜ(G) = ⨁_{λ∈P^{+}} V_{λ} ⊗ V_{λ}^{∗}.  (0.1)

where V_{λ}^{∗} is the dual representation of G. The multiplication in ℜ(G) can be described in representation-theoretic terms as a pairing of intertwining operators for the left and right 𝔤actions with appropriate structural coefficients. Thus the algebra structure on ℜ(G) encodes the information about the tensor category of finite-dimensional 𝔤-modules.

The representations of G can also be viewed as modules over the simple complex Lie algebra 𝔤 associated with G. In the case of the Lie algebra 𝔤 it is natural to consider a larger collection of modules - namely, the Bernstein-Gelfand-Gelfand category O. Infinitedimensional 𝔤-modules from the category O are not integrable, and therefore their matrix elements cannot be regarded as functions on G. However, one can interpret them as functions on the open dense subset G^{o} ⊂ G, given by the Gauss decomposition

G^{o} = N_{−} · T · N_{+},  (0.2)

where N_{±} is the upper and lower triangular unipotent subgroup of G, and T is the diagonal maximal abelian subgroup. The space ℜ(G^{o}) of regular functions on G^{o} does not have the structure of a representation of G. Nevertheless, the left and right infinitesimal actions of the Lie algebra 𝔤 on this space are well-defined, and can be expressed in terms of explicit differential operators in the parameters of the Gauss decomposition (0.2). The enlarged regular representation ℜ(G^{o}) decomposes into the direct sum of bimodules spanned by the matrix coefficients of all ”big” projective 𝔤-modules P_{λ}, indexed by strictly anti-dominant highest weights λ ∈ −P^{++} = −(P^{+} + ρ), where ρ is the half-sum of all positive roots of 𝔤. Thus we obtain an isomorphism of 𝔤-bimodules

ℜ(G^{o}) ≅ ⨁_{λ∈−P^{++}} (P_{λ} ⊗ P_{λ}^{∗})/I_{λ},  (0.3)

where P_{λ}^{∗} is the module dual to P_{λ}, and I_{λ} is the sub-bimodule of the matrix coefficients which vanish identically on the universal enveloping algebra U(𝔤).

It is important to note that the dual modules P_{λ}^{∗} do not belong to the category O, but to its “mirror image”, in which all highest weight modules are replaces by lowest weight modules. In order to stay in the category O we replace the open subset G^{o} coming from the Gauss decomposition by the maximal cell in the Bruhat decomposition

G_{0} = N_{+} · w_{0} · T · N_{+},  (0.4)

where w_{0} is the longest element of the Weyl group W. Then we obtain a version of the isomorphism (0.3),

ℜ(G_{0}) ≅ ⨁_{λ∈−P^{++}} (P_{λ} ⊗ P_{λ}^{⋆})/I_{λ},  (0.5)

where the ’twisted’ duals P_{λ}^{⋆} differs from P_{λ}^{∗} by the automorphism ω of 𝔤, which is induced by w_{0} and interchanges the positive and negative roots.

The theorems of Peter-Weyl type and the Gauss decomposition can be extended to the central extension of the loop group Ĝ associated to G, and to the corresponding affine Lie algebra 𝔤̂ and its universal enveloping algebra U(𝔤̂). In this infinite-dimensional case the space ℜ(Ĝ) of regular functions on Ĝ is decomposed into the direct sum of subspaces ℜ_{k}(Ĝ), corresponding to the value k ∈ ℤ of the central charge. Using the version of the Gauss decomposition (0.2) known as the Birkhoff decomposition, one can show (see [PS]) that for any k ∈ ℤ there is an isomorphism

ℜ_{k}(Ĝ) ≅ ⨁_{λ∈P_{+}^{k}} V̂_{λ,k} ⊗ V̂_{λ,k}^{∗},  (0.6)

where λ runs over the truncated alcove P_{k}^{+} ⊂ P^{+}, depending on k, and V̂_{λ,k} are the corresponding irreducible modules. Similarly, one can obtain decompositions of ℜ_{k}(Ĝ^{o}) analogous to (0.3), where Ĝ^{o} is the maximal cell in the Birkhoff decomposition. Viewing the decomposition (0.6) in terms of the Lie algebra 𝔤̂ allows to extend it for all values of k, with P_{k}^{+} = P^{+} for k ∉ ℚ.

To transform the dual module V̂_{λ,k}^{∗} into a module from the category O for 𝔤̂, one might apply again an automorphism of 𝔤̂ which interchanges the positive and negative affine roots. However, it no longer belongs to the affine Weyl group, and the Bruhat decomposition for Ĝ does not have a maximal cell. To overcome this obstacle we consider instead an intermediate between the Birkhoff and the affine Bruhat decompositions - the loop version of the finitedimensional Bruhat decomposition, and the corresponding big cell

Ĝ_{0} = LN_{+} · w_{0} · L̂T · LN_{+},  (0.7)

where LN_{±} denote the loop groups with values in 𝔫_{±}, and L̂T is the central extension of the loop group with values in T. The decomposition (0.7) is especially useful for explicit realizations of the left and right regular 𝔤̂-actions in terms of differential operators. However, we are still in “semi-infinite” distance from the category O, and need to further apply a well-known procedure of “changing the vacuum”, which has originated from the free field realizations of the Wakimoto modules and the irreducible representations V̂_{λ,k} (see [FeFr1, BF]). As a result of this procedure we obtain a modified affine version of the extended regular representation (0.5),

ℜ′_{k}(Ĝ_{0}) ≅ ⨁_{λ∈−P^{++}} (P̂_{λ,k−h^{∨}} ⊗ P̂_{λ,−k−h^{∨}}^{⋆})/Î_{λ,k},  (0.8)

where P̂_{λ,k−h^{∨}} and P̂_{λ,−k−h^{∨}}^{⋆} are the projective 𝔤̂-modules and their ’twisted’ duals, Î_{λ,k} are appropriate sub-bimodules, and we assume k ∉ ℚ. The levels are shifted by the dual Coxeter number h^{∨}, so that the diagonal 𝔤̂-action has the level −2h^{∨}.

Like the Wakimoto modules, the bimodule ℜ′_{k}(Ĝ_{0}) is realized as a certain Fock space, with two commuting 𝔤̂-actions described explicitly (cf. [FeP]). This realization is similar to the standard realization of the Wakimoto modules, but the actions of 𝔤̂ contain a crucial new ingredient - the vertex operators, directly related to the screening operators used to construct intertwining operators for the affine Lie algebra. We also establish that for k ∉ ℚ the structure of the socle filtration of the non-semisimple bimodule ℜ′_{k}(Ĝ_{0}) is the same as in the finite-dimensional case. In particular, ℜ′_{k}(Ĝ_{0}) contains the distinguished sub-bimodule

ℜ′_{k}(Ĝ) ≅ ⨁_{λ∈P^{+}} V̂_{λ,k−h^{∨}} ⊗ V̂_{λ,−k−h^{∨}}^{⋆}.  (0.9)

The shifts of the central charge by the dual Coxeter number no longer allow the interpretation of the bimodules in (0.8) and (0.9) as spaces of matrix elements of 𝔤̂-modules. Nevertheless, the structures of these bimodules are completely analogous to those of bimodules ℜ(G_{0}) and ℜ(G)!

The vacuum module V̂_{0,k} of the affine Lie algebra 𝔤̂ carries an extremely rich additional structure of a vertex operator algebra (VOA); other 𝔤̂-modules V̂_{λ,k} become its representations (see [FZ]). We show in this paper that the bimodules ℜ′_{k}(Ĝ) and ℜ′_{k}(Ĝ_{0}) also admit a vertex operator structure, compatible with 𝔤̂-actions. In the proof we use the explicit Fock space realization of these bimodules. An alternative approach is given by the geometric construction of the algebras of chiral differential operators (cdo) over G, studied in [GMS] and [AG]. ^{1} Thus besides the vacuum modules there is a class of vertex operator algebras associated to affine Lie algebras with fixed central charges. On the other hand, the original regular representation ℜ_{k}(Ĝ) does not seem to have a VOA structure. Instead it has the structure of a two-dimensional conformal field theory, which is an object of a different nature despite having local properties similar to those of a VOA. The bimodule ℜ_{k}(Ĝ_{0}) has a structure of a generalized (non-semisimple!) conformal field theory.

It is well-known that the representation theory of 𝔤̂ is closely related to the representation theory of the corresponding W-algebra via a functorial cohomological construction, called the quantum Drinfeld-Sokolov reduction [FeFr2]. For the generic central charge, this functor gives an equivalence of categories, and all VOA constructions and structural results in the affine case are automatically transformed into their W-algebra counterparts. In particular, we get W-algebra analogues of the Peter-Weyl theorem and its non-semisimple extension.

In this paper we consider in full detail the simplest case of 𝔤 = 𝔰𝔩(2, ℂ), when the corresponding W-algebra is the infinite-dimensional Virasoro algebra. We give explicit realizations of its modified regular representations, analogous to (0.8) and (0.9), and equip them with compatible VOA structures. We describe their bimodule structures, which are completely parallel to those in the finite-dimensional and affine cases. Using the approach presented in this paper, it is easy to generalize the bimodule and vertex algebra constructions to the case of affine Lie algebras of higher rank and the associated W-algebras. However, the characterization of ℜ(G_{0}) as the algebra of matrix elements of all modules from the category O, and the description of the corresponding bimodule structures, require more general methods and will be given in the forthcoming paper [S2].

A remarkable feature of all the modified bimodules is that the central charge for the diagonal action is always equal to the special values that appear in the semi-infinite cohomology theory [Fe, FGZ] - namely, −2h^{∨} for the affine Lie algebras and 26 for Virasoro. Moreover, thanks to a general result of [LZ1], the corresponding semi-infinite cohomology with coefficients in the modified regular representations inherits a VOA structure; in our case it degenerates into a supercommutative associative superalgebra. For generic central charge we establish isomorphisms between the semi-infinite cohomology spaces for the affine and Virasoro algebras and their finite-dimensional counterparts. Moreover, their superalgebra structures retain the information about the fusion tensor categories of affine and Virasoro modules, built into the corresponding modified regular representations. In particular, we show that the 0th semi-infinite cohomologies of the affine and Virasoro algebras are isomorphic to the Grothendieck ring of finite-dimensional representations of G. We conjecture that for integral k they lead to the Verlinde algebra and its projective counterpart.

This paper is organized as follows. In Section 1 we consider the Bruhat decomposition and the Peter-Weyl theorems in the finite-dimensional case with G = SL(2, ℂ). We give a Fock space realization of the algebra ℜ(G_{0}), and obtain explicit formulas for the 𝔤-actions and decomposition theorems, which will later be used as prototypes of the infinite-dimensional case. In the last subsection we compute the Lie algebra cohomology with coefficients in ℜ(G) and ℜ(G_{0}). In Section 2 we study the affine case, and use the loop version of the finite-dimensional Bruhat decomposition to obtain the modified Peter-Weyl theorems for the spaces ℜ′_{k}(Ĝ) and ℜ′_{k}(Ĝ_{0}). The Fock space realization of these spaces equips them with VOA structures compatible with the regular 𝔤̂-actions. The semi-infinite cohomology of 𝔤̂ with coefficients in the modified regular representations ℜ′_{k}(Ĝ) and ℜ′_{k}(Ĝ_{0}) for generic central charge is shown to be isomorphic to its finite-dimensional counterpart. In Section 3 we construct the analogues of the modified regular representations of the Virasoro algebra using the quantum Drinfeld-Sokolov reduction. We compute the corresponding semi-infinite cohomology groups using methods developed in string theory, and prove that they are isomorphic to their affine counterparts. Finally, in Section 4 we describe another class of vertex operator algebras obtained by the pairing of 𝔰̂𝔩(2, ℂ) and Virasoro modules. We also discuss generalizations of our results to Lie algebras of other types, and to the integral values of the central charge. We conclude with conjectures on relations of the semi-infinite cohomology of ℜ′_{k}(Ĝ) and ℜ′_{k}(Ĝ_{0}) for k ∈ ℤ_{>0} with the Verlinde algebra, its projective counterpart and twisted equivariant K-theory.

1We were informed by F. Malikov and S. Arkhipov that their cdo algebras could be shown to have the same bimodule structure as ℜ′_{k}(Ĝ), establishing the equivalence of our realization and their constructions.

We thank G. Zuckerman for sharing his expertise on semi-infinite cohomology. We are grateful to F. Malikov, who told us about the papers [FeP, GMS, AG] when this work was nearing completion. We also thank S. Arkhipov and F. Malikov for valuable comments. I.B.F. is supported in part by NSF grant DMS-0070551.

1. Regular representation of 𝔰𝔩(2, ℂ) on the big cell.

1.1. Regular representations of 𝔰𝔩(2, ℂ). Let G = SL(2, ℂ). We define the left and right regular actions of G on the space ℜ(G) of regular functions on G by

(π_{l}(g)ψ)(h) = ψ(g^{−1}h),  (π_{r}(g)ψ)(h) = ψ(h g),  g, h ∈ G.  (1.1)

The multiplication in ℜ(G) intertwines both left and right regular actions.

Let T, N_{+} denote the diagonal and unipotent upper-triangular subgroups of G. The group W = Norm(T)/T is called the Weyl group. The Bruhat decomposition G = N_{+} · W · T · N_{+} implies that every g ∈ G can be factored as g = n·w·t·n′ for some n, n′ ∈ N_{+}, t ∈ T, w ∈ W.

We denote by G_{0} the big cell of the Bruhat decomposition, corresponding to the longest Weyl group element w_{0}. Explicitly, G_{0} is the dense open subset of G, consisting of g ∈ G,

g = (\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}ζ & 0 \\ 0 & ζ^{−1} \end{matrix}) (\begin{matrix}1 & y \\ 0 & 1 \end{matrix})  (1.2)

for some x, y ∈ ℂ and ζ ∈ ℂ^{×}. The variables x, y, ζ can be viewed as coordinates on G_{0}, and thus the algebra ℜ(G_{0}) of regular functions on G_{0} is identified with the space ℂ[x, y, ζ^{±1}].

Let 𝔤 = 𝔰𝔩(2, ℂ) be the Lie algebra of G, with the standard basis

e = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ,  h = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ,  f = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) ,

satisfying the commutation relations [h, e] = 2e, [h, f] = −2f, [e, f] = h. The nilpotent subalgebras 𝔫_{±} and the Cartan subalgebra 𝔥 of 𝔤 are defined by 𝔫_{+} = ℂe, 𝔥 = ℂh, 𝔫_{−} = ℂf. The element w_{0} ∈ W determines a Lie algebra involution ω of 𝔤, such that ω(𝔫_{±}) = 𝔫_{∓} and ω(𝔥) = 𝔥, defined by

ω(e) = −f,  ω(h) = −h,  ω(f) = −e.  (1.3)

The infinitesimal regular actions of 𝔤 on ℜ(G), corresponding to (1.1), are given by

(π_{l}(x)ψ)(g) = \frac{d}{dt}ψ(e^{−t x}g)|_{t=0},  (π_{r}(x)ψ)(g) = \frac{d}{dt}ψ(g e^{t x})|_{t=0},  x ∈ 𝔤, g ∈ G. (1.4)

These formulas also define left and right infinitesimal actions of 𝔤 on the space ℜ(G_{0}). (These actions cannot be lifted to the group G, because G_{0} is not invariant under left and right shifts). Elementary calculations yield the following explicit description of the regular 𝔤-actions (cf. [FeP]).

Proposition 1.1. The regular 𝔤-actions on ℜ(G_{0}) are given by

\begin{aligned}π_{l}(e) & = −∂_{x}, \\ π_{l}(h) & = ζ∂_{ζ} − 2x∂_{x}, \\ π_{l}(f) & = −xζ∂_{ζ} + x^{2}∂_{x} + ζ^{−2}∂_{y}. \end{aligned}  (1.5)

\begin{aligned}π_{r}(e) & = ∂_{y}, \\ π_{r}(h) & = ζ∂_{ζ} − 2y∂_{y}, \\ π_{r}(f) & = yζ∂_{ζ} − y^{2}∂_{y} − ζ^{−2}∂_{x}. \end{aligned}  (1.6)

1.2. Bosonic realizations. We now reformulate the constructions of the previous section in terms of Fock modules for certain Heisenberg algebras. These realizations admit generalizations to the affine and Virasoro cases, where the geometric approach to the regular representations becomes more subtle.

The operators β = x, γ = −∂_{x} acting on polynomials in y give a representation of the Heisenberg algebra with generators β, γ and relation [β, γ] = 1. The polynomial space ℂ[y] is then identified with its irreducible representation F (β, γ), generated by a vector 1 satisfying γ 1 = 0.

The operators β̄ = −y, γ̄ = ∂_{y} generate a second Heisenberg algebra, acting irreducibly in the space F (β̄, γ̄) ≅ ℂ[x]. Here and everywhere else in this paper the ’bar’ notation is used to denote the second copies of algebras and their generators; it does not denote the complex conjugation.

We identify 𝔥^{∗} ≅ ℂ so that P ≅ ℤ. Whenever possible, we use the more invariant notation in order to avoid possible numeric coincidences. The operators 1_{λ} = ζ^{λ} and a = ζ∂_{ζ} gives rise to the semi-direct product ℂ[a] ⋉ ℂ[P], with ℂ[a] acting on ℂ[P] by derivations: a1_{λ} = λ1_{λ}.

Thus, we get a realization of the algebra ℜ(G_{0}) of regular functions on G_{0}, with the 𝔤 ⊕ 𝔤-action described by the abstract versions of the formulas (1.5), (1.6).

Theorem 1.2. The space 𝔽 = F (β, γ) ⊗ F (β̄, γ̄) ⊗ ℂ[P] gives a realization of the algebra ℜ(G_{0}). In particular,

(1) The space 𝔽 has a 𝔤 ⊕ 𝔤-module structure, given by

\begin{aligned}e & = γ, \\ h & = 2 βγ + a, \\ f & = −β^{2}γ − β a + γ̄ 1_{−2}, \end{aligned}  (1.7)

\begin{aligned}ē & = γ̄, \\ h̄ & = 2 β̄γ̄ + a, \\ f̄ & = −β̄^{2}γ̄ − β̄ a + γ 1_{−2}. \end{aligned}  (1.8)

(2) The space 𝔽 has a compatible commutative algebra structure (i.e. the multiplication in 𝔽 intertwines the 𝔤 ⊕ 𝔤-action).

By specializing the action (1.7) to the subspace ker γ̄ ⊂ 𝔽, we get the following well-known realizations of 𝔤-action in the spaces F_{λ} = F (β, γ) ⊗ ℂ1_{λ}:

\begin{aligned}e & = γ, \\ h & = 2βγ + λ, \\ f & = −β^{2}γ − λ β. \end{aligned}  (1.9)

Remark 1. Simultaneous rescaling of the extra terms in (1.7),(1.8), involving the shift 1_{−2}, by any multiple ϵ would preserve all the 𝔤 ⊕ 𝔤 commutation relations. For ϵ = 0 such 𝔤 ⊕ 𝔤-action degenerates into the product of two standard 𝔤-actions (1.9). However, the multiplication in this naïve bimodule loses much of its rich structure, and no longer encodes the information about the fusion rules in the tensor category of finite-dimensional 𝔤-modules.

1.3. 𝔤 ⊕ 𝔤-module structure of the modified regular representation. In this subsection we describe the socle filtration of the 𝔤 ⊕ 𝔤-module 𝔽.

For any λ ∈ 𝔥^{∗}, we denote by V_{λ} the irreducible 𝔤-module, generated by a highest weight vector v_{λ} satisfying e v_{λ} = 0 and h v_{λ} = λ v_{λ}.

Recall that a 𝔤-module V is said to have a weight space decomposition, if

V = ⨁_{μ∈𝔥^{∗}} V [μ], V [μ] = {v ∈ V | h v = μ v} .

The restricted dual space V′ = ⨁_{μ∈𝔥^{∗}} V [μ]′ can be equipped with a 𝔤-action, defined by

⟨g v′, v⟩ = −⟨v′, ω(g) v⟩, g ∈ 𝔤, v ∈ V, v′ ∈ V′,

where ω is as in (1.3). We denote the resulting dual module V^{⋆}.

We have an involution λ ↦ λ^{⋆} of 𝔥^{∗}, determined by the condition (V_{λ})^{⋆} ≅ V_{λ^{⋆}}. This involution can also be defined by λ^{⋆} = −w_{0}(λ), where w_{0} is the longest Weyl group element.

For 𝔤 = 𝔰𝔩(2, ℂ), we have λ^{⋆} = λ. However, we keep the notation λ^{⋆}, to indicate how our constructions generalize to Lie algebras of higher rank, where the involution is nontrivial.

Theorem 1.3. There exists a filtration

0 ⊂ 𝔽^{(0)} ⊂ 𝔽^{(1)} ⊂ 𝔽^{(2)} = 𝔽  (1.10)

of 𝔤 ⊕ 𝔤-submodules of 𝔽, such that

𝔽^{(2)}/𝔽^{(1)} ≅ ⨁_{λ∈P^{+}} V_{−λ−2} ⊗ V_{−λ−2}^{⋆},  (1.11)
𝔽^{(1)}/𝔽^{(0)} ≅ ⨁_{λ∈P^{+}} (V_{λ} ⊗ V_{−λ−2}^{⋆} ⊕ V_{−λ−2} ⊗ V_{λ}^{⋆}) ,  (1.12)
𝔽^{(0)} ≅ ⨁_{λ∈P} V_{λ} ⊗ V_{λ}^{⋆}.  (1.13)

Proof. We introduce a filtration of 𝔤 ⊕ 𝔤-submodules of 𝔽

⋯ ⊂ 𝔽_{≤−2} ⊂ 𝔽_{≤−1} ⊂ 𝔽_{≤0} ⊂ 𝔽_{≤1} ⊂ 𝔽_{≤2} ⊂ …,  (1.14)

satisfying ⋂_{λ∈P} 𝔽_{≤λ} = 0 and ⋃_{λ∈P} 𝔽_{≤λ} = 𝔽, where

𝔽_{≤λ} = F (β, γ) ⊗ F (β̄, γ̄) ⊗ ⨁_{μ≤λ} ℂ1_{μ}, λ ∈ P.

It is clear that 𝔽_{≤λ}/𝔽_{<λ} ≅ F_{λ} ⊗ F_{λ^{⋆}}; moreover, for λ < 0 we have F_{λ} ≅ V_{λ}, and for λ ≥ 0 there is a short exact sequence 0 → V_{λ} → F_{λ} → V_{−λ−2} → 0. The linking principle for 𝔤-modules implies that the successive quotients 𝔽_{≤λ}/𝔽_{<λ} and 𝔽_{≤μ}/𝔽_{<μ} of this filtration may be non-trivially linked only if μ = −λ − 2.

Thus we see that the 𝔤 ⊕ 𝔤-module 𝔽 splits into the direct sum of blocks

𝔽 = 𝔽(−1) ⊕ ⨁_{λ∈P^{+}} 𝔽(λ),  (1.15)

where 𝔽(−1) ≅ V_{−1} ⊗ V_{−1}^{⋆}, and 𝔽(λ) ≅ (V_{−λ−2} ⊗ V_{−λ−2}^{⋆}) + (F_{λ} ⊗ F_{λ^{⋆}}) for λ ∈ P^{+}; another way to obtain the decomposition (1.15) is by using the Casimir operator.

It remains to describe the structure of 𝔽(λ) for each λ ∈ P^{+}. By construction, 𝔽(λ) can be included in a short exact sequence 0 → V_{−λ−2} ⊗ V_{−λ−2}^{⋆} → 𝔽(λ) → F_{λ} ⊗ F_{λ^{⋆}} → 0. We conclude that there exists a filtration 0 ⊂ 𝔽(λ)^{(0)} ⊂ 𝔽(λ)^{(1)} ⊂ 𝔽(λ)^{(2)} = 𝔽(λ), such that

𝔽(λ)^{(2)}/𝔽(λ)^{(1)} ≅ V_{−λ−2} ⊗ V_{−λ−2}^{⋆},
𝔽(λ)^{(1)}/𝔽(λ)^{(0)} ≅ (V_{λ} ⊗ V_{−λ−2}^{⋆}) ⊕ (V_{−λ−2} ⊗ V_{λ}^{⋆}) ,
𝔽(λ)^{(0)} ≅ (V_{−λ−2} ⊗ V_{−λ−2}^{⋆}) + (V_{λ} ⊗ V_{λ}^{⋆}) .

In fact, the linking principle implies that the sum in 𝔽(λ)^{(0)} is direct:

𝔽(λ)^{(0)} ≅ (V_{−λ−2} ⊗ V_{−λ−2}^{⋆}) ⊕ (V_{λ} ⊗ V_{λ}^{⋆}) .

Finally, we construct the filtration (1.10) by setting

𝔽^{(0)} = 𝔽(−1) ⊕ ⨁_{λ∈P^{+}} 𝔽(λ)^{(0)}, 𝔽^{(1)} = 𝔽(−1) ⊕ ⨁_{λ∈P^{+}} 𝔽(λ)^{(1)},

which obviously satisfies the required conditions (1.11), (1.12), (1.13). □

Remark 2. The analogous filtration for a Lie algebra 𝔤 of higher rank has blocks 𝔽(λ), whose structure is governed by the endomorphism ring End_{𝔤}(P (λ)) of the corresponding projective generator P (λ) = ⨁_{μ∈W ·λ} P_{μ}.

The natural inclusion of algebras ℜ(G) ⊂ ℜ(G_{0}) can be seen in the Fock space realizations.

Corollary 1.4. There exists a subspace F ⊂ 𝔽 satisfying the following properties.

(1) F is a subalgebra of 𝔽, and is generated by the elements from the submodule V_{1} ⊗ V_{1}^{⋆}, corresponding to the matrix elements of the canonical representation of G.

(2) F is a 𝔤 ⊕ 𝔤-submodule of 𝔽, and is generated by the vectors {1_{λ}}_{λ∈P^{+}}. We have

F = ⨁_{λ∈P^{+}} F(λ) ≅ ⨁_{λ∈P^{+}} V_{λ} ⊗ V_{λ}^{⋆}.  (1.16)

(3) The space F is a realization of the algebra ℜ(G).

In the polynomial realization, the generators of F from V_{1} ⊗ V_{1}^{⋆} are identified with functions

ψ_{11} = ζ,  ψ_{12} = xζ,  ψ_{21} = yζ,  ψ_{22} = xyζ + ζ^{−1},

which satisfy the relation ψ_{11}ψ_{22} − ψ_{12}ψ_{21} = 1. This establishes a very direct connection with the space of regular functions on the group G = SL(2, ℂ).

1.4. The generalized Peter-Weyl theorem. In this section we interpret the space ℜ(G_{0}) of regular functions on G_{0} and its Fock space realization 𝔽 as the algebra of matrix elements of all modules from the category O.

Recall that the Bernstein-Gelfand-Gelfand category O consists of all finitely generated, locally 𝔫_{+}-nilpotent 𝔤-modules. In particular, V_{λ} ∈ O for any λ. If V ∈ O, then V^{⋆} ∈ O.

For any 𝔤-module V we define 𝕄(V) to be the subspace of U(𝔤)′, spanned by functionals

ϕ_{v,v′}(x) = ⟨v′, x v⟩,  v ∈ V, v′ ∈ V′, x ∈ U(𝔤),  (1.17)

where ⟨·, ·⟩ stands for the natural pairing between V and V′. The functionals (1.17) are called matrix elements of the representation V.

Proposition 1.5. Introduce a 𝔤 ⊕ 𝔤-module structure on the restricted dual U(𝔤)′ by

(π_{l}(g)ϕ)(x) = ϕ(xg),  (π_{r}(g)ϕ)(x) = −ϕ(ω(g)x)  (1.18)

for any ϕ ∈ U(𝔤)′, g ∈ 𝔤, x ∈ U(𝔤). Then (1) For any 𝔤-module V, the space 𝕄(V) is a 𝔤 ⊕ 𝔤-submodule of U(𝔤)′.

(2) For any φ ∈ U(𝔤)′, there exists a 𝔤-module V, such that φ ∈ 𝕄(V). Moreover, if φ is 𝔫_{+} ⊕ 𝔫_{+}-nilpotent, then V can be chosen from the category O.

Proof. To show that 𝕄(V) is invariant under the left action of 𝔤, we compute

(π_{l}(g)ϕ_{v,v′})(x) = ϕ_{v,v′}(xg) = ⟨v′, xg v⟩ = ϕ_{gv,v′}(x),

for any x ∈ U(𝔤), g ∈ 𝔤, v ∈ V, v′ ∈ V′. This shows that yϕ_{v,v′} ∈ 𝕄(V). The invariance under the right action follows from the computation

(π_{r}(g)ϕ_{v,v′})(x) = −ϕ_{v,v′}(ω(g)x) = −⟨v′, ω(g)x v⟩ = ⟨yv′, x v⟩ = ϕ_{v,yv′}(x).

For the second part, assume φ ∈ U(𝔤)′. Denote by V the subspace of U(𝔤)′, generated from φ by the left action of 𝔤. Let φ′ be the restriction to V of the unit 1 ∈ U(𝔤) = U(𝔤)′′. Equivalently, φ′ is the linear functional on V, determined by ⟨φ′, ψ⟩ = ψ(1), for any ψ ∈ V ⊂ U(𝔤)′. We claim that φ = ϕ_{φ,φ′} ∈ 𝕄(V). Indeed, for any x ∈ U(𝔤) we have

ϕ_{φ,φ′}(x) = ⟨φ′, xφ⟩ = (xφ)(1) = φ(x).

Finally, if φ is left-𝔫_{+}-nilpotent, then V is locally 𝔫_{+}-nilpotent. Since V is generated by a single element φ, it belongs to category O. The right-𝔫_{+}-nilpotency condition guarantees that φ′ belongs to the restricted dual space V′. □

The elements of the universal enveloping algebra U(𝔤) may be regarded as the differential operators, acting on ℜ(G). This gives an interpretation of the regular functions on G (or even on G_{0}) as linear functionals on U(𝔤), and thus to identifications of the spaces ℜ(G) and ℜ(G_{0}) with certain subspaces of U(𝔤)′. In the explicit realizations F and 𝔽 this correspondence is constructed using the algebraic analogue of the ”co-unit” element of the Hopf algebra ℜ(G)

- the linear functional ⟨·⟩ : 𝔽 → ℂ, defined by

⟨β^{m}β̄^{n}1_{λ}⟩ = δ_{m,0}δ_{n,0}.  (1.19)

Proposition 1.6. The linear map ϑ : 𝔽 → U(𝔤)′, defined by v ↦ ϑ_{v},

ϑ_{v}(x) = ⟨π_{l}(x)v⟩,  v ∈ 𝔽, x ∈ U(𝔤).  (1.20)

is an injective 𝔤 ⊕ 𝔤-homomorphism.

Proof. In terms of the polynomial realization, ⟨·⟩ corresponds to evaluating a function ψ(x, y, ζ) ∈ ℜ(G_{0}) at the element w_{0}: ⟨ψ⟩ = ψ(0, 0, 1). This implies that for any v ∈ 𝔽

⟨ev⟩ = −⟨f̄v⟩,  ⟨hv⟩ = −⟨h̄v⟩,  ⟨fv⟩ = −⟨ēv⟩.  (1.21)

Therefore, for any g ∈ 𝔤 and x ∈ U(𝔤) we have

ϑ_{gv}(x) = ⟨x gv⟩ = ϑ_{v}(xg) = (π_{l}(g)ϑ_{v})(x),
ϑ_{ḡv}(x) = ⟨x ḡv⟩ = ⟨ḡ xv⟩ = −⟨ω(g)xv⟩ = −ϑ_{v}(ω(g)x) = (π_{r}(g)ϑ_{v})(x).

We conclude that the map ϑ is a 𝔤 ⊕ 𝔤-homomorphism. To prove that it is injective, we need to show that for any nonzero v ∈ 𝔽 there exists an element x ∈ 𝔤 ⊕ 𝔤 such that ⟨xv⟩ ≠ 0.

Since 𝔽 is locally 𝔫_{+}-nilpotent, we can pick k ≥ 0 such that e^{k}v ≠ 0, but e^{k+1}v = 0. Replacing v by e^{k}v, we see that it suffices consider the case of v ≠ 0 such that ev = 0. Similarly, we may assume that ēv = 0.

A vector v satisfying ev = 0 = ēv must have the form v = ∑_{λ∈P} c_{λ}1_{λ} with only finitely many c_{λ} ≠ 0. Using the formula for the Vandermonde determinant and the fact that

⟨h^{m}v⟩ = ∑_{λ∈P} c_{λ}λ^{m}, m ≥ 0,

we conclude that ⟨h^{k}v⟩ = 0 for all k ≥ 0 if and only if all c_{λ} vanish. Thus, θ_{v} = 0 is equivalent to v = 0, which means that ϑ is an injection. □

The following statement is an algebraic version of the classical Peter-Weyl theorem.

Theorem 1.7. The space ℜ(G) of regular functions on G is spanned by the matrix elements of finite-dimensional irreducible 𝔤-modules,

ℜ(G) ≅ ⨁_{λ∈P^{+}} 𝕄(V_{λ}).

The decomposition of ℜ(G) as a 𝔤 ⊕ 𝔤-module is given by

ℜ(G) ≅ ⨁_{λ∈P^{+}} V_{λ} ⊗ V_{λ}^{⋆}.

Remark 3. The subspace of U(𝔤)′, corresponding to ℜ(G), is invariantly characterized as the restricted Hopf dual U(𝔤)′_{Hopf} ⊂ U(𝔤)′, defined by

U(𝔤)′_{Hopf} = {ϕ ∈ U(𝔤)′|∃ two-sided ideal J ⊂ U(𝔤) such that ϕ(J) = 0 and codim J < ∞}.

The extended space ℜ(G_{0}) corresponds to a larger subalgebra of U(𝔤)′, spanned by the matrix elements of all modules in the category O.

Recall that the category O has enough projectives; we denote by P_{λ} the indecomposable projective cover of the irreducible module V_{λ}. It is known that every indecomposable module in the category O with integral weights is isomorphic to a subfactor of the projective module, corresponding to some anti-dominant integral weight λ. In particular, this means that it suffices to consider the matrix elements of the big projective modules {P_{λ}}_{λ<0}.

The following result can be regarded as a non-semisimple generalization of the Peter-Weyl theorem.

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 11

Theorem 1.8. The space ℜ(G_{0}) of regular functions on G_{0} is spanned by the matrix elements of all big projective modules in the category O,

ℜ(G_{0}) ≅ ⨁_{λ∈P^{+}} 𝕄(P_{λ}).

As a 𝔤 ⊕ 𝔤-module, ℜ(G_{0}) is given by

ℜ(G_{0}) ≅ ⨁_{λ∈−P^{++}} (P_{λ} ⊗ P_{λ}^{⋆})/I_{λ}

where I_{λ}’s are the 𝔤 ⊕ 𝔤-submodules of P_{λ}⊗P_{λ}^{⋆}, corresponding to identically vanishing matrix elements.

Proof. We use the realization of ℜ(G_{0}) in the Fock space 𝔽. The inclusion (1.20) provides the identification of 𝔽 with a subspace of U(𝔤)′. Since 𝔽 is locally 𝔫_{+} ⊕𝔫_{+}-nilpotent, Proposition 1.5 implies that for any v ∈ 𝔽 there exists a 𝔤-module W ∈ O such that ϑ_{v} ∈ 𝕄(W).

Let W = W_{1} ⊕ W_{2} ⊕ ⋯ ⊕ W_{m} be the decomposition of W into a direct sum of indecomposable submodules. Each indecomposable component W_{i}, i = 1, …, m, is a subfactor of some big projective module P_{λ_{i}}. Then 𝕄(W_{i}) ⊂ 𝕄(P_{λ_{i}}), and therefore we have

𝕄(W) = 𝕄(W_{1}) + 𝕄(W_{2}) + ⋯ + 𝕄(W_{m}) ⊂ ⨁_{λ∈−P^{++}} 𝕄(P_{λ}),

which shows that ϑ(𝔽) ⊂ ⨁_{λ∈−P^{++}} 𝕄(P_{λ}). To prove that in fact ϑ(𝔽) = ⨁_{λ∈−P^{++}} 𝕄(P_{λ}), we compare the characters of the two spaces, and show that they have the same size.

For any λ ∈ P^{+} the 𝔤 ⊕ 𝔤-module 𝕄(P_{−λ−2}) is isomorphic to the quotient of the product P_{−λ−2} ⊗ P_{−λ−2}^{⋆} by the kernel of the map

Θ_{λ} : P_{−λ−2} ⊗ P_{−λ−2}^{⋆} → U(𝔤)′,  Θ_{λ}(v ⊗ v′) = ϕ_{v,v′}.  (1.22)

Obviously, I_{λ} = ker Θ_{λ} is a 𝔤 ⊕ 𝔤-submodule of P_{−λ−2} ⊗P_{−λ−2}^{⋆}; we describe it more explicitly. It is known that the module P_{−λ−2} has a filtration 0 ⊂ P^{(0)} ⊂ P^{(1)} ⊂ P_{−λ−2} such that

P^{(0)} ≅ V_{−λ−2},  P^{(1)}/P^{(0)} ≅ V_{λ},  P_{−λ−2}/P^{(1)} ≅ V_{−λ−2},

and the dual filtration of the module P_{−λ−2}^{⋆} is given by

0 ⊂ Ann(P^{(1)}) ⊂ Ann(P^{(0)}) ⊂ P_{−λ−2}^{⋆}.

They determine a filtration of the tensor product

0 ⊂ P^{(0)} ⊗ Ann(P^{(1)}) ⊂ P^{(0)} ⊗ Ann(P^{(0)}) + P^{(1)} ⊗ Ann(P^{(1)}) ⊂
⊂ P^{(0)} ⊗ P_{−λ−2}^{⋆} + P^{(1)} ⊗ Ann(P^{(0)}) + P_{−λ−2} ⊗ Ann(P^{(1)}) ⊂
⊂ P^{(1)} ⊗ P_{−λ−2}^{⋆} + P_{−λ−2} ⊗ Ann(P^{(0)}) ⊂ P_{−λ−2} ⊗ P_{−λ−2}^{⋆}.

If v ∈ P^{(0)} and v′ ∈ Ann(P^{(0)}), then ϕ_{v,v′} is the zero functional, since for any x ∈ U(𝔤) we have x v ∈ P^{(0)} and ϕ_{v,v′}(x) = ⟨v′, x v⟩ = 0. Hence the submodule P^{(0)} ⊗ Ann(P^{(0)}) lies in the kernel of the map Θ_{λ}, and similarly does P^{(1)} ⊗ Ann(P^{(1)}). One can easily see that

Θ_{λ} (P^{(1)} ⊗ Ann(P^{(0)})) = 𝕄(V_{λ}),
Θ_{λ} (P^{(0)} ⊗ P_{−λ−2}^{⋆}) = Θ_{λ} (P_{−λ−2} ⊗ Ann(P^{(1)})) = 𝕄(V_{−λ−2}).

It follows that the 𝔤 ⊕ 𝔤-module 𝕄(P_{−λ−2}) has a filtration

0 ⊂ 𝕄^{(0)} ⊂ 𝕄^{(1)} ⊂ 𝕄^{(2)} = 𝕄(P_{−λ−2})

such that

𝕄^{(2)}/𝕄^{(1)} ≅ V_{−λ−2} ⊗ V_{−λ−2}^{⋆},
𝕄^{(1)}/𝕄^{(0)} ≅ (V_{λ} ⊗ V_{−λ−2}^{⋆}) ⊕ (V_{−λ−2} ⊗ V_{λ}^{⋆}),
𝕄^{(0)} ≅ (V_{λ} ⊗ V_{λ}^{⋆}) ⊕ (V_{−λ−2} ⊗ V_{−λ−2}^{⋆}).

Thus, the block 𝔽(λ) of (1.15) is identified with the subspace, spanned by the matrix elements of the big projective module P_{−λ−2}. Taking direct sums over all λ ∈ P^{+}, adding the 𝔤 ⊕ 𝔤-module 𝕄(P_{−1}) ≅ V_{−1} ⊗ V_{−1}^{⋆}, and comparing with Theorem 1.3, we see that ⨁_{λ∈−P^{++}} 𝕄(P_{λ}) and 𝔽 have the same characters. The statement of the theorem follows. □

1.5. Cohomology of 𝔤 with coefficients in regular representations. The algebra ℜ(G) contains the subalgebra ℜ(G)^{G} of the conjugation-invariant functions on G, which is linearly spanned by the characters of the irreducible finite-dimensional representations. The subalgebra ℜ(G)^{G} is thus isomorphic to the Grothendieck ring of the finite-dimensional representations of G.

There is an isomorphism ℜ(G)^{G} ≅ ℂ[P]^{W}, obtained by restricting the group characters to 𝔥 and taking its Fourier expansion. Finally, the algebra ℜ(G)^{G} also admits a cohomological interpretation, which will be instrumental for further generalizations to the regular representations of the affine and Virasoro algebras. We briefly recall the definition of the cohomology of 𝔤.

Proposition 1.9. Let Λ = ⋀ 𝔤′ be the exterior algebra of 𝔤′ with unit 1. Then (1) The Clifford algebra, generated by {ι(g), ε(g′)}_{g∈𝔤,g′∈𝔤′} with relations

{ι(x), ι(y)} = {ε(x′), ε(y′)} = 0,  {ι(x), ε(y′)} = ⟨y′, x⟩,  (1.23)

acts irreducibly on Λ, so that for any ω ∈ Λ we have

ι(g)1 = 0,  ε(g′)ω = g′ ∧ ω,  g ∈ 𝔤, g′ ∈ 𝔤′, ω ∈ Λ.

(2) Λ is a commutative superalgebra,

ω_{1} ∧ ω_{2} = (−1)^{|ω_{1}|·|ω_{2}|} ω_{2} ∧ ω_{1}, ω_{1}, ω_{2} ∈ Λ,

where | · | is the natural grading on Λ satisfying |1| = 0, |ι(g)| = −1, |ε(g′)| = 1.

(3) The 𝔤-module structure on Λ is given by

π_{Λ}(x) = ∑_{i} ε(g′_{i})ι([g_{i}, x]),

where {g_{i}} is any basis of 𝔤, and {g′_{j}} is the corresponding dual basis of 𝔤′.

Definition 1. The cohomology H^{•}(𝔤; V) of 𝔤 with coefficients in a 𝔤-module V is the cohomology of the graded complex C^{•}(𝔤; V) = Λ^{•} ⊗ V, with the differential

d = ∑_{i} ε(g′_{i})π_{V} (g_{i}) − \frac{1}{2} ∑_{i,j} ε(g′_{i})ε(g′_{j})ι([g_{i}, g_{j}]),  (1.24)

where {g_{i}} is any basis of 𝔤, and {g′_{i}} is the dual basis of 𝔤′.

The following is one of the fundamental results in Lie algebra cohomology, (see e.g. [HS]).

Theorem 1.10. For any finite-dimensional 𝔤-module V we have

H^{•}(𝔤; V) ≅ V^{𝔤} ⊗ H_{DR}^{•}(G),  (1.25)

where H_{DR}^{•}(G) denotes the holomorphic de Rham cohomology H_{DR}^{•}(G) of the Lie group G.

If V is a commutative algebra with a compatible 𝔤-action, then its cohomology inherits the multiplication from V and Λ, and H^{•}(𝔤; V) becomes itself a commutative superalgebra. Moreover, the isomorphism (1.25) becomes an isomorphism of superalgebras, with respect to the cup product in H_{DR}^{•}(G).

The diagonal 𝔤-action in F corresponds to the coadjoint action of G in ℜ(G); thus, we get

Corollary 1.11. There is an isomorphism of commutative superalgebras

H^{•}(𝔤; F) = ℂ[P]^{W} ⊗ H_{DR}^{•}(G).

Our next goal is to study the cohomology of 𝔤 with coefficients in the extended regular representation 𝔽 ≅ ℜ(G_{0}). For infinite-dimensional 𝔤-modules Theorem 1.10 does not hold, and the cohomology H^{•}(𝔤; 𝔽) does not reduce to 𝔽^{𝔤} ⊗ H_{DR}^{•}(G). We have instead

Theorem 1.12. There is an isomorphism of commutative superalgebras

H^{•}(𝔤; 𝔽) ≅ ℂ[P]^{W} ⊗ ^{⋀•} ℂ^{2}.

Proof. It is easy to show using the results of [W] that for λ ≥ −1

H^{n}(𝔤; V_{λ} ⊗ V_{−λ−2}^{⋆}) = H^{n}(𝔤; V_{−λ−2} ⊗ V_{λ}^{⋆}) = \begin{cases}ℂ, & n = 1, 2 \\ 0, & otherwise \end{cases}

and that for λ ≥ 0 we have H^{n}(𝔤; V_{−λ−2} ⊗ V_{−λ−2}^{⋆}) = 0 for all n. The spectral sequence associated with the filtration of Theorem 1.3 can be used to show that

H^{n}(𝔤; 𝔽(−1)) = \begin{cases}ℂ, & n = 1, 2 \\ 0, & otherwise \end{cases} ,  H^{n}(𝔤; 𝔽(λ)) = \begin{cases}ℂ, & n = 0, 2 \\ ℂ^{2}, & n = 1 \\ 0, & otherwise \end{cases} , λ ≥ 0. (1.26)

Also, this spectral sequence shows that we have a natural isomorphism H^{0}(𝔤; 𝔽) ≅ H^{0}(𝔤; F).

To explicitly get the generators of the commutative superalgebra H^{•}(𝔤; 𝔽), we pick nonzero elements

χ ∈ H^{0}(𝔤; 𝔽(1)), ξ_{−1} ∈ H^{1}(𝔤; 𝔽(−1)), η_{0} ∈ H^{1}(𝔤; 𝔽(0)),

such that η_{0} is not proportional to χ ξ_{−1}. It is known that H^{0}(𝔤; 𝔽) ≅ ℂ[P]^{W} is isomorphic to the polynomial algebra ℂ[χ]. It is also clear that H^{•}(𝔤; 𝔽) is a free ℂ[χ]-module. For each λ ≥ 0, the set

B_{≤λ} = {ξ_{−1}, χξ_{−1}, …, χ^{λ+1} ξ_{−1}} ⋃{η_{0}, χ η_{0}, …, χ^{λ} η_{0}}

consists of 2λ+3 linearly independent elements, and in view of (1.26) is a basis of H^{1}(𝔤; 𝔽_{≤λ}). Finally, one can check that η_{0} ξ_{−1} ≠ 0, and thus the elements {η_{0} ξ_{−1}, χ η_{0} ξ_{−1}, …, χ^{λ+1} η_{0} ξ_{−1}} give a basis of H^{2}(𝔤; 𝔽_{≤λ}) for each λ ≥ −1.

It follows that H^{•}(𝔤; 𝔽) ≅ ℂ[χ] ⊗ ⋀^{•}[ξ_{−1}, η_{0}], and the theorem is proven. □

Remark 4. One of the ingredients in the exterior algebra part of the cohomology H^{•}(𝔤; 𝔽) is the exterior algebra ⋀^{•} 𝔥, corresponding to ⋀^{•}[η_{0}] above. It would be interesting to obtain an invariant characterization of the remaining part of H^{•}(𝔤; 𝔽) for arbitrary 𝔤.

Remark 5. In each of the two-dimensional spaces H^{1}(𝔤; 𝔽(λ)) there is a unique up to proportionality cohomology class ξ_{λ} divisible by ξ_{−1}; the elements \frac{ξ_{λ}}{ξ_{−1}} constitute a basis of H^{0}(𝔤; 𝔽) ≅ ℂ[P]^{W}, associated with the characters of big projective modules (cf. [La]).

2. Modified regular representations of the affine Lie algebra 𝔰̂𝔩(2, ℂ).

2.1. Regular representations of 𝔰̂𝔩(2, ℂ). Let Ĝ be the central extension of the loop group LG, associated with G = SL(2, ℂ) (see [PS]), and let 𝔤̂ be the corresponding Lie algebra. As we discussed in the introduction, there is no maximal cell in the affine Bruhat decomposition, and thus we will use the loop version (0.7) of the finite-dimensional one. An additional advantage is that we get an explicit realization of the left and right regular 𝔤̂-actions, analogous to the finite-dimensional case.

The standard basis of 𝔤̂ consists of the elements {e_{n}, h_{n}, f_{n}}_{n∈ℤ} and the central element k, subject to the commutation relations

[h_{m}, e_{n}] = 2e_{m+n},  [h_{m}, f_{n}] = −2f_{m+n},  [h_{m}, h_{n}] = 2mδ_{m+n,0}k,
[e_{m}, f_{n}] = h_{m+n} + m δ_{m+n,0}k, [e_{m}, e_{n}] = [f_{m}, f_{n}] = 0.

The Lie algebra 𝔤̂ has a ℤ-grading 𝔤̂ = ⨁_{n∈ℤ} 𝔤̂[n], determined by

deg f_{n} = deg h_{n} = deg e_{n} = −n, deg k = 0,

We introduce subalgebras 𝔤̂_{±} = ⨁_{±n>0} 𝔤[n]; the finite-dimensional Lie algebra 𝔤 is naturally identified with a subalgebra in 𝔤̂[0].

The element w_{0} of the classical Weyl group defines an involution ω̂ of 𝔤̂, such that

ω̂(e_{n}) = −f_{n}, ω̂(h_{n}) = −h_{n}, ω̂(f_{n}) = −e_{n}, ω̂(k) = −k.  (2.1)

We use the loop version of the finite-dimensional Bruhat decomposition (0.4), and factorize the central extension L̂T into the product of loops that extend holomorphically inside and outside of the unit circle. The analogue of (1.2) is the formal decomposition

g = exp (∑_{n∈ℤ}x_{n}e_{n}) w_{0} τ^{k} exp (∑_{m<0} ζ_{m}h_{m}) ζ^{h_{0}} exp (∑_{m>0} ζ_{m}h_{m}) exp (∑_{n∈ℤ}y_{n}e_{n}) .

The polynomial algebra ℜ_{0}(Ĝ_{0}) = ℂ[{x_{n}}, {y_{n}}, {ζ_{n≠0}}, ζ^{±1}] can be thought of as the algebra of regular functions on the big cell of the loop group, and for ℜ(Ĝ_{0}) we get

ℜ(Ĝ_{0}) = ℜ_{0}(Ĝ_{0}) ⊗ ℂ[τ^{±1}] = ⨁_{κ∈ℤ}ℜ_{κ}(Ĝ_{0}), ℜ_{κ}(Ĝ_{0}) = ℜ_{0}(Ĝ_{0}) ⊗ ℂτ^{κ}

Note that for each κ the subspace ℜ_{κ}(Ĝ_{0}) is a 𝔤̂ ⊕ 𝔤̂-submodule of ℜ(Ĝ_{0}), but it is not a subalgebra of ℜ(Ĝ_{0}) when κ ≠ 0 ! It is easy to see that the infinitesimal regular 𝔤̂-actions of the central element k on ℜ_{κ}(Ĝ_{0}) are given by

π_{l}(k) = −κ · Id,  π_{r}(k) = κ · Id .  (2.2)

As vector spaces, all ℜ_{κ}(Ĝ_{0}) are identified with the same polynomial space, and one can compute the infinitesimal regular actions of 𝔤̂ by treating κ as a complex parameter. In particular, the regular actions of 𝔤̂ make sense for arbitrary κ ∈ ℂ. Computations yield the following description, analogous to Proposition 1.1.

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 15

Theorem 2.1. The regular actions of 𝔤̂ on ℜ_{κ}(Ĝ_{0}) are given by (2.2) and

\begin{aligned}π_{l}(e_{n}) & = −∂_{x_{n}}, \\ π_{l}(h_{n}) & = −2 ∑_{i∈ℤ}x_{i}∂_{i_{n+n}} + \begin{cases}∂_{ζ_{n}} + 2nκ ζ_{−n}, & n > 0 \\ ζ ∂_{ζ}, & n = 0 \\ ∂_{ζ_{n}}, & n < 0 \end{cases} , \\ π_{l}(f_{n}) & = ∑_{i,i′∈ℤ}x_{i}x_{i′}∂_{x_{i+i′+n}} − ∑_{j<0} x_{j−n}∂_{ζ_{j}} − x_{−n}ζ ∂_{ζ} − ∑_{j>0} x_{j−n} (∂_{ζ_{j}} + 2jκ ζ_{−j}) − \\ & − κ nx_{−n} + ζ^{−2} ∑_{j,j′>0} s_{j′}(−2ζ_{1}, −2ζ_{2}, …) s_{j}(−2ζ_{−1}, −2ζ_{−2}, …)∂_{y_{n−j+j′}}, \end{aligned} (2.3)

\begin{aligned}π_{r}(e_{n}) & = ∂_{y_{n}}, \\ π_{r}(h_{n}) & = −2 ∑_{i∈ℤ}y_{i}∂_{y_{i+n}} + \begin{cases}∂_{ζ_{n}} & n > 0, \\ ζ ∂_{ζ}, & n = 0, \\ ∂_{ζ_{n}} − 2nκ ζ_{−n}, & n < 0. \end{cases} , \\ π_{r}(f_{n}) & = − ∑_{i,i′∈ℤ}y_{i}y_{i′}∂_{y_{i+i′+n}} + ∑_{j>0} y_{j−n}∂_{ζ_{j}} + y_{−n}ζ ∂_{ζ} + ∑_{j<0} y_{j−n} (∂_{ζ_{j}} − 2jκ ζ_{−j}) − \\ & − κ ny_{−n} − ζ^{−2} ∑_{j,j′>0} s_{j′}(−2ζ_{−1}, −2ζ_{−2}, …) s_{j}(−2ζ_{1}, −2ζ_{2}, …)∂_{x_{n+j−j′}}, \end{aligned} (2.4)

where the Schur polynomials s_{k}(α_{1}, α_{2}, …) are defined by

s_{m}(α_{1}, α_{2}, …) = ∑_{\begin{matrix}l_{1},l_{2},...≥0 \\ l_{1}+2l_{2}+...=m \end{matrix}} \frac{α_{1}^{l_{1}}α_{2}^{l_{2}} …}{l_{1}!l_{2}! …}.

Proof. The presence of the central extension requires the use of some elementary cases of the Campbell-Hausdorff formula in our computations; we use the identity

exp(B) exp(tA) ≡^{mod t^{2}} exp (t ∑_{j=0}^{∞} \frac{1}{j!} \underbrace{[B, …, [B, [B, A]] …]}_{j commutators}) exp(B).

For example, to derive the last of (2.4), we use the formulas:

exp (∑_{i∈ℤ}y_{i}e_{i}) exp (tf_{n}) ≡^{mod t^{2}} exp (tf_{n}) exp (−tny_{−n}k + t ∑_{i∈ℤ}y_{i}h_{i+n}) ×

× exp (−t ∑_{i,i′∈ℤ}y_{i}y_{i′}e_{i+i′+n}) exp (∑_{i∈ℤ}y_{i}e_{i}) ,

exp (∑_{m>0} ζ_{m}h_{m}) exp (tf_{n}) ≡^{mod t^{2}} exp (t ∑_{j>0} s_{j}(−2ζ_{1}, −2ζ_{2}, …)f_{n+j}) exp (∑_{m>0} ζ_{m}h_{m}) ,

ζ^{h_{0}} exp (tf_{n}) ≡^{mod t^{2}} exp (tζ^{−2} f_{n}) ζ^{h_{0}},
exp (∑_{m<0} ζ_{m}h_{m}) exp (tf_{n}) ≡^{mod t^{2}} exp (t ∑_{j′>0} s_{j′}(−2ζ_{−1}, −2ζ_{−2}, …)f_{n−j′}) exp (∑_{m<0} ζ_{m}h_{m}) ,

w_{0} exp (tf_{n}) ≡^{mod t^{2}} exp (−t e_{n}) w_{0}.

Combining these equations, we get the desired formulas. We leave the technical calculations to the reader. □

2.2. Vertex operator algebras: review and useful examples. We aim to endow ℜ_{κ}(Ĝ_{0}) (or its modification) with a structure similar to that of an associative commutative algebra. The relevant formalism is provided by the vertex algebra theory.

We recall the definitions of vertex and vertex operator algebras in the most convenient to us form. For more details and equivalent alternative definitions, we refer the reader to the books on the subject [FLM, FrB].

Let V be a vector space, equipped with a linear correspondence

v ↦ Y(v, z) = ∑_{n∈ℤ}v_{(n)}z^{−n−1},  v_{(n)} ∈ End(V).  (2.5)

We refer to such formal End(V)-valued generating functions as ’quantum fields’. We say that V satisfies the locality property, if for any a, b ∈ V

(z − w)^{N}[Y(a, z), Y(b, w)] = 0 for N ≫ 0  (2.6)

in the ring of End(V)-valued formal Laurent series in two variables z, w.

A vector 1 ∈ V is called the vacuum vector, if it satisfies

Y(1, z) = Id_{V},  Y(v, z)1|_{z=0} = v.  (2.7)

An element D ∈ End(V), is called the infinitesimal translation operator, if it satisfies

D 1 = 0,  [D, Y(v, z)] = \frac{d}{dz} Y(v, z),  for all v ∈ V.  (2.8)

Definition 2. The space V is called a vertex algebra, if it is equipped with a linear map (2.5), vacuum vector 1, and infinitesimal translation operator D, satisfying the axioms (2.6), (2.7), (2.8) above.

Vertex superalgebras are defined as usual by inserting ± signs according to parity. A vertex superalgebra V is called bi-graded, if it has ℤ-gradings, | · | and deg,

V = ⨁_{m,n∈ℤ}V^{m}[n], V^{m}[n] = {v ∈ V | |v| = m and deg v = n} ,

such that the parity in superalgebra is determined by | · |, and for any homogeneous v

v ↦ Y(v, z) = ∑_{n∈ℤ}v_{(n)}z^{−n−1}, with |v_{(n)}| = |v| and deg v_{(n)} = deg v − n − 1.

In particular, for the vacuum we must have |1| = deg 1 = 0. Also, we write |Y(v, z)| = |v| and deg Y(v, z) = deg v for the quantum field Y(v, z), if the above conditions are satisfied.

A vertex algebra V is called a vertex operator algebra (VOA) of rank c ∈ ℂ, if there exists an element ω ∈ V, usually called the Virasoro element, such that the operators {L_{n}}_{n∈ℤ} defined by

Y(ω, z) = ∑_{n∈ℤ}L_{n}z^{−n−2},

satisfy L_{−1} = D, and the Virasoro commutation relations

[L_{m}, L_{n}] = (m − n)L_{m+n} + δ_{m+n,0} \frac{m^{3} − m}{12} c.

We define the the normal ordered product of two quantum fields X(z) and Y (z) by

: X(z)Y (w) := X_{−}(z)Y (w) + Y (w)X_{+}(z),

where X_{±}(z) are the regular and principal parts of X(z) = ∑_{n∈ℤ} X_{(n)}z^{−n−1},

X_{+}(z) = ∑_{n≥0} X_{(n)}z^{−n−1}, X_{−}(z) = ∑_{n<0} X_{(n)}z^{−n−1}.

For products of three or more quantum fields, the normal ordered product is defined inductively, starting from the left. In general, the normal ordered product is neither commutative nor associative.

The following ’reconstruction theorem’ is an effective tool for constructing vertex algebras.

Proposition 2.2. Let V be a vector space with a distinguished vector 1 and a family of pairwise local End(V)-valued quantum fields {X^{α}(z) = ∑_{n∈ℤ} X_{(n)}^{α}z^{−n−1}}_{α∈ℑ}. Suppose V is generated from 1 by the action of the Laurent coefficients of quantum fields X^{α}(w), and that the vectors {X^{α}(z)1|_{z=0}}_{α∈ℑ} are linearly independent in V. Then the operators

Y (X_{(−n_{1}−1)}^{α_{1}} … X_{(−n_{k}−1)}^{α_{k}}1, z) = : X^{α_{1}}(z)^{(n_{1})} … X^{α_{k}}(z)^{(n_{k})} :,

where X(z)^{(n)} = \frac{1}{n!} \frac{d^{n}}{dz^{n}} X(z), satisfy (2.6) and (2.7).

If a linear operator D ∈ End(V) satisfies D1 = 0 and [D, X^{α}(z)] = \frac{d}{dz}X^{α}(z) for every α ∈ ℑ, then [D, Y(v, z)] = \frac{d}{dz}Y(v, z) for any v ∈ V.

We say that a vertex algebra V has a PBW basis, associated with quantum fields {X^{α}(z)}_{α∈ℑ}, if the index set ℑ is ordered, and we have a linear basis of V, formed by the vectors

{X_{(−n_{1}−1)}^{α_{1}} … X_{(−n_{k}−1)}^{α_{k}}1 | n_{1} ≥ n_{2} ≥ ⋯ ≥ n_{k} ≥ 0, and if n_{i} = n_{i+1}, then α_{i} ⪯ α_{i+1}} .

For two mutually local quantum fields X(z), Y (w) we introduce the operator product expansion (OPE) formalism, and write

X(z)Y (w) ∼ ∑_{j} \frac{C_{j}(w)}{(z − w)^{j}} ,

if for a finite collection of quantum fields {C_{j}(w)}_{j=1,2,...} we have the equality

X(z)Y (w) = ∑_{j} \frac{C_{j}(w)}{(z − w)^{j}} + : X(z)Y (w) :

where \frac{1}{(z−w)^{j}} should be expanded into the Laurent series in non-negative powers of \frac{w}{z}. The importance of OPE lies in the fact that all commutators [X_{m}, Y_{n}] of Laurent coefficients of quantum fields X(z), Y (w) are completely encoded by the collection {C_{j}(w)}.

The remainder of this subsection presents some examples of vertex algebras, which will be used in this paper. All of these algebras are bi-graded and have a PBW basis associated with given quantum fields, for which we specify the OPEs.

Example 1. We denote by F̂(β, γ) the vertex algebra generated by quantum fields

β(z) = ∑_{n∈ℤ}β_{n}z^{−n},  |β(z)| = 0,  deg β(z)= 0,
γ(z) = ∑_{n∈ℤ}γ_{n}z^{−n−1},  |γ(z)| = 1,  deg γ(z)= 0,

with the operator product expansions

β(z)γ(w) ∼ \frac{1}{z − w},  β(z)β(w) ∼ γ(z)γ(w) ∼ 0.  (2.9)

The commutation relations for the underlying Heisenberg algebra are

[β_{m}, γ_{n}] = δ_{m+n,0},  [β_{m}, β_{n}] = [γ_{m}, γ_{n}] = 0.  (2.10)

Example 2. We denote by Λ̂(ψ, ψ^{∗}) the vertex superalgebra generated by quantum fields

ψ(z) = ∑_{n∈ℤ}ψ_{n}z^{−n−1},  |ψ(z)| = −1,  deg ψ(z) = 1,
ψ^{∗}(z) = ∑_{n∈ℤ}ψ_{n}^{∗}z^{−n},  |ψ^{∗}(z)| = 1,  deg ψ^{∗}(z) = 0,

with the operator product expansions

ψ(z)ψ(w) ∼ ψ^{∗}(z)ψ^{∗}(w) ∼ 0, ψ(z)ψ^{∗}(w) ∼ \frac{1}{z − w}.

The (anti)-commutation relations for the underlying Clifford algebra are

{ψ_{m}, ψ_{n}} = {ψ_{m}^{∗}, ψ_{n}^{∗}} = 0,  {ψ_{m}, ψ_{n}^{∗}} = δ_{m+n,0}.  (2.11)

REGULAR REPRESENTATIONS, VERTEX ALGEBRAS AND SEMI-INFINITE COHOMOLOGY 19

Example 3. We denote by 𝔤̂_{k} the vertex algebra generated by quantum fields

X_{n} = ∑_{n∈ℤ}X_{n}z^{−n−1}, |X(z)| = 0, deg X(z) = 1, X ∈ 𝔤,

with the operator product expansions

X(z)Y (w) ∼ \frac{[X, Y](w)}{z − w} + k \frac{⟨X, Y ⟩}{(z − w)^{2}} , k ∈ ℂ

where ⟨·, ·⟩ is the Killing form on 𝔤. The number k is called the level of 𝔤̂_{k}.

We note that a module for the vertex algebra 𝔤̂_{k} is a ℤ-graded 𝔤̂-module V̂ = ⨁_{n≥n_{0}} V̂ [n], such that πV̂ (k) = k · IdV̂ and 𝔤̂[m]V̂ [n] ⊂ V̂ [m + n] for any m, n ∈ ℤ. Example 4. We denote by Vir_{c} the vertex algebra generated by the quantum field

L(z) = ∑_{n∈ℤ}L_{n}z^{−n−2}, |L(z)| = 0, deg L(z) = 2,

with the operator product expansion

L(z)L(w) ∼ \frac{c/2}{(z − w)^{4}} + \frac{2L(w)}{(z − w)^{2}} + \frac{L′(w)}{z − w}, c ∈ ℂ.

The number c is called the central charge of Vir_{c}.

A module for the vertex algebra Vir_{c} is a ℤ-graded Vir-module Ṽ = ⨁_{n≥n_{0}} Ṽ [n], such that πṼ (c) = c · IdṼ and L_{−m}Ṽ [n] ⊂ Ṽ [m + n] for any m, n ∈ ℤ. Example 5. We denote by F̂_{κ}(a) the vertex algebra generated by the quantum field

a(z) = ∑_{n∈ℤ}a_{n}z^{−n−1}, |a(z)| = 0, deg a(z) = 1,

with the operator product expansion

a(z)a(w) ∼ \frac{2κ}{(z − w)^{2}} , κ ∈ ℂ. (2.12)

The commutation relations for the underlying Heisenberg algebra H(a) are

[a_{m}, a_{n}] = 2κ m δ_{m+n,0}.  (2.13)

Note that the operator a_{0} is central and kills the vacuum.

Below we give the construction of a vertex algebra, which will be crucial for our future considerations. Let F̂_{−κ}(ā) be defined similarly to F̂_{κ}(a), so that

[ā_{m}, ā_{n}] = −2κ m δ_{m+n,0}, ā(z)ā(w) ∼ −\frac{2κ}{(z − w)^{2}} . (2.14)

Theorem 2.3. Let κ ≠ 0. The space 𝔽̃_{κ} = F̂_{κ}(a) ⊗ F̂_{−κ}(ā) ⊗ ℂ[P] has a vertex algebra structure, extending those of F̂_{κ}(a) and F̂_{−κ}(ā), and such that a_{0}1_{λ} = ā_{0}1_{λ} = λ1_{λ}.

Proof. Introduce the quantum fields {𝕐(μ, w)}_{μ∈P} by

𝕐(μ, z) = exp (\frac{μ}{2κ} ∑_{n<0} \frac{a_{n}}{−n}z^{−n}) exp (\frac{μ}{2κ} ∑_{n>0} \frac{a_{n}}{−n}z^{−n}) ×

× exp (−\frac{μ}{2κ} ∑_{n<0} \frac{ā_{n}}{−n}z^{−n}) exp (−\frac{μ}{2κ} ∑_{n>0} \frac{ā_{n}}{−n}z^{−n}) 1_{μ}.

(2.15)

Straightforward computations lead to the operator product expansions

a(z)ā(w) ∼ ā(z)a(w) ∼ 𝕐(μ, z)𝕐(ν, w) ∼ 0,

a(z)𝕐(μ, w) ∼ ā(z)𝕐(μ, w) ∼ \frac{μ 𝕐(μ, w)}{z − w},

and establish mutual pairwise locality for the quantum fields a(z), ā(z), Y(μ, z).

The vacuum is, of course, the vector 1 ⊗ 1 ⊗ 1_{0} ∈ 𝔽_{κ}. We set Y(1_{λ}, z) = 𝕐(λ, z) for any λ ∈ P. The spanning and linear independence conditions of Proposition 2.2 are immediate. Finally, we set D1_{λ} = \frac{λ}{2κ}(a_{−1} − ā_{−1})1_{λ}. The conditions on D amount to

𝕐′(λ, z) = \frac{λ}{2κ} ( : a(z)𝕐(λ, z) : − : ā(z)𝕐(λ, z) :),  (2.16)

which is checked directly. Applying Proposition 2.2, we get the desired statement. □

Theorem 2.3 should be compared with the construction of lattice vertex algebras. It is known that the space F̂_{κ}(a) ⊗ ℂ[P] carries a vertex algebra structure only for special values of κ, satisfying certain integrality conditions.

2.3. Bosonic realizations. We now proceed to study the generalizations of the algebra ℜ(G_{0}). As in the finite-dimensional case, we study modules for the Lie algebra 𝔤̂ ⊕ 𝔤̂, which is equivalent to having two commuting actions of 𝔤̂ on the same space.

As in the classical case, the regular 𝔤̂-actions on ℜ_{κ}(Ĝ_{0}), described in Theorem 2.1, can be reformulated in terms of representations of Heisenberg algebras. We note that the operators

\begin{aligned}β_{n} & = −y_{−n} \\ γ_{n} & = ∂_{y_{n}} \end{aligned}, a_{n} = \begin{cases}∂_{ζ_{n}}, & n > 0 \\ ζ∂_{ζ}, & n = 0 \\ ∂_{ζ_{n}} − 2nκ ζ_{−n}, & n < 0 \end{cases}

satisfy the commutation relations (2.10),(2.13), and similarly for

\begin{aligned}β̄_{n} & = x_{−n}, \\ γ̄_{n} & = −∂_{x_{n}} \end{aligned},  ā_{n} = \begin{cases}∂_{ζ_{n}} + 2nκ ζ_{−n} & n > 0 \\ ζ∂_{ζ}, & n = 0 \\ ∂_{ζ_{n}} & n < 0 \end{cases} .

Note also that ℂ[ζ^{±1}] ≅ ℂ[P], and a_{0} = ā_{0} = a act on ℂ[P] by derivations a1_{λ} = λ1_{λ}.

The formulas of Theorem 2.1 are particularly simple, when written for the generating series e(z), h(z), f(z). For example, (2.4) becomes

\begin{aligned}π_{r}(e(z)) & = γ(z), \\ π_{r}(h(z)) & = 2β(z)γ(z) + a(z), \\ π_{r}(f(z)) & = −β(z)^{2}γ(z) − β(z)a(z) − κβ′(z) + exp (\frac{1}{κ} ∑_{n≠0} \frac{a_{n} − ā_{n}}{n} z^{−n}) γ̄(z)1_{−2}. \end{aligned} (2.17)

Note that in this polynomial realization the constants are annihilated by {e_{n}}_{n∈ℤ} and {h_{n}}_{n≥0}. The vertex algebra formalism requires a different choice of vacuum, and the introduction of normal ordering to make products of quantum fields well-defined. This procedure is well-known in the theory of Wakimoto modules (see [FrB] and references therein), for which the 𝔤̂-action is constructed by modifying the formulas originating from the semi-infinite flag variety. In particular, one expects the shifts of the levels of the representations by the dual Coxeter number h^{∨} = 2.

The modifications of the formulas (2.17) leads to the following result.

Theorem 2.4. Let κ ≠ 0, and let k = κ − h^{∨} and k̄ = −κ − h^{∨}, and let

𝔽̂_{κ} = F̂(β, γ) ⊗ F̂(β̄, γ̄) ⊗ 𝔽̃_{κ}.

(1) The space 𝔽̂_{κ} has a 𝔤̂_{k} ⊕ 𝔤̂k̄-module structure, defined by

\begin{aligned}e(z) & = γ(z), \\ h(z) & = 2 : β(z)γ(z) : +a(z), \\ f(z) & = − : β(z)^{2}γ(z) : −β(z)a(z) − kβ′(z) + 𝕐(−2, z)γ̄(z), \end{aligned}  (2.18)

\begin{aligned}ē(z) & = γ̄(z), \\ h̄(z) & = 2 : β̄(z)γ̄(z) : +ā(z), \\ f̄(z) & = − : β̄(z)^{2}γ̄(z) : −β̄(z)ā(z) − k̄β̄′(z) + 𝕐(−2, z)γ(z). \end{aligned}  (2.19)

(2) The space 𝔽̂_{κ} has a compatible VOA structure with rank 𝔽̂_{κ} = 6. (Compatible means that the operators Y(v, z) are 𝔤̂_{k} ⊕ 𝔤̂k̄-intertwining operators in the VOA sense).

Similar formulas for the two commuting actions of 𝔤̂ were suggested in [FeP], by analogy with the finite-dimensional Gauss decomposition of G. However, in order to get a meaningful VOA structure - and the corresponding semi-infinite cohomology theory! - one must incorporate the twist by w_{0}, built into the Bruhat decomposition.

One can recover the original Wakimoto realization from (2.18) by properly discarding the ’bar’ variables. We use superscripts ’W’ to distinguish the Wakimoto 𝔤̂_{k}-action from (2.18).

Corollary 2.5. The space Ŵ_{λ,k} = F̂(β, γ) ⊗ F̂_{κ}(a) ⊗ ℂ1_{λ} has the structure of a 𝔤̂_{k}-module with k = κ − h^{∨}, defined by the formulas

\begin{aligned}e^{W} (z) & = γ(z), \\ h^{W} (z) & = 2 : β(z)γ(z) : +a(z), \\ f^{W} (z) & = − : β(z)^{2}γ(z) : −β(z)a(z) − k β′(z). \end{aligned}  (2.20)

The 𝔤̂_{k}-module Ŵ_{λ,k} is called the Wakimoto module.

Proof of Theorem 2.4. It suffices to show that modifying the standard Wakimoto actions by the extra terms

δf(z) = f(z) − f^{W} (z) = 𝕐(−2, z)γ̄(z),
\overline{δf}(z) = f̄(z) − f̄^{W} (z) = 𝕐(−2, z)γ(z),

does not destroy the operator product expansions.

We begin by showing that the commutation relations for 𝔤̂_{k} hold. Only those involving the modified quantum field f(z) need to be considered. We have:

e^{W} (z) δf(w) = γ(z)𝕐(−2, w)γ̄(w) ∼ 0,
h^{W} (z) δf(w) = (2 : β(z)γ(z) : +a(z)) 𝕐(−2, w)γ̄(w) ∼

∼ a(z)𝕐(−2, w)γ̄(w) ∼ −\frac{2𝕐(−2, w)}{z − w}γ̄(w) = −\frac{2 δf(w)}{z − w},

f^{W} (z) δf(w) = −β(z)a(z)γ(z)𝕐(−2, w)γ̄(w) ∼ \frac{2 𝕐(−2, w)}{z − w}β(w)γ̄(w),
δf(z) δf(w) = 𝕐(−2, z)γ̄(z)𝕐(−2, w)γ̄(w) ∼ 0.

Using the operator product expansions above we immediately check that

e(z)f(w) = e^{W} (z)f^{W} (w) + e^{W} (z) δf(w) ∼ (\frac{k}{(z − w)^{2}} + \frac{h^{W} (w)}{z − w}) + 0 = \frac{k}{(z − w)^{2}} + \frac{h(w)}{z − w},
h(z)f(w) = h^{W} (z)f^{W} (w) + h^{W} (z) δf(w) ∼ −\frac{2 f^{W} (w)}{z − w} + 0 = −\frac{2 f(w)}{z − w},
f(z)f(w) = f^{W} (z)f^{W} (w) + f^{W} (z) δf(w) + δf(z) f^{W} (w) + δf(z) δf(w) ∼

∼ 0 + \frac{2 𝕐(−2, w)}{z − w}β(w)γ̄(w) + \frac{2 𝕐(−2, z)}{w − z}β(z)γ̄(z) + 0 ∼ 0,

and since the operator product expansions not involving f(z) are unchanged, we have proved the commutation relations for the (left) 𝔤̂_{k}-action. Similarly, one verifies the commutation relations for the (right) 𝔤̂k̄-action.

We now prove that the two actions of 𝔤̂_{k} and 𝔤̂k̄ commute. We have

ē^{W} (z) δf(w) = γ̄(z)𝕐(−2, w)γ̄(w) ∼ 0,
h̄^{W} (z) δf(w) = (2 : β̄(z)γ̄(z) : +ā(z)) 𝕐(−2, w)γ̄(w) ∼

∼ 2\frac{γ̄(z)}{z − w}𝕐(−2, w) − \frac{2𝕐(−2, w)}{z − w}γ̄(w) ∼ 0,

which implies that ē(z)f(w) ∼ h̄(z)f(w) ∼ 0. Finally, we compute

f̄^{W} (z) δf(w) =(− : β̄(z)^{2}γ̄(z) : −k̄β̄′(z) − β̄(z)ā(z))(𝕐(−2, w)γ̄(w)) ∼

∼ −2 \frac{β̄(z)γ̄(z)}{z − w}𝕐(−2, w) + \frac{k̄}{(z − w)^{2}} 𝕐(−2, w)−

− (−\frac{2𝕐(−2, w)}{z − w} : β̄(z)γ̄(w) : +\frac{: ā(w)𝕐(−2, w) :}{z − w} − \frac{2𝕐(−2, w)}{(z − w)^{2}}) ∼

∼ \frac{(k̄ + 2)𝕐(−2, w)}{(z − w)^{2}} − \frac{: ā(w)𝕐(−2, w) :}{z − w} = −κ \frac{𝕐(−2, w)}{(z − w)^{2}} − \frac{: ā(w)𝕐(−2, w) :}{z − w}.

and similarly

\overline{δf}(z)f^{W} (w) ∼ κ \frac{𝕐(−2, w)}{(z − w)^{2}} + \frac{: ā(w)𝕐(−2, w) :}{z − w} ∼ − f̄^{W} (z) δf(w).

Therefore,

f̄(z)f(w) = f̄^{W} (z)f^{W} (w) + f̄^{W} (z) δf(w) + \overline{δf}(z) f^{W} (w) + \overline{δf}(z) δf(w) ∼ 0,

and we have established the commutativity of the two 𝔤̂-actions.

Proposition 2.2 implies that 𝔽̂_{κ} is a vertex algebra. The formulas (2.18) can be written as

\begin{aligned}e(z) & = Y(γ_{−1}1_{0}, z), \\ h(z) & = Y(2β_{0}γ_{−1}1_{0} + a_{−1}1_{0}, z), \\ f(z) & = Y(−(β_{0})^{2}γ_{−1}1_{0} − a_{−1}β_{0}1_{0} − kβ_{−1}1_{0} − γ̄_{−1}1_{−2}, z), \end{aligned}

which means that the quantum fields (2.18) are special cases of the operators Y(·, z). The same is true for the quantum fields (2.19). Therefore, the vertex algebra structure is compatible (in the vertex algebra sense) with the 𝔤̂_{k} ⊕ 𝔤̂k̄-module structure on 𝔽̂_{κ}.

To give 𝔽̂_{κ} a VOA structure we need to introduce the Virasoro element. The Sugawara construction for the affine algebra 𝔤̂_{k} gives a Virasoro quantum field with central charge c = \frac{3k}{k+h^{∨}} = 3 − \frac{6}{κ}:

\begin{aligned}L(z) & = \frac{1}{2κ} (\frac{: h^{2}(z) :}{2}+ : e(z)f(z) : + : f(z)e(z) :) = \\ & = \frac{: a(z)^{2} :}{4κ} − \frac{a′(z)}{2κ} − : β′(z)γ(z) : +\frac{1}{κ} 𝕐(−2, z)γ(z)γ̄(z). \end{aligned}  (2.21)

We also note that

L(z) = L^{W} (z) − \frac{1}{κ} 𝕐(−2, z)γ(z)γ̄(z),

where L^{W} (z) is the Virasoro quantum field given by the Sugawara construction for the standard Wakimoto realization (2.20).

Similarly, the affine algebra 𝔤̂k̄ produces another Virasoro quantum field with central charge c̄ = \frac{3k̄}{k̄+h^{∨}} = 3 + \frac{6}{κ}:

\begin{aligned}L̄(z) & = −\frac{1}{κ} (\frac{: h̄^{2}(z) :}{2}+ : ē(z)f̄(z) : + : f̄(z)ē(z) :) = \\ & = −\frac{: ā(z)^{2} :}{4κ} + \frac{ā′(z)}{2κ} − : β̄′(z)γ̄(z) : −\frac{1}{κ} 𝕐(−2, z)γ(z)γ̄(z). \end{aligned}  (2.22)

We set L(z) = L(z) + L̄(z) = L^{W} (z) + L̄^{W} (z). To show that L_{−1} = D, we check that

Y(L_{−1}v, z) = \frac{d}{dz} Y(v, z), v ∈ 𝔽̂_{κ},  (2.23)

which for all the generating quantum fields follows from straightforward computations. Finally, the rank of the VOA 𝔽̂_{κ} is equal to

rank 𝔽̂_{κ} = c + c̄ = (3 − \frac{6}{κ} ) + (3 + \frac{6}{κ} ) = 6.

This concludes the proof of the theorem. □

2.4. 𝔤̂_{k} ⊕ 𝔤̂k̄-module structure of 𝔽̂_{κ} for generic κ. We now prove the analogue of the Theorem 1.3, describing the structure of the 𝔤̂_{k} ⊕ 𝔤̂k̄-module 𝔽̂_{κ} for generic values of the parameter κ.

For λ ∈ 𝔥^{∗}, k ∈ ℂ we denote by V̂_{λ,k} the irreducible 𝔤̂_{k}-module, generated by a vector v̂ satisfying 𝔤_{+}v̂ = 𝔫_{+}v̂ = 0 and hv̂ = λ v̂.

For any 𝔤̂_{k}-module V̂, the restricted dual space V̂′ can be equipped with a 𝔤̂_{k}-action by

⟨g_{n} v′, v⟩ = −⟨v′, ω̂(g_{−n})v⟩, v ∈ V̂ , v′ ∈ V̂′, g ∈ 𝔤,

where ω̂ is as in (2.1). We denote the resulting dual module by V̂^{⋆}.

An important source of 𝔤̂_{k}-modules is the induced module construction. Any 𝔤-module V may be regarded as a module for the subalgebra 𝔭 = ⨁_{n≥0} 𝔤̂[n], with 𝔤[n] acting trivially for n > 0 and k acting as the multiplication by a scalar k ∈ ℂ. The induced 𝔤̂_{k}-module V̂_{k} is defined as the space

V̂_{k} = U(𝔤̂) ⊗_{U(𝔭)} V,  (2.24)

with 𝔤̂_{k} acting by left multiplication.

For the remainder of this section, we will assume that complex numbers κ, k, k̄ satisfy

κ ∉ ℚ,  k = κ − h^{∨},  k̄ = −κ − h^{∨}.  (2.25)

Theorem 2.6. There exists a filtration

0 ⊂ 𝔽̂_{κ}^{(0)} ⊂ 𝔽̂_{κ}^{(1)} ⊂ 𝔽̂_{κ}^{(2)} = 𝔽̂_{κ}  (2.26)

of 𝔤̂_{k} ⊕ 𝔤̂k̄-submodules of 𝔽̂_{κ} such that

𝔽̂_{κ}^{(2)}/𝔽̂_{κ}^{(1)} ≅ ⨁_{λ∈P^{+}} V̂_{−λ−2,k} ⊗ V̂_{−λ−2,k̄}^{⋆},  (2.27)
𝔽̂_{κ}^{(1)}/𝔽̂_{κ}^{(0)} ≅ ⨁_{λ∈P^{+}} (V̂_{λ,k} ⊗ V̂_{−λ−2,k̄}^{⋆} ⊕ V̂_{−λ−2,k} ⊗ V̂_{λ,k̄}^{⋆}) ,  (2.28)
𝔽̂_{κ}^{(0)} ≅ ⨁_{λ∈P} V̂_{λ,k} ⊗ V̂_{λ,k̄}^{⋆}.  (2.29)

Proof. The operator L_{0} determines a ℤ-grading deg of 𝔽̂_{κ}, which is explicitly described by

deg 1_{λ} = 0,  deg X_{n} = −n for X = a, ā, β, γ, β̄, γ̄.  (2.30)

The lowest graded subspace 𝔽̂_{κ}[0] = F (β_{0}, γ_{0}) ⊗ F (β̄_{0}, γ̄_{0}) ⊗ ℂ[P] of the vertex algebra 𝔽̂_{κ} is identified with the Fock space 𝔽 for the finite-dimensional Lie algebra 𝔤. Moreover, since κ is generic, 𝔽̂_{κ} can be constructed as the induced 𝔤̂_{k} ⊕ 𝔤̂k̄-module from the 𝔤 ⊕ 𝔤-module 𝔽:

𝔽̂_{κ} = U(𝔤̂ ⊕ 𝔤̂) ⊗_{U(𝔭⊕𝔭)} 𝔽.

We construct the filtration (2.26) by inducing it from the finite-dimensional one (1.10):

𝔽̂_{κ}^{(0)} = U(𝔤̂ ⊕ 𝔤̂) ⊗_{U(𝔭⊕𝔭)} 𝔽^{(0)},  𝔽̂_{κ}^{(1)} = U(𝔤̂ ⊕ 𝔤̂) ⊗_{U(𝔭⊕𝔭)} 𝔽^{(1)}.

It is easy to check that (1.11),(1.12),(1.13) respectively imply (2.27),(2.28),(2.29), which proves the theorem. □

The analogue of the Corollary 1.4, describing the realization of the subalgebra ℜ(G) ⊂ ℜ(G_{0}), is given below.

Theorem 2.7. There exists a subspace F̂_{κ} ⊂ 𝔽̂_{κ}, satisfying

(1) F̂_{κ} is a vertex operator subalgebra of 𝔽̂_{κ}, and is generated by the quantum fields (2.18), (2.19) and 𝕐(1, z). In particular, F is a 𝔤̂_{k} ⊕ 𝔤̂k̄-submodule of 𝔽̂_{κ}.

(2) As a 𝔤̂_{k} ⊕ 𝔤̂k̄-module, F̂_{κ} is generated by the vectors {1_{λ}}_{λ∈P^{+}}, and we have

F̂_{κ} ≅ ⨁_{λ∈P^{+}} V̂_{λ,k} ⊗ V̂_{λ,k̄}^{⋆}.  (2.31)

Proof. As before, we identify the lowest graded subspace 𝔽̂_{κ}[0] ⊂ 𝔽̂_{κ} with the Fock space 𝔽 for the finite-dimensional Lie algebra 𝔤. Recall from Corollary 1.4 that the 𝔤 ⊕ 𝔤-module 𝔽 contains the distinguished submodule F. We define the subspace F̂_{κ} as the 𝔤̂_{k} ⊕ 𝔤̂k̄- submodule of 𝔽̂_{κ}, induced from F:

F̂_{κ} = U(𝔤̂ ⊕ 𝔤̂) ⊗_{U(𝔭⊕𝔭)} F.

It immediately follows from Corollary 1.4 that F̂_{κ} is generated by the vectors {1_{λ}}_{λ∈P^{+}}, and has the decomposition (2.31).

Next, we need to show that F̂_{κ} is a vertex subalgebra. Let F̂′_{κ} denote the space, spanned by the Laurent coefficients of 𝔽̂_{κ}-valued fields Y(a, z)b for all possible a, b ∈ F̂_{κ}. We will establish that F̂′_{κ} = F̂_{κ}.

Indeed, F̂′_{κ} is a 𝔤̂_{k} ⊕ 𝔤̂k̄-submodule of 𝔽̂_{κ}, and can be induced from its lowest graded component F′ = F̂′_{κ}[0], which is a 𝔤 ⊕ 𝔤-submodule of 𝔽. It suffices to prove that F′ = F.

For any a, b ∈ F, the lowest graded component of Y(a, z)b is equal to the product ab in the algebra 𝔽, and since F is a subalgebra, we have ab ∈ F. (Note that any element a ∈ F can be obtained this way, for example, by taking b = 1). Using the commutation relations with the two copies of 𝔤̂, we can prove that the lowest graded component of Y(a, z)b lies in F for any a, b ∈ F̂_{κ}.

It follows that F′ = F and hence F̂′_{κ} = F̂_{κ}, which means that the restrictions of the operators Y(·, z), corresponding to the subspace F̂_{κ}, are well-defined. Thus F̂_{κ} is a vertex subalgebra of 𝔽̂_{κ}. It is clear that as a vertex subalgebra F̂_{κ} is generated by the quantum fields (2.18), (2.19) and {𝕐(λ, z)}_{λ∈P^{+}}, and the latter are generated by the single operator 𝕐(1, z).

Finally, F̂_{κ} contains both L^{W} (z) and L̄^{W} (z) - hence also L(z) - and therefore is a vertex operator subalgebra of 𝔽̂_{κ}. □

Remark 6. The vertex operator algebras F̂_{κ} and 𝔽̂_{κ} give explicit realizations of the modified regular representations ℜ′_{κ}(Ĝ) and ℜ′_{κ}(Ĝ_{0}) we discussed in the introduction. It would be interesting to construct them invariantly by using the correlation functions approach [FZ], interpreting the rational functions ⟨1′, Y(v_{1}, z_{1}) … Y(v_{n}, z_{n})1⟩ for v_{1}, …, v_{n} ∈ 𝔽 ≅ 𝔽̂_{κ}[0] as solutions of differential equations similar to the Knizhnik-Zamolodchikov equations.

2.5. Semi-infinite cohomology of 𝔤̂. The fact that the level of the diagonal action of 𝔤̂ in the modified regular representations is equal to the special value −2h^{∨} allows us to introduce the semi-infinite cohomology of 𝔤̂ with coefficients in 𝔽̂_{κ} and in F̂_{κ}. In this section we show that for generic values of κ these cohomologies lead to the same algebras of formal characters as in the finite-dimensional case.

We recall the definition of the semi-infinite cohomology [Fe, FGZ]. The main new ingredient is the ”space of semi-infinite forms” Λ̂^{\frac{∞}{2}} , which replaces the finite-dimensional exterior algebra Λ. We summarize its properties in the following

Proposition 2.8. Let Λ̂^{\frac{∞}{2}} = ⋀ 𝔤̂_{−} ⊗ ⋀(𝔤̂′_{+} ⊕ 𝔤′). Then (1) The Clifford algebra, generated by {ι(g_{n}), ε(g′_{n})}_{g∈𝔤,g′∈𝔤′,n∈ℤ} with relations

{ι(x_{m}), ι(y_{n})} = {ε(x′_{m}), ε(y′_{n})} = 0,  {ι(x_{m}), ε(y′_{n})} = δ_{m,n} ⟨y′, x⟩.  (2.32)

acts irreducibly on Λ̂^{\frac{∞}{2}} , so that for any ω_{−} ∈ ⋀ 𝔤̂_{−}, ω_{+} ∈ ⋀(𝔤̂′_{+} ⊕ 𝔤′) we have

ι(x_{n})(ω_{−} ⊗ 1) = \begin{cases}0, & n ≥ 0 \\ (x_{n} ∧ ω_{−}) ⊗ 1, & n < 0 \end{cases} ,  ε(x′_{n})(1 ⊗ ω_{+}) = \begin{cases}1 ⊗ (x′_{n} ∧ ω_{+}), & n ≥ 0 \\ 0, & n < 0 \end{cases} .

(2) Λ̂^{\frac{∞}{2}} is a bi-graded vertex superalgebra, with vacuum 1 = 1 ⊗ 1, and generated by

ι(x, z) = ∑_{n∈ℤ}ι(x_{n})z^{−n−1},  |ι(x, z)|= − 1, deg ι(x, z) = 1, x ∈ 𝔤,
ε(x′, z) = ∑_{n∈ℤ}ε(x′_{−n})z^{−n},  |ε(x′, z)|=1,  deg ε(x′, z) = 0, x′ ∈ 𝔤′.

(3) Λ̂^{\frac{∞}{2}} has a 𝔤̂-module structure on the level k = 2h^{∨}, defined by

π(x_{n}) = ∑_{m∈ℤ} ∑_{i} : ε((g′_{i})_{m})ι([g_{i}, x]_{n+m}) :, x ∈ 𝔤.

One can think of ⋀ 𝔤̂_{−} as the space spanned by formal “semi-infinite” forms

ω = ξ′_{i_{1}} ∧ ξ′_{i_{2}} ∧ ξ′_{i_{3}} ∧ …, i_{n+1} = i_{n} + 1 for n ≫ 0,

where {ξ_{j}}_{j∈ℕ} is a homogeneous basis of 𝔤̂_{−}. A monomial ξ_{j_{1}} ∧ ⋯ ∧ ξ_{j_{m}} ∈ ⋀ 𝔤̂_{−} is identified with the semi-infinite form with the corresponding factors missing:

ω = ± ξ′_{1} ∧ ξ′_{2} ∧ ⋯ ∧ ξ′_{j_{1}−1} ∧ ξ̂′_{j_{1}} ∧ ξ′_{j_{1}+1} ∧ ⋯ ∧ ξ_{j_{m}−1} ∧ ξ̂_{j_{m}} ∧ ξ_{j_{m}+1} ∧ ….

In other words, if ξ_{j} ∈ 𝔤̂, then ι(ξ_{j}) operates as usual by eliminating the factor ξ′_{j}.

Definition 3. The BRST complex, associated with a 𝔤̂-module V̂ on the level k = −2h^{∨}, is the complex C^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂) = Λ̂^{\frac{∞}{2} +•} ⊗ V̂, with the differential

d̂ = ∑_{n∈ℤ} ∑_{i} ε((g′_{i})_{n})πV̂ ((g_{i})_{n}) − \frac{1}{2} ∑_{m,n∈ℤ} ∑_{i,j} : ε((g_{i})′_{m})ε((g_{j})′_{n})ι([g_{i}, g_{j}]_{m+n}) :,  (2.33)

where {g_{i}} is any basis of 𝔤, and {g′_{i}} is the dual basis of 𝔤′. The corresponding cohomology is denoted H^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂).

The BRST complex above gives the relative (to the center) version of the semi-infinite cohomology. We don’t consider any other type of cohomology, and thus simply drop the word ’relative’ everywhere. The condition k = −2h^{∨} is equivalent to d̂^{2} = 0.

If V̂ is a vertex algebra, then its semi-infinite cohomology inherits a vertex superalgebra structure [LZ1].

The following theorem is similar to the reduction theorem of [FGZ] (see also [L]), and relates the semi-infinite cohomology for generic values of κ with the classical cohomology of Lie algebras.

Theorem 2.9. Let V be a 𝔤 ⊕ 𝔤-module, and let κ ∈ ℂ be generic. Set k = κ − h^{∨} and k̄ = −κ −h^{∨}, and denote V̂ be the induced 𝔤̂_{k} ⊕𝔤̂k̄-module. Then with respect to the diagonal 𝔤-action V̂ is a level k = −2h^{∨} module, and

H^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂) ≅ H^{•}(𝔤, V).

Proof. As a vector space, the module V̂ has a decomposition

V̂ = U(𝔤̂_{−}) ⊗ V ⊗ U(𝔤̂_{+})′,

where we identified the factor U(𝔤̂_{−}), coming from the right induced action of 𝔤̂k̄, with U(𝔤̂_{+})′ using the non-degenerate (since κ is generic!) contravariant pairing. Therefore, as vector spaces

C^{•}(𝔤̂, ℂk; V̂) = C_{−}^{•} ⊗ C_{0}^{•} ⊗ C_{+}^{•},  (2.34)

where

C_{−}^{•} = ⋀ 𝔤̂_{−} ⊗ U(𝔤̂_{−}),  C_{0}^{•} = ⋀ 𝔤′ ⊗ V,  C_{+}^{•} = ⋀ 𝔤̂′_{+} ⊗ U(𝔤̂_{+})^{∗}

We write the differential d̂ as

d̂ = d_{−} + d_{0} + d_{+} + δ,

where d_{±} are the BRST differentials for 𝔤̂_{±},

d_{−} = ∑_{n<0} ∑_{i} ε((g′_{i})_{n})π_{l}((g_{i})_{n}) − \frac{1}{2} ∑_{m,n<0} ∑_{i,j} : ε((g_{i})′_{m})ε((g_{j})′_{n})ι([g_{i}, g_{j}]_{m+n}) :,

d_{+} = ∑_{n>0} ∑_{i} ε((g′_{i})_{n})π_{r}((g_{i})_{n}) − \frac{1}{2} ∑_{m,n>0} ∑_{i,j} : ε((g_{i})′_{m})ε((g_{j})′_{n})ι([g_{i}, g_{j}]_{m+n}) :,

the differential d_{0} is defined as in (1.24) with the 𝔤-action π_{V} replaced by

π(x) = πV̂ (x) + ∑_{n≠0} : ε((g_{j})′_{n})ι([x, g_{j}]_{n}) :,  x ∈ 𝔤,  (2.35)

and δ includes all the remaining terms:

δ = ∑_{n>0} ∑_{i} ε((g′_{i})_{n})π_{l}((g_{i})_{n}) + ∑_{n<0} ∑_{i} ε((g′_{i})_{n})π_{r}((g_{i})_{n})−

− ∑_{m>0,n<0} ∑_{i,j} : ε((g_{i})′_{m})ε((g_{j})′_{n})ι([g_{i}, g_{j}]_{m+n}) : .

Following [FGZ], we introduce the skewed degree f deg by

f deg(w_{−} ⊗ w_{0} ⊗ w_{+}) = deg w_{+} − deg w_{−}, w_{±} ∈ C_{±}, w_{0} ∈ C_{0},

where the ’deg’ gradings in the complexes C_{±} are inherited from C^{\frac{∞}{2}} (𝔤̂, k; V̂). We set

𝔅^{p} = \{v ∈ C^{\frac{∞}{2}} (𝔤̂, k; V̂) | f deg v ≥ p\} .

One can check that d_{±} and d_{0} preserve the filtered degree, and that δ(𝔅^{p}) ⊂ 𝔅^{p+1}. Thus, \{𝔅^{p}\}_{p∈ℤ} is a decreasing filtration of the complex C^{\frac{∞}{2}} (𝔤̂, k; V̂), and the associated graded complex has the reduced differential

d_{red} = d_{−} + d_{0} + d_{+}.

We now compute the corresponding reduced cohomology, which will provide a bridge to H^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂).

It is clear that d_{±}^{2} = {d_{+}, d_{−}} = 0, and that the differentials d_{±} : C_{±}^{•} → C_{±}^{•+1} act in their respective factors of (2.34). One can also check that (d_{0})^{2} = {d_{0}, d_{±}} = 0; moreover,

d_{0}(C_{−}^{•} ⊗ C_{0}^{•} ⊗ C_{+}^{•}) ⊂ (C_{−}^{•} ⊗ C_{0}^{•+1} ⊗ C_{+}^{•})

despite the fact that d_{0} does not act in C_{0}^{•}.

It is a well-known fact in homological algebra that

H^{n}(C_{+}, d_{+}) = H^{n}(𝔤̂_{+}; U(𝔤̂_{+})′) = δ_{n,0} ℂ,

with 1 ⊗ 1′ ∈ C_{+} representing the non-trivial cohomology class. Similarly, one has

H^{n}(C_{−}, d_{−}) = H_{−n}(𝔤̂_{−}; U(𝔤̂_{−})) = δ_{n,0} ℂ,

and 1 ⊗ 1 ∈ C_{−} represents the non-trivial cohomology. Further, one can check that the subspace 1 ⊗ C_{0}^{•} ⊗ 1 ⊂ C^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂) is stabilized by d_{0}, and that the 𝔤-action (2.35) on that subspace reduces to the 𝔤 action 1 ⊗ π_{V} ⊗ 1. It follows that

H_{red}^{•}(𝔤̂, ℂk; V̂) ≅ H^{•}(1 ⊗ C_{0} ⊗ 1, d_{0}) ≅ H^{•}(C_{0}, d) ≅ H^{•}(𝔤, V).

We now return to the cohomology of C^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂). Since d̂ preserves the ’deg’ grading, it can be computed separately for each subcomplex C^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂)[m], m ∈ ℤ. The filtration {𝔅^{p}[m]}_{p∈ℤ} of this complex is finite for each m, and leads to a finitely converging spectral sequence with E_{1}^{p,q}[m] = H_{red}^{q}(𝔅^{p}[m]/𝔅^{p+1}[m]).

For m ≠ 0 we have H_{red}^{q}(𝔅^{p}[m]/𝔅^{p+1}[m]) = 0 for all p, hence the spectral sequence is zero, and H^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂)[m] = 0. For m = 0 we note that

𝔅^{0}[0] = C^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂)[0], 𝔅^{1}[0] = 0,

which means that E_{1}^{p,q}[0] = 0 unless p = 0, and the collapsing spectral sequence implies

H^{\frac{∞}{2} +•}(𝔤̂, ℂk; V̂)[0] ≅ H_{red}^{•}(𝔅^{0}[0]/𝔅^{1}[0]) ≅ H^{•}(𝔤; V).

This completes the proof of the theorem. □

Corollary 2.10. The vertex superalgebras H^{\frac{∞}{2} +•}(𝔤̂, ℂk; F̂_{κ}), H^{\frac{∞}{2} +•}(𝔤̂, ℂk; 𝔽̂_{κ}) degenerate into commutative superalgebras. Moreover, we have commutative superalgebra isomorphisms

H^{\frac{∞}{2} +•}(𝔤̂, ℂk; F̂_{κ}) ≅ H^{•}(𝔤; F),  H^{\frac{∞}{2} +•}(𝔤̂, ℂk; 𝔽̂_{κ}) ≅ H^{•}(𝔤; 𝔽).  (2.36)

In particular,

H^{\frac{∞}{2} +0}(𝔤̂, ℂk; F̂_{κ}) ≅ H^{\frac{∞}{2} +0}(𝔤̂, ℂk; 𝔽̂_{κ}) ≅ ℂ[P]^{W} .

Proof. Theorem 2.9 gives us isomorphisms (2.36) on the level of vector spaces. It is also clear from its proof that the semi-infinite cohomology is concentrated in the subspace of deg = 0, and thus the operators Y(·, z) on cohomology are reduced to their constant terms. In particular, they are independent of z, which means that the vertex superalgebra degenerates into a commutative superalgebra. The multiplication is easily traced back to the multiplications in F ≅ F̂_{κ}[0] and in the exterior algebra Λ = ⋀ 𝔤′, which shows that (2.36) are superalgebra isomorphisms. □

3. Modified regular representations of the Virasoro algebra.

3.1. Virasoro algebra and the quantum Drinfeld-Sokolov reduction. In this section we present a construction of the regular representation of the Virasoro algebra, which goes in parallel with constructions in the previous sections. However, instead of beginning with a space of functions on the corresponding group (which is, strictly speaking, a semigroup in the complex case), we will use the quantum Drinfeld-Sokolov reduction [FeFr2] (see also [FrB] and references therein), applied to the modified regular representations of 𝔤̂ constructed in Section 1. As a result we obtain certain bimodules over the Virasoro algebra, which have the structure similar to their affine counterparts. The result of the quantum Drinfeld-Sokolov reduction applied to the actual regular representation of 𝔤̂ should have a standard interpretation in terms of the space of functions on the Virasoro semigroup, but we will not need this fact for our purposes.

Recall that the Virasoro algebra Vir is the infinite-dimensional complex Lie algebra, generated by {L_{n}}_{n∈ℤ} and a central element c, subject to the commutation relations

[L_{m}, L_{n}] = (m − n)L_{m+n} + \frac{m^{3} − m}{12} c.

The Virasoro algebra has a ℤ-grading Vir = ⊕_{n∈ℤ} Vir[n], determined by

deg L_{n} = −n, deg c = 0.

There is a functorial correspondence between certain representations of affine Lie algebras and their W-algebra counterparts, called the quantum Drinfeld-Sokolov reduction [FeFr2] (see also [FrB] and references therein). We review this procedure for the case 𝔤̂ = 𝔰̂𝔩(2, ℂ), when the corresponding W-algebra is identified with the Virasoro algebra.

Definition 4. For any 𝔤̂_{k}-module V̂, the complex (C_{DS}(V̂), d_{DS}),

C_{DS}(V̂) = V̂ ⊗ Λ̂(ψ, ψ^{∗}), d_{DS} = ∑_{n∈ℤ}ψ_{n}^{∗}πV̂ (e_{n}) + ψ_{1}^{∗},

is called the BRST complex of the quantum Drindeld-Sokolov reduction. The corresponding cohomology is denoted H_{DS}(V̂).

The BRST complex above is very similar to the semi-infinite cohomology complex for the nilpotent loop algebra 𝔫̂_{+} = ⨁_{n∈ℤ} ℂe_{n}. Indeed, the corresponding space of semi-infinite forms Λ^{\frac{∞}{2}} (𝔫̂_{+}) is identified with Λ̂(ψ, ψ^{∗}) by ι(e_{n}) ≡ ψ_{n}, ε(e′_{n}) ≡ ψ_{−n}^{∗}, and the only modification is the additional term ψ_{1}^{∗} in the differential.

The BRST complex inherits the gradings |·| and deg from Λ̂(ψ, ψ^{∗}) and V̂. Since |d_{DS}| = 1, the grading | · | descends to the cohomology H_{DS}(V̂). However, with respect to the other grading, the differential d_{DS} is not homogeneous. We introduce a modified grading deg′ by

deg′ e(z) = 0,  deg′ h(z) = 1,  deg′ f(z) = 2,
deg′ ψ(z) = 0, deg′ ψ^{∗}(z) = 1.

The differential d_{DS} then satisfies deg′ d_{DS} = 0, and the grading deg′ descends to H_{DS}(V̂).

The cohomology H_{DS}^{0}(𝔤̂_{k}) of the vacuum module inherits a vertex algebra structure. We have the following result (details of the proof can be found in [FrB]).

Proposition 3.1. For k ≠ −h^{∨} we have H_{DS}^{0}(𝔤̂_{k}) ≅ Vir_{c}, where c = 1 − \frac{6}{k+h^{∨}} − 6k.

For any 𝔤̂_{k}-module V̂, the vertex algebra Vir_{c} ≅ H_{DS}^{0}(𝔤̂_{k}) acts on H_{DS}^{0}(V̂). For κ ≠ 0, set F̃_{λ,κ} = H_{DS}^{0}(Ŵ_{λ,κ−h^{∨}}). The following identifies the Vir_{c}-module structure on F̃_{λ,κ}.

Proposition 3.2. Let κ ≠ 0, and let c = 13 − 6κ − \frac{6}{κ}. Then F̃_{λ,κ} ≅ F̂_{κ}(a) ⊗ ℂ1_{λ} as a vector space, and the Vir_{c}-action is given by

L^{F} (z) = \frac{1}{4 κ} : a(z)^{2} : +\frac{κ − 1}{2 κ} a′(z).  (3.1)

Proof. In the vector space factorization of Ŵ_{λ,κ−h^{∨}} = F̂(β, γ) ⊗ F̂_{κ}(a) ⊗ ℂ1_{λ}, the differential d_{DS} acts only in the first component. Therefore, we must have

F̃_{λ,κ} = H_{DS}(F̂(β, γ)) ⊗ F̂_{κ}(a) ⊗ ℂ1_{λ}.

A spectral sequence reduces the cohomology H_{DS}^{0}(F̂(β, γ)) to the cohomology of the semiinfinite Weil complex F̂(β, γ) ⊗ Λ̂(ψ, ψ^{∗}). The latter splits into an infinite product of finitedimensional Weil complexes, and thus has one-dimensional cohomology, concentrated in degree 0.

The inclusion of vertex algebras 𝔤̂_{κ−h^{∨}} ↪ Ŵ_{0,κ−h^{∨}} induces an inclusion Vir_{c} ↪ F̃_{0,κ}, and the explicit formula (3.1) for L(z) in terms of a(z) is a result of a direct computation. □

The realization (3.1) of Virasoro modules was known long before the quantum Drinfeld-Sokolov reduction, and is called the Feigin-Fuks construction in the literature. We use the superscript ”F” to distinguish this standard action from the modified Virasoro actions, which we will be considering later.

3.2. Bosonic realization of the regular representation. The Virasoro analogue of the Peter-Weyl theorem is more subtle than in the case of classical and affine Lie algebras. There is no clear way to calculate the two commuting Vir-actions in a way similar to Theorem 1.2 and Theorem 2.1. However, there exists a Fock space realization analogous to Theorem 2.4, which we will call the regular representation of the Virasoro algebra.

Theorem 3.3. Let κ ≠ 0, and let c = 13 − 6κ − \frac{6}{κ} and c̄ = 13 + 6κ + \frac{6}{κ}.

(1) The space 𝔽̃_{κ} has a Vir_{c} ⊕ Virc̄-module structure, defined by

L(z) = \frac{1}{4κ} : a(z)^{2} : +\frac{κ − 1}{2κ}a′(z) − \frac{1}{κ} 𝕐(−2, z),  (3.2)
L̄(z) = −\frac{1}{4κ} : ā(z)^{2} : +\frac{κ + 1}{2κ}ā′(z) + \frac{1}{κ} 𝕐(−2, z).  (3.3)

(2) The space 𝔽̃_{κ} has a compatible VOA structure with rank 𝔽̃_{κ} = 26.

Proof. The formulas (3.2),(3.3) are nothing else but the result of the two-sided quantum Drinfeld-Sokolov reduction, which consists of two reductions applied separately to the two commuting 𝔤̂-actions of Theorem 2.4, cf. formulas (2.21),(2.22) and Proposition 3.2.

Rather than give detailed proof of this fact, we choose to verify the commutation relations directly. Introduce notation

δL(z) = L(z) − L^{F} (z) = \frac{1}{κ} 𝕐(−2, z),
\overline{δL}(z) = L̄(z) − L̄^{F} (z) = −\frac{1}{κ} 𝕐(−2, z).

Without the additional terms δL(z), \overline{δL}(z), both (3.2) and (3.3) give two commuting copies of the standard construction (3.1) with the specified central charges. Therefore, it suffices to show that the presence of these extra terms does not violate the commutation relations for Vir_{c} ⊕ Virc̄.

Straightforward computations immediately show that

δL(z) δL(w) ∼ δL(z) \overline{δL}(w) ∼ \overline{δL}(z) \overline{δL}(w) ∼ 0.

L^{F} (z)𝕐(−2, w) ∼ \frac{𝕐(−2, w)}{(z − w)^{2}} − \frac{1}{κ} \frac{: a(w)𝕐(−2, w) :}{z − w},

L̄^{F} (z)𝕐(−2, w) ∼ \frac{𝕐(−2, w)}{(z − w)^{2}} + \frac{1}{κ} \frac{: ā(w)𝕐(−2, w) :}{z − w}.

We now prove the commutation relations for the action (3.2). We have

L(z)L(w) − L^{F} (z)L^{F} (w) = L^{F} (z) δL(w) + δL(z)L^{F} (w) + δL(z) δL(w) ∼

∼ \frac{1}{κ} (\frac{𝕐(−2, w)}{(z − w)^{2}} − \frac{1}{κ} \frac{: a(w)𝕐(−2, w) :}{z − w}) + \frac{1}{κ} (\frac{𝕐(−2, z)}{(z − w)^{2}} + \frac{1}{κ} \frac{: a(z)𝕐(−2, w) :}{z − w}) ∼

∼ \frac{1}{κ} (\frac{2 𝕐(−2, w)}{(z − w)^{2}} + \frac{𝕐′(−2, w)}{z − w}) ∼ \frac{2 δL(w)}{(z − w)^{2}} + \frac{(δL)′(w)}{z − w},

and therefore

L(z)L(w) ∼ L^{F} (z)L^{F} (w) + \frac{2 δL(w)}{(z − w)^{2}} + \frac{(δL)′(w)}{z − w} ∼ (\frac{c/2}{(z − w)^{4}} + \frac{2L^{F} (w)}{(z − w)^{2}} + \frac{(L^{F})′(w)}{z − w}) +

+ \frac{2 δL(w)}{(z − w)^{2}} + \frac{(δL)′(w)}{(z − w)^{2}} = \frac{c/2}{(z − w)^{4}} + \frac{2L(w)}{(z − w)^{2}} + \frac{L′(w)}{z − w}.

We have established that adding the extra term δL(z) to the action (3.1) preserves the commutation relations for Vir_{c} . Similarly, the formula (3.3) gives a representation of Virc̄ .

We now show that the two actions of Vir_{c} and Virc̄ commute. Using (2.16), we get

δL(z)L̄^{F} (w) ∼ \frac{1}{κ} (\frac{𝕐(−2, z)}{(z − w)^{2}} − \frac{1}{κ} \frac{: ā(z)𝕐(−2, z) :}{z − w}) ∼

∼ \frac{1}{κ} (\frac{𝕐(−2, w)}{(z − w)^{2}} + \frac{𝕐′(−2, w)}{z − w} − \frac{1}{κ} \frac{: ā(w)𝕐(−2, w) :}{z − w}) ∼

∼ \frac{1}{κ} (\frac{𝕐(−2, w)}{(z − w)^{2}} − \frac{1}{κ} \frac{: a(w)𝕐(−2, w) :}{z − w}) .

Note that (2.16) implies δL(z)L̄^{F} (w) ∼ −L^{F} (z) \overline{δL}(w), and thus

L(z)L̄(w) = L^{F} (z)L̄^{F} (w) + L^{F} (z) \overline{δL}(w) + δL(z)L̄^{F} (w) + δL(z)\overline{δL}(w) ∼ 0,

which means that the two Virasoro actions commute.

It is easy to see that the formula (3.2) can be written as

L(z) = Y (\frac{(a_{−1})^{2}}{4κ} 1_{0} + \frac{κ − 1}{2κ}a_{−2}1_{0} + \frac{1}{κ} 1_{−2}, z) ,

and similarly for (3.3), which means that the vertex algebra structure is compatible with Vir_{c} ⊕ Virc̄-module structure on 𝔽̃_{κ}.

We introduce the VOA structure in 𝔽̃_{κ} by setting L(z) = L(z) + L̄(z) = L^{F} (z) + L̄^{F} (z). One immediately checks that L(z) is a Virasoro quantum field with central charge 26, and satisfies L_{−1}1_{0} = 0. It suffices to check the remaining relation

[L_{−1}, Y(v, z)] = \frac{d}{dz} Y(v, z), v ∈ 𝔽̃_{κ},  (3.4)

for each of the generating quantum fields, which is done by direct computations. □

3.3. Vir_{c} ⊕ Virc̄-module structure of 𝔽̃_{κ} for generic κ. We now describe the socle filtration of the Vir_{c} ⊕ Virc̄-module 𝔽̃_{κ} for generic κ, when it is completely analogous to the finite-dimensional and affine cases, given by Theorem 1.3 and Theorem 2.6. In this subsection we assume that

κ ∉ ℚ,  c = 13 − \frac{6}{κ} − 6κ,  c̄ = 13 + \frac{6}{κ} + 6κ.

For a Vir_{c}-module Ṽ, the restricted dual space Ṽ′ can be equipped with a Vir_{c}-action by

⟨L_{n} v′, v⟩ = ⟨v′, L_{−n}v⟩.

We denote the resulting dual module by Ṽ^{⋆}.

We denote by Ṽ_{Δ,c} the irreducible Vir_{c}-module, generated by a highest weight vector ṽ satisfying L_{0} ṽ = Δ ṽ and L_{n}ṽ = 0 for n > 0. For any λ ∈ 𝔥^{∗}, set

Δ(λ) = \frac{λ(λ + 2)}{4κ} − \frac{λ}{2},  Δ̄(λ) = −\frac{λ(λ + 2)}{4κ} − \frac{λ}{2}.

Theorem 3.4. There exists a filtration

0 ⊂ 𝔽̃_{κ}^{(0)} ⊂ 𝔽̃_{κ}^{(1)} ⊂ 𝔽̃_{κ}^{(2)} = 𝔽̃_{κ}  (3.5)

of Vir_{c} ⊕ Virc̄-submodules of 𝔽̃_{κ} such that

𝔽̃_{κ}^{(2)}/𝔽̃_{κ}^{(1)} ≅ ⨁_{λ∈P^{+}} Ṽ_{Δ(−λ−2),c} ⊗ Ṽ_{Δ̄(−λ−2),c̄}^{⋆},  (3.6)
𝔽̃_{κ}^{(1)}/𝔽̃_{κ}^{(0)} ≅ ⨁_{λ∈P^{+}} (Ṽ_{Δ(λ),c} ⊗ Ṽ_{Δ̄(−λ−2),c̄}^{⋆} ⊕ Ṽ_{Δ(−λ−2),c} ⊗ Ṽ_{Δ̄(λ),c̄}^{⋆}) ,  (3.7)
𝔽̃_{κ}^{(0)} ≅ ⨁_{λ∈P} Ṽ_{Δ(λ),c} ⊗ Ṽ_{Δ̄(λ),c̄}^{⋆}.  (3.8)

Proof. One can derive from Proposition 3.2 that the for generic κ the reduction sends exact sequences of 𝔤̂_{k}-modules to exact sequences of Vir_{c}-modules, which implies in particular that

H_{DS}^{n}(V̂_{λ,k}) = \begin{cases}Ṽ_{Δ(λ),c}, & n = 0 \\ 0, & n ≠ 0 \end{cases} ,

It is then easy to check that the images 𝔽̃_{κ}^{(0,1,2)} of the 𝔤̂_{k} ⊕ 𝔤̂k̄-submodules 𝔽̂_{κ}^{(0,1,2)} from Theorem 2.6 under the two-sided quantum Drinfeld-Sokolov reduction satisfy the required properties. □

An alternative direct approach repeats the steps in the proof of Theorem 1.3. In particular, we get a decomposition into blocks,

𝔽̃_{κ} = 𝔽̃_{κ}(−1) ⊕ ⨁_{λ∈P^{+}} 𝔽̃_{κ}(λ).  (3.9)

We also have the following Virasoro analogue of Corollary 1.4 and Theorem 2.7.

Theorem 3.5. There exists a subspace F̃_{κ} ⊂ 𝔽̃_{κ}, satisfying

(1) F̃_{κ} is a vertex operator subalgebra of 𝔽̃_{κ}, and is generated by the quantum fields (2.18), (2.19) and 𝕐(1, z). In particular, F̃_{κ} is a Vir_{c} ⊕ Virc̄-submodule of 𝔽̃_{κ}.

(2) As a Vir_{c} ⊕ Virc̄-module, F̃_{κ} is generated by the vectors {1_{λ}}_{λ∈P^{+}}, and we have

F̃_{κ} ≅ ⨁_{λ∈P^{+}} Ṽ_{Δ(λ),c} ⊗ Ṽ_{Δ̄(λ),c̄}^{⋆}.  (3.10)

Proof. The desired subspace F̃_{κ} is the image of the vertex subalgebra F̂_{κ} under the two-sided quantum Drinfeld-Sokolov reduction. We leave technical details to the reader. □

3.4. Semi-infinite cohomology of Vir. The central charge for the diagonal action of Vir in the modified regular representations is equal to the special value 26. In this section we study the semi-infinite cohomology of Vir with coefficients in 𝔽̃_{κ} and in F̃_{κ}.

The properties of the appropriate ”space of semi-infinite forms” Λ̃^{\frac{∞}{2}} for the Virasoro algebra are summarized in the following

Proposition 3.6. Set Λ̃^{\frac{∞}{2}} = ⋀ Vir_{−} ⊗ ⋀ Vir′_{+}, where Vir_{−} = ⨁_{n≤−2} ℂL_{n} and Vir_{+} = ⨁_{n≥−1} ℂL_{n}. Then

(1) The Clifford algebra, generated by {b_{n}, c_{n}}_{n∈ℤ} with relations

{b_{m}, b_{n}} = {c_{m}, c_{n}} = 0,  {b_{m}, c_{n}} = δ_{m+n,0}.  (3.11)

acts irreducibly on Λ̃^{\frac{∞}{2}} , so that for any ω_{−} ∈ ⋀ Vir_{−}, ω_{+} ∈ ⋀ Vir′_{+} we have

b_{n}(1 ⊗ ω) = \begin{cases}0, & n ≥ −1 \\ 1 ⊗ (L_{n} ∧ ω), & n ≤ −2 \end{cases} ,  c_{n}(ω ⊗ 1) = \begin{cases}(L′_{−n} ∧ ω) ⊗ 1, & n ≤ 1 \\ 0, & n ≥ 2 \end{cases} .

(2) Λ̃^{\frac{∞}{2}} is a bi-graded vertex superalgebra, with vacuum 1 = 1 ⊗ 1, generated by

b(z) = ∑_{n∈ℤ}b_{n}z^{−n−2},  |b(z)| = −1, deg b(z) = 2,
c(z) = ∑_{n∈ℤ}c_{n}z^{−n+1},  |c(z)| = 1, deg c(z) = −1.

(3) Λ̃^{\frac{∞}{2}} has a Vir-module structure with central charge c = −26, defined by

π(L_{n}) = ∑_{m∈ℤ}(m − n) : c_{−m}b_{n+m} : .

Definition 5. The BRST complex, associated with a Vir-module Ṽ with central charge c = 26, is the complex C^{\frac{∞}{2} +•}(Vir, ℂc; Ṽ) = Λ̃^{\frac{∞}{2} +•} ⊗ Ṽ, with the differential

d̃ = ∑_{n∈ℤ}c_{−n}πṼ (L_{n}) − \frac{1}{2} ∑_{m,n∈ℤ}(m − n) : c_{−m}c_{−n}b_{m+n} : .  (3.12)

The corresponding cohomology is denoted H^{\frac{∞}{2} +•}(Vir, ℂc; Ṽ).

As in the affine case, the special value c = 26 of the central charge is required to ensure that d̃^{2} = 0.

Theorem 3.7. The vertex superalgebras H^{\frac{∞}{2} +•}(Vir, ℂc; F̃_{κ}) and H^{\frac{∞}{2} +•}(Vir, ℂc; 𝔽̃_{κ}) degenerate into the commutative superalgebras, and we have commutative algebra isomorphisms

H^{\frac{∞}{2} +•}(Vir, ℂc; F̃_{κ}) ≅ H^{\frac{∞}{2} +•}(𝔤̂, ℂk; F̂_{κ}), H^{\frac{∞}{2} +•}(Vir, ℂc; 𝔽̃_{κ}) ≅ H^{\frac{∞}{2} +•}(𝔤̂, ℂk; 𝔽̂_{κ}). (3.13)

In particular,

H^{\frac{∞}{2} +0}(Vir, ℂc; F̃_{κ}) ≅ H^{\frac{∞}{2} +0}(Vir, ℂc; 𝔽̃_{κ}) ≅ ℂ[P]^{W} .

Proof. The problem of computing the semi-infinite cohomology of Vir, as well as its inherited algebra structure, has been extensively studied by mathematicians and physicists working in the string theory. We take advantage of these results, and construct our proof by combining entire blocks from previous papers.

We note that for both F̃_{κ} and 𝔽̃_{κ} the diagonal action of Vir does not contain additional vertex operator shifts, and is equal to the sum of two standard Feigin-Fuks actions.

The comprehensive answer for the cohomology of tensor products of Feigin-Fuks and/or irreducible modules was given in [LZ2] for the most difficult case of the central charge c = c_{p,q}, corresponding to κ = \frac{p}{q} ∈ ℚ. Simplified (for the case of generic κ) version of their computations, and the spectral sequence associated with filtrations of Theorem 3.4, yield

H^{\frac{∞}{2} +n}(Vir, ℂc; F̃_{κ}(λ)) = \begin{cases}ℂ, & n = 0, 3 \\ 0, & otherwise \end{cases} ,

H^{\frac{∞}{2} +n}(Vir, ℂc; 𝔽̃_{κ}(−1)) = \begin{cases}ℂ, & n = 1, 2 \\ 0, & otherwise \end{cases} , H^{\frac{∞}{2} +n}(Vir, ℂc; 𝔽̃_{κ}(λ)) = \begin{cases}ℂ, & n = 0, 2 \\ ℂ^{2}, & n = 1 \\ 0, & otherwise \end{cases} ,

for each λ ∈ P^{+}, as well as natural isomorphisms

H^{\frac{∞}{2} +0}(Vir, ℂc; F̃_{κ}(λ)) ≅ H^{\frac{∞}{2} +0}(Vir, ℂc; 𝔽̃_{κ}(λ)).

The algebra structure of H^{\frac{∞}{2} +0}(Vir, ℂc; 𝔽̃_{κ}) is in fact independent of κ, as can be seen from the change of variables

p_{n} = \frac{a_{n} + ā_{n}}{2} , q_{n} = \frac{a_{n} − ā_{n}}{2κ} .

Indeed, the new commutation relations become [p_{m}, p_{n}] = [q_{m}, q_{n}] = 0 and [p_{m}, q_{n}] = δ_{m+n,0}, and the diagonal Virasoro action is given by

L(z) =: p(z)q(z) : +p(z) − q(z).

For the special case κ = 1, corresponding to the pairing of c = 1 and c̄ = 25 modules, the cohomology of a bigger vertex algebra A_{2D} = ⨁_{λ,μ∈ℤ} F̃_{λ,1} ⊗F̃_{μ,−1} was identified in [WZ] with the polynomial algebra ℂ[x, y] in two variables. The subalgebra H^{\frac{∞}{2} +0}(Vir, ℂc; 𝔽̃_{κ}) is therefore isomorphic to the polynomial algebra ℂ[χ], and we can take any nonzero cohomology class χ ∈ H^{\frac{∞}{2} +0}(Vir, ℂc; 𝔽̃_{κ}(1)) as the generator.

The vertex superalgebra structures on H^{\frac{∞}{2} +•}(Vir, ℂc; F̃_{κ}) and H^{\frac{∞}{2} +•}(Vir, ℂc; 𝔽̃_{κ}) degenerate into commutative superalgebras. It is clear that both are free ℂ[χ]-modules.

It follows immediately that H^{\frac{∞}{2} +•}(Vir, ℂc; F̃_{κ}) ≅ ℂ[χ] ⊗ ⋀^{•}[η], where we can pick any non-zero element η ∈ H^{\frac{∞}{2} +3}(Vir, ℂc; F̃_{κ}(0)). This settles the case of F̃_{κ}.

To get the generators of H^{\frac{∞}{2} +•}(Vir, ℂc; 𝔽̃_{κ}), we pick non-zero representatives

ξ_{−1} ∈ H^{\frac{∞}{2} +1}(Vir, ℂc; 𝔽̃_{κ}(−1)), η_{0} ∈ H^{\frac{∞}{2} +1}(Vir, ℂc; 𝔽̃_{κ}(0)),

such that η_{0} is not proportional to χ · ξ_{−1}. One can check that η_{0}ξ_{−1} ≠ 0, and as in Theorem 1.12 it follows that H^{\frac{∞}{2} +•}(Vir, ℂc; 𝔽̃_{κ}) ≅ ℂ[χ]⊗⋀^{•}[ξ_{−1}, η_{0}]. The statement now follows from Theorem 1.12 and Corollary 2.10. □

Remark 7. It would be nice to get a direct proof of isomorphisms (3.13) by using the techniques of the quantum Drinfeld-Sokolov reduction.

4. Extensions, generalizations, conjectures

4.1. Heterogeneous vertex operator algebra. As we mentioned above, the vertex algebra construction for the Virasoro algebra can be obtained from their affine analogues by applying the two-sided quantum Drinfeld-Sokolov reduction to the left and right 𝔤̂-action. One can consider a similar construction where the reduction is only applied to the affine action on one side, thus leading to a vertex operator algebra with two commuting actions of 𝔤̂_{k} and Virc̄ with appropriate k, c̄. Indeed, one can see that the following gives a direct realization of such a vertex algebra.

Theorem 4.1. Let κ ≠ 0. Set k = κ − h^{∨}, c̄ = 13 + 6κ + \frac{6}{κ}, and 𝔽̌_{κ} = F̂(β, γ) ⊗ 𝔽̃_{κ}. Then (1) The space 𝔽̌_{κ} has a 𝔤̂_{k} ⊕ Virc̄-module structure, defined by

\begin{aligned}e(z) & = γ(z), \\ h(z) & = 2 : β(z)γ(z) : +a(z), \\ f(z) & = − : β(z)^{2}γ(z) : −β(z)a(z) − kβ′(z) − 𝕐(−2, z), \end{aligned}  (4.1)

L̄(z) = −\frac{: ā(z)^{2} :}{4κ} + \frac{κ + 1}{2κ}ā′(z) + \frac{1}{κ} 𝕐(−2, z)γ(z).  (4.2)

(2) The space 𝔽̌_{κ} has a compatible VOA structure with rank 𝔽̌_{κ} = 28.

Proof. The verification of commutation relations is straightforward. We define the Virasoro quantum field by

L(z) = \frac{1}{2κ} (\frac{: h(z)^{2} :}{2}+ : e(z)f(z) : + : f(z)e(z) :) + \frac{h′(z)}{2} + L̄(z)  (4.3)

The central charge for the Sugawara construction modified by \frac{h′(z)}{2} is equal to \frac{3k}{k+h^{∨}} − 6k, and we compute

rank 𝔽̌_{κ} = (\frac{3(κ − h^{∨})}{κ} − 6(κ − h^{∨})) + (13 + \frac{6}{κ} + 6κ) = 28.

□

We will call the vertex operator algebra of Theorem 4.1 the heterogeneous VOA. Note (see [L] and references therein) that the central charge c = 28 appears as the critical value in the study of 2D gravity in the light-cone gauge!

The structure of the bimodule 𝔽̌_{κ} in the generic case is again quite similar to the nonsemisimple bimodule ℜ(G_{0}). From now on we assume that

κ ∉ ℚ,  k = κ − h^{∨},  c̄ = 13 + \frac{6}{κ} + 6κ.

Theorem 4.2. There exists a filtration

0 ⊂ 𝔽̌_{κ}^{(0)} ⊂ 𝔽̌_{κ}^{(1)} ⊂ 𝔽̌_{κ}^{(2)} = 𝔽̌_{κ}

of 𝔤̂_{k} ⊕ Virc̄-submodules of 𝔽̌_{κ}, such that

𝔽̌_{κ}^{(2)}/𝔽̌_{κ}^{(1)} ≅ ⨁_{λ∈P^{+}} V̂_{−λ−2,k} ⊗ Ṽ_{Δ̄(−λ−2),c̄}^{⋆},  (4.4)
𝔽̌_{κ}^{(1)}/𝔽̌_{κ}^{(0)} ≅ ⨁_{λ∈P^{+}} (V̂_{λ,k} ⊗ Ṽ_{Δ̄(−λ−2),c̄}^{⋆} ⊕ V̂_{−λ−2,k} ⊗ Ṽ_{Δ̄(λ),c̄}^{⋆}) ,  (4.5)
𝔽̌_{κ}^{(0)} ≅ ⨁_{λ∈P} V̂_{λ,k} ⊗ Ṽ_{Δ̄(λ),c̄}^{⋆}.  (4.6)

The heterogeneous VOA contains a vertex operator subalgebra, analogous to the classical Peter-Weyl subalgebra ℜ(G) ⊂ ℜ(G_{0}).

Theorem 4.3. There exists a subspace F̌_{κ} ⊂ 𝔽̌_{κ}, satisfying

(1) F̌_{κ} is a vertex operator subalgebra of 𝔽̌_{κ}, and is generated by the fields (4.1), (4.2), and 𝕐(1, z). In particular, F̌_{κ} is a 𝔤̂_{k} ⊕ Virc̄-submodule of 𝔽̌_{κ}.

(2) As a 𝔤̂_{k} ⊕ Virc̄-module, F̌_{κ} is generated by the vectors {1_{λ}}_{λ∈P^{+}}, and we have

F̌_{κ} ≅ ⨁_{λ∈P^{+}} V̂_{λ,k} ⊗ Ṽ_{Δ̄(λ),c̄}.  (4.7)

The proofs of the above theorems are obtained from their affine counterparts by applying the quantum Drinfeld-Sokolov reduction to (right) 𝔤̂k̄-action, similarly to the Virasoro case.

The fact that the ranks of VOAs 𝔽̌_{κ} and F̌_{κ} are equal to 28 naturally leads to the consideration of the semi-infinite cohomology of these modules. We note that although the total Virasoro quantum field L(z) does not commute with 𝔤̂, the spaces 𝔽̌_{κ} and F̌_{κ} can be regarded as modules over the semi-direct product Vir ⋉𝔤̂, such that

\begin{aligned}[L_{m}, e_{n}] & = −(n + m + 1) e_{m+n}, \\ [L_{m}, f_{n}] & = (m − n + 1) f_{m+n}, \\ [L_{m}, h_{n}] & = −n h_{m+n} + m(m + 1)δ_{m+n,0} k. \end{aligned}  (4.8)

The semi-infinite cohomology is defined by the BRST complex of the subalgebra Vir ⋉𝔫̂_{+}, where as before 𝔫̂_{+} = ⨁_{n∈ℤ} ℂe_{n} is the nilpotent loop subalgebra of 𝔤̂. The condition c = 28 ensures that the differential squares to zero.

Definition 6. The BRST complex, associated with a Vir ⋉𝔫̂_{+}-module V̌ with Virasoro central charge c = 28 is the complex C^{\frac{∞}{2} +•}(Vir, ℂc; V̌) = Λ̃^{\frac{∞}{2} +•} ⊗ Λ̂(ψ, ψ^{∗}) ⊗ V̌, with the differential

\begin{aligned}ď & = ∑_{n∈ℤ}c_{−n}πV̌ (L_{n}) − \frac{1}{2} ∑_{m,n∈ℤ}(m − n) : c_{−m}c_{−n}b_{m+n} : + \\ & + ∑_{n∈ℤ}ψ_{−n}^{∗}πV̌ (e_{n}) − ∑_{m,n∈ℤ}−(m + n + 1)c_{−m} : ψ_{−n}^{∗}ψ_{m+n} : \end{aligned}  (4.9)

The corresponding cohomology is denoted H^{\frac{∞}{2} +•}(Vir ⋉𝔫̂_{+}, ℂc; V̌).

Proposition 4.4. The vertex algebras H^{\frac{∞}{2} +•}(Vir ⋉𝔫̂_{+}, ℂc; F̌_{κ}) and H^{\frac{∞}{2} +•}(Vir ⋉𝔫̂_{+}, ℂ; F̌_{κ}) degenerate into commutative algebra structures, and we have isomorphisms

H^{\frac{∞}{2} +•}(Vir ⋉𝔫̂_{+}, ℂc; F̌_{κ}) ≅ H^{\frac{∞}{2} +•}(Vir, ℂc; F̃_{κ}),
H^{\frac{∞}{2} +•}(Vir ⋉𝔫̂_{+}, ℂc; 𝔽̌_{κ}) ≅ H^{\frac{∞}{2} +•}(Vir, ℂc; 𝔽̃_{κ}).

In particular,

H^{\frac{∞}{2} +0}(Vir ⋉𝔫̂_{+}, ℂc; F̌_{κ}) ≅ H^{\frac{∞}{2} +0}(Vir ⋉𝔫̂_{+}, ℂc; 𝔽̌_{κ}) ≅ ℂ[P]^{W} .

Proof. We use the technique from [L], where similar isomorphisms were established for relative cohomology spaces. Let d_{𝔫_{+}} = ∑_{n∈ℤ} ψ_{−n}^{∗}π(e_{n}) be the BRST differential for 𝔫_{+}; one can show that d_{𝔫_{+}}^{2} = 0 = {d_{𝔫_{+}}, ď}, which leads to the spectral sequence associated with decomposition ď = d_{𝔫_{+}} + (ď − d_{𝔫_{+}}). Computing the cohomology with respect to d_{𝔫_{+}} first, and using Proposition 3.2, we get the desired statement. For full technical details (there is a slight difference between BRST reduction for 𝔫_{+} and quantum Drinfeld-Sokolov reduction, but it doesn’t affect the outcome) we refer the reader to [L]. □

4.2. General construction of vertex operator algebras and equivalence of categories. The vertex operator algebras constructed in the previous sections can be built, like conformal field theories, by pairing the left and right modules from certain equivalent categories of representations of infinite-dimensional Lie algebras. The operators Y(·, z) are then constructed by pairing the left and right intertwining operators. There is a unique choice of the structural coefficients for such pairing that would ensure the locality condition for the vertex operator algebras; these coefficients are determined by the tensor structure on the category of representations. Conversely, a natural VOA structure on a bimodule can be used to establish the equivalence of the left and right tensor categories.

The vertex operator algebra constructions in this paper deal with the pairings of different categories of modules. In the affine case, we pair the 𝔤̂ modules on levels k and k̄ = −2h^{∨}−k, symmetric with respect to the critical level −h^{∨}; the equivalence of the corresponding tensor categories was studied in [Fi]. In the Virasoro case, we pair the modules with central charges c and c̄ = 26 − c.

The theorems of Peter-Weyl type can be extended to the quantum group U_{q}(𝔤), associated with G. On one hand, the modules from the category O can be q-deformed into modules over U_{q}(𝔤); on the other hand, one can define q-deformations ℜ_{q}(G) and ℜ_{q}(G_{0}) of the algebras of regular functions, which have especially simple description for 𝔤 = 𝔰𝔩(2, ℂ). When q is not a root of unity, we have the quantum analogues of isomorphisms (0.1), (0.5).

The Drinfeld-Kohno theorem establishes an isomorphism of tensor categories of representations of the quantum groups and affine Lie algebras when q = exp (\frac{πi}{k+h^{∨}} ). This equivalence, also extended to W-algebras, was made explicit in [S1], where intertwining operators for U_{q}(𝔤) were directly identified with their VOA counterparts for 𝔤̂_{k} and W(𝔤̂_{k}); the key ingredients were the geometric results in [V] on the homology of configuration spaces. The construction in [S1] built conformal field theories, associated to affine Lie algebras and Walgebras based on their quantum group counterparts, and can be modified to produce the vertex algebras discussed in this paper.

The Drinfeld-Kohno equivalence also allows to couple categories of different types, producing in particular the heterogeneous VOA of the previous subsection. Another important case is the Frenkel-Kac construction, which corresponds to the pairing of modules for 𝔤 and W(𝔤̂) with central charge c = dim 𝔥 (see [F]). However, in general pairings between the quantum groups and the affine Lie or W-algebras lead to the generalized vertex algebra structures, satisfying a braided version of the commutativity axiom. An example of such structure was proposed in [MR].

4.3. Integral central charge, semi-infinite cohomology and Verlinde algebras. In this work we studied the structure of the generalized Peter-Weyl bimodules for 𝔤̂ only for the generic values k ∉ ℚ of the central charge. The structure of these bimodules when k is integral is more complex and undoubtedly even more interesting. In the most special case when k = k̄ = −h^{∨}, we get a regular representation of the affine Lie algebra 𝔤̂ at the critical level, which can be viewed as the direct counterpart of the finite-dimensional case. This space admits a realization as a certain space of meromorphic functions on the affine Lie group Ĝ, and subspaces of spherical functions with respect to conjugation give rise to solutions of the quantum elliptic Calogero-Sutherland system, generalizing the trigonometric analogue in the finite-dimensional case [EFK].

Another special case is k = −h^{∨}+1, k̄ = −h^{∨}−1, when the quantum group degenerates into its classical counterpart. In this case the left and right Fock spaces used in our construction each have separate vertex algebra structures, and the operators 𝕐(−2, w), which play an important role in this paper, are factored into products of left and right vertex operators used in the basic representations of 𝔤̂. The semi-infinite cohomology of the corresponding W-algebras is fundamental to the string theory, and was studied in [WZ] for the Virasoro algebra, and in [BMP] for W_{3}.

For positive integral k one expects the existence of truncated versions F̂_{k+h^{∨}} and 𝔽̂_{k+h^{∨}} of our vertex operators algebras, similar to the truncation in the conformal field theory, where the positive dominant cone P^{+} is replaced by the alcove P_{k}^{+} ⊂ P^{+}. Then the relative semi-infinite cohomology of 𝔤̂ with coefficients in F̂_{k+h^{∨}} and 𝔽̂_{k+h^{∨}} with respect to the center should be truncated correspondingly. The identification of the zero semi-infinite cohomology groups with the representation ring of G in Corollary 2.10 leads to the following conjecture.

Conjecture 1. For positive integral k, let V_{k}(𝔤̂) denote the Verlinde algebra associated with integrable level k representations of 𝔤̂, and let 𝕍_{k}(𝔤̂) denote its counterpart associated to the big projective modules (see [La]). Then we have commutative algebra isomorphisms

H^{\frac{∞}{2} +0}(𝔤̂, ℂk; F̂_{k+h^{∨}}) ≅ V_{k}(𝔤̂), H^{\frac{∞}{2} +0}(𝔤̂, ℂk; 𝔽̂_{k+h^{∨}}) ≅ 𝕍_{k}(𝔤̂).

In other words, the most essential part of the VOA structure, embodied in the 0th cohomology, is equivalent to the structure of the fusion rules of the tensor category of 𝔤̂-modules, encoded in the Verlinde algebra.

It was also realized recently (see [FHT] and references therein) that the Verlinde algebra V_{k}(𝔤̂) admits an alternative realization in terms of twisted equivariant K-theory k+h^{∨}K_{G}^{dim G}(G) of a compact simple Lie group G (which, in the notations of [FHT], is the compact form of the complex Lie group which we denoted by G in this paper).

Thanks to the results of [FHT], the first isomorphism of our conjecture can be restated in a more invariant form.

Conjecture 2. We have a natural commutative algebra isomorphism

H^{\frac{∞}{2} +0}(𝔤̂, ℂk; ℜ′_{k+h^{∨}}(Ĝ)) ≅ ^{k+h^{∨}}K_{G}^{dim G}(G).

A conceivable direct geometric proof of the last isomorphism might combine the realization of the left hand side using the works [GMS] and [AG] with the interpretation of the right hand side given in the works [FHT] and [AS]. A similar K-theoretic interpretation of H^{\frac{∞}{2} +0}(𝔤̂, ℂk; 𝔽̂_{k+h^{∨}}) in our second conjecture might add another twirl to the twisted equivariant K-theory.

References

[AG] S. Arkhipov, D. Gaitsgory, Differential operators on the loop group via chiral algebras. Int. Math. Res. Not. 2002, no. 4, 165-210

[AS] M. Atiyah, G. Segal. Twisted K-theory, math.KT/0407054

[BF] D. Bernard, G. Felder, Fock representations and BRST cohomology in SL(2) current algebra. Comm. Math. Phys. 127 (1990), no. 1, 145–168.

[BGG] J. Bernstein, I. Gelfand, S. Gelfand, A certain category of 𝔤-modules. (Russian) Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1–8.

[BMP] P. Bouwknegt, J. McCarthy, K. Pilch, The W_{3} algebra. Modules, semi-infinite cohomology and BV algebras. Lecture Notes in Physics. Monographs, 42. Springer-Verlag, Berlin, 1996.

[EFK] P. Etingof, I. Frenkel, A. Kirillov Jr., Spherical functions on affine Lie groups. Duke Math. J. 80 (1995), no. 1, 59–90.

[Fe] B. Feigin, Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras. (Russian) Uspekhi Mat. Nauk 39 (1984), no. 2(236), 195–196.

[FeFr1] B. Feigin, E. Frenkel, Representations of affine Kac-Moody algebras and bosonization. Physics and mathematics of strings, 271–316, World Sci. Publishing, Teaneck, NJ, 1990.

[FeFr2] B. Feigin, E. Frenkel, Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras. Infinite analysis, Part A, B (Kyoto, 1991), 197–215, Adv. Ser. Math. Phys., 16, World Sci. Publishing, River Edge, NJ, 1992.

[FeFu] B. Feigin, D. Fuks, Verma modules over a Virasoro algebra. Funktsional. Anal. i Prilozhen. 17 (1983), no. 3, 91–92.

[FeP] B. Feigin, S. Parkhomenko, Regular representation of affine Kac-Moody algebras. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), 415–424, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996.

[Fi] M. Finkelberg, An equivalence of fusion categories. Geom. Funct. Anal. 6 (1996), no. 2, 249–267.

[FHT] D. Freed, M. Hopkins, C. Teleman, Twisted K-theory and loop group representations I. math.AT/0312155

[FrB] E. Frenkel, D. Ben-Zvi, Vertex algebras and algebraic curves. Mathematical Surveys and Monographs,

88. American Mathematical Society, Providence, RI, 2001

[F] I. Frenkel, Representations of Kac-Moody algebras and dual resonance models, Applications of group theory in physics and mathematical physics (Chicago, 1982), 325–353, [FGZ] I. Frenkel, H. Garland, G. Zuckerman, Semi-infinite cohomology and string theory. Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 22, 8442–8446.

[FLM] I. Frenkel, J. Lepowsky, A. Meurman Vertex operator algebras and the Monster. Pure and Applied Mathematics, 134. Academic Press, Inc., Boston, MA, 1988.

[FZ] I. Frenkel, Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66 (1992), no. 1, 123–168.

[GMS] V. Gorbounov, F. Malikov, V. Schechtman, On chiral differential operators over homogeneous spaces. Int. J. Math. Math. Sci. 26 (2001), no. 2, 83–106.

[HS] P. Hilton, U. Stammbach, A course in homological algebra. Graduate Texts in Mathematics, Vol. 4. Springer-Verlag, New York-Berlin, 1971.

[La] A. Lachowska, A counterpart of the Verlinde algebra for the small quantum group. Duke Math. J. 118 (2003), no. 1, 37–60.

[L] B. Lian, Semi-infinite homology and 2D quantum gravity. PhD thesis, Yale University, (1991).

[LZ1] B. Lian, G. Zuckerman, New perspectives on the BRST-algebraic structure of string theory. Comm. Math. Phys. 154 (1993), no. 3, 613–646.

[LZ2] B. Lian, G. Zuckerman, Semi-infinite homology and 2D gravity. I. Comm. Math. Phys. 145 (1992), no. 3, 561–593.

[MR] G. Moore, N. Reshetikhin, A comment on quantum group symmetry in conformal field theory. Nuclear Phys. B 328 (1989), no. 3, 557–574.

[PS] A. Pressley, G. Segal, Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1986.

[S1] K. Styrkas, Quantum groups, conformal field theories, and duality of tensor categories. PhD thesis, Yale University, (1998).

[S2] K. Styrkas, Regular representations on the big cells and projective modules in the category O. In preparation.

[V] A. Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups. Advanced Series in Mathematical Physics, 21. World Scientific Publishing Co., Inc., River Edge, NJ, 1995

[W] F. Williams, The cohomology of semisimple Lie algebras with coefficients in a Verma module. Trans. Amer. Math. Soc. 240 (1978), 115–127.

[WZ] E.Witten, B.Zwiebach, Algebraic structures and differential geometry in 2d string theory. Nucl. Phys. B377 (1992) , 55–112

Department of Mathematics, Yale University, New Haven, CT 06520, USA

Max-Planck-Institut für Mathematik, D-53111 Bonn, Germany