Bosonic and Fermionic Representations of Lie Algebra Central Extensions

Michael Lau^{∗}

Abstract. Given any representation of an arbitrary Lie algebra 𝔤 over a field 𝕂 of characteristic 0, we construct representations of a central extension of 𝔤 on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in H^{2}(𝔤; 𝕂). We illustrate these techniques with several concrete examples.

MSC: 17B10, 17B56, 17B65, 17B68. Keywords: Lie algebras, Fock space, central extensions, cohomology.

1 Introduction

Clifford and Weyl algebras have natural representations on exterior and symmetric algebras, respectively. In the early 1980s, I.B. Frenkel, V.G. Kac, and D.H. Peterson ([7],[12]) explicitly constructed the orthogonal affine Lie algebra 𝔰̂𝔬_{N} using the quadratic elements of a Clifford algebra C. These elements were viewed as quadratic operators on a certain highest weight C-module, called fermionic Fock space. A.J. Feingold and I.B. Frenkel [5] later gave an analogous construction of the symplectic affine Lie algebra \begin{matrix} \\ 𝔰𝔭 & 2N \end{matrix} from quadratic elements of a Weyl algebra A acting on bosonic Fock space, a highest weight A-module. The natural inclusion of 𝔤̂𝔩_{N} into both \begin{matrix} \\ 𝔰𝔬 & 2N \end{matrix} and \begin{matrix} \\ 𝔰𝔭 & 2N \end{matrix} meant that the affine Lie algebra of type A_{N−1}^{(1)} admits a uniform construction of both fermionic and bosonic modules. Notably, the bosonic construction (which has level −1) was the first known construction of a nonstandard irreducible highest weight representation^{1} for an affine Lie algebra. Seligman [18] later modified the Feingold-Frenkel construction to produce a large class of irreducible integrable highest-weight modules for 𝔰̂𝔬_{N} and \begin{matrix} \\ 𝔰𝔭 & 2N \end{matrix}. The fermionic construction (for types A and D) is isomorphic to the vertex operator construction (in [8],[14] for instance), giving a boson-fermion correspondence. A generalization of this boson-fermion correspondence appears in [15].

∗The author wishes to thank Professor G. Benkart for her many helpful suggestions, and Professor B.N. Allison for his comments on a previous draft. This work was partially supported by N.S.A. grant MDA 904-03-1-0068.

E-mail: mlau@uottawa.ca Current address: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa, Ontario K1N 6N5, Canada

In 2002, Y. Gao [10] used techniques similar to those of Feingold and Frenkel [5] to construct bosonic and fermionic representations of the extended affine Lie algebra { \atop 𝔤𝔩_{N}(ℂ_{q})}, where ℂ_{q} is the quantum torus in two variables. This was accomplished by defining an interesting module for { \atop 𝔤𝔩_{N}(ℂ_{q})}, a central extension of 𝔤𝔩_{N}(ℂ_{q}).

The Feingold-Frenkel construction for { \atop 𝔤𝔩_{N}(ℂ)} and Gao’s construction for { \atop 𝔤𝔩_{N}(ℂ_{q})} are special cases of a more general phenomenon addressed in this paper. Both constructions define a Weyl or Clifford algebra 𝔄 from generators that we view as basis elements of modules for the Lie algebras. These modules, ℂ^{N} ⊗ ℂ[t, t^{−1}] and its dual module (ℂ^{N} ⊗ ℂ[t, t^{−1}])^{∗}, can be thought of as natural modules for the Lie algebras 𝔤𝔩_{N}(ℂ[t, t^{−1}]) and 𝔤𝔩_{N}(ℂ_{q}). Selecting some of these generators to be “positive”, Feingold-Frenkel and Gao define an associative subalgebra 𝔄^{+} of 𝔄. Their Fock spaces are trivial 𝔄^{+}-modules induced to all of 𝔄.

Our construction replaces 𝔤𝔩_{N}(ℂ[t, t^{−1}]) and 𝔤𝔩_{N}(ℂ_{q}) with any Lie algebra 𝔤 over a field 𝕂 of characteristic 0, and replaces ℂ^{N} ⊗ ℂ[t, t^{−1}] with an arbitrary 𝔤-module W. We use Weyl- (resp. Clifford-) type relations to generate a unital associative algebra 𝔞. This algebra is constructed from a basis of W ⊕ W′ where W′ is the 𝔤-submodule of the dual space W^{∗} generated by the restricted dual Span_{𝕂}{w_{α}^{∗} | α ∈ I}, where {w_{α} | α ∈ I} is a basis for W. The generators for 𝔞 are partitioned into “positive” and “negative” elements, and the division is used to define a vacuum vector and an induced module for 𝔞, called bosonic (resp. fermionic) Fock space V. Some care is needed so that V remains a well-defined module under the action of quadratic operators f_{x} defined for each element x in 𝔤. We treat the operators f_{x} as elements of a completion of 𝔞. The fact that the completion is itself an associative algebra simplifies some of the most difficult computations of Feingold-Frenkel and Gao.

1The level k of a representation of an affine Lie algebra is the constant by which the canonical central element acts. If k is anything other than a nonnegative integer, the representation is said to be nonstandard.

The assignment x ↦ f_{x} extends to a representation of a certain onedimensional central extension 𝔤̃ of 𝔤. For Feingold-Frenkel, 𝔤̃ is simply the affine Lie algebra { \atop 𝔤𝔩_{N}(ℂ)} and for Gao, 𝔤̃ is a homomorphic image of { \atop 𝔤𝔩_{N}(ℂ_{q})}. The representation he constructs for { \atop 𝔤𝔩_{N}(ℂ_{q})} is the pullback of the Fock representation we construct for 𝔤̃.

When this representation is faithful, it gives an embedding of 𝔤̃ into a Lie algebra of “infinite matrices”, analogous to that found by Kac-Peterson [12] for { \atop 𝔤𝔩_{N}(ℂ)}. It also affords a simple formula for the 2-cocycle defining the central extension, and we use it to explicitly compute the 2-cocycle of the (universal) central extension of the Lie algebra of N × N matrices over the ring of differential operators of the form ∑_{n=0}^{∞} f_{n}(t)(\frac{d}{dt})^{n} (where f_{n}(t) ∈ ℂ[t, t^{−1}] is 0 for n ≫ 0).

2 Bosonic and Fermionic Realizations

Assume ρ = 1 or −1. If ρ = −1 (resp. +1), we call the resulting constructions bosonic (resp. fermionic). For elements a, b of any associative algebra A, let {a, b}_{ρ} = ab + ρba. Note that {a, b}_{ρ} = ρ{b, a}_{ρ}, and [ab, c] = a{b, c}_{ρ} − ρ{a, c}_{ρ}b for a, b, c ∈ A, where [a, b] is the usual commutator ab − ba.

Definition 2.1 Let 𝔤 be a Lie algebra over a field 𝕂 of characteristic 0, and let W be an arbitrary 𝔤-module with 𝕂-basis 𝔅 = {w_{α} | α ∈ I}. The universal enveloping algebra U(𝔤) has a natural action on the dual module W^{∗} = Hom_{𝕂}(W, 𝕂) coming from the 𝔤-action (x.λ)(w) = −λ(x.w) for all x ∈ 𝔤, λ ∈ W^{∗}, w ∈ W. Let W′ = ∑_{α∈I} U(𝔤).w_{α}^{∗} where the linear functionals w_{α}^{∗} : W → 𝕂 are defined by w_{α}^{∗}(w_{β}) = δ_{α,β} ∀α, β ∈ I. Note that if W is infinite-dimensional, it is possible that W′ ⊋ Span_{𝕂}{w_{α}^{∗} | α ∈ I}, so we fix a 𝕂-basis 𝔅′ = {w_{α}^{∗}, λ_{β} | α ∈ I, β ∈ I′} for W′. The choice of 𝔅, 𝔅′, and a subset J ⊆ I is called a realization of (𝔤, W) if card{α ∈ J | x.w_{α} ∉ S_{J} } < ∞ for each x ∈ 𝔤, where S_{J} = Span_{𝕂}{w_{β} | β ∈ J }.

Realizations always exist—for example, any finite subset J ⊆ I will trivially satisfy the “finiteness condition”, card{α ∈ J | x.w_{α} ∉ S_{J} } < ∞. The purpose of a realization is to define an associative algebra 𝔞, with “positive” and “negative” parts, 𝔞^{+} and 𝔞^{−}, for later use in constructing bosonic (resp. fermionic) Fock space.

Given a realization of a Lie algebra 𝔤 and representation W, let 𝔞 = 𝔞(𝔤, W, I, I′, ρ) be the (unital) associative algebra generated by {w_{α}, w_{α}^{∗}, λ_{β} | α ∈ I, β ∈ I′}, modulo the relations

(R1) {v, w}_{ρ} = {λ, η}_{ρ} = 0

(R2) {λ, w}_{ρ} = λ(w)

for all v, w ∈ W, λ, η ∈ W′. Let 𝔞^{+} ⊆ 𝔞 denote the (unital) subalgebra generated by those elements w_{α} ∈ 𝔅, λ ∈ 𝔅′ such that α ∈ J and λ(w_{β}) = 0 for all β ∈ J. Likewise, let 𝔞^{−} be the (unital) subalgebra generated by {w_{α} ∈ 𝔅, λ ∈ 𝔅′ | α ∈ I \ J , and λ(w_{β}) ≠ 0 for some β ∈ J }. We will sometimes use the restricted algebra 𝔞_{res}, defined as the (unital) subalgebra generated by {w_{α}, w_{α}^{∗} | α ∈ I}. Analogously, 𝔞_{res}^{+} and 𝔞_{res}^{−} are the (unital) subalgebras generated by {w_{α} ∈ 𝔅, w_{β}^{∗} ∈ 𝔅′ | α ∈ J , β ∈ I \ J } and {w_{α} ∈ 𝔅, w_{β}^{∗} ∈ 𝔅′ | α ∈ I \ J , β ∈ J }, respectively. Note that in many interesting cases (see Examples 6.1-6.7 below), I′ = ∅, so 𝔞_{res} = 𝔞 and 𝔞_{res}^{±} = 𝔞^{±}.^{2}

We will use the multiindex notation w\underline{α} = w_{α_{1}}w_{α_{2}} ⋯ w_{α_{r}},

w∗ ; \underline{α} = w_{α_{1}}^{∗}w_{α_{2}}^{∗} ⋯ w_{α_{r}}^{∗}, and λ\underline{β} = λ_{β_{1}}λ_{β_{2}} ⋯ λ_{β_{s}} where \underline{α} = (α_{1}, …, α_{r}) and

\underline{β} = (β_{1}, …, β_{s}). Fix a total ordering ⪯ on I and I′. We say that w\underline{α}λ\underline{β}w∗ ; \underline{γ} ≠ 0 is a standard monomial if α_{i} ⪯ α_{j}, β_{i} ⪯ β_{j}, and γ_{i} ⪯ γ_{j}whenever i < j. By relations (R1) and (R2), the standard monomials span 𝔞. The length ℓ(\underline{α}) of a multiindex \underline{α} = (α_{1}, …, α_{r}) is r, the number of its entries. If \underline{α} = ∅, we say that ℓ(\underline{α}) = 0, and we let w\underline{α} = λ\underline{α} = w∗ ; \underline{α} = 1.

2An example where I′ ≠ ∅ is where 𝔤 = Span_{𝕂}\{∑_{n=0}^{∞} f_{n}(t)( \frac{d}{dt})^{n} | f_{n}(t) ∈ 𝕂[t]\}, viewed as a Lie algebra of infinite series of differential operators on W = 𝕂[t].

Let 𝔞̂ be the 𝕂-vector space of linear combinations of (possibly infinitely many) distinct standard monomials w\underline{α}λ\underline{β}w∗ ; \underline{γ} of 𝔞. That is, the elements of 𝔞̂ are those that can be expressed as (possibly infinite) sums ∑_{\underline{α},\underline{β},\underline{γ}} c_{\underline{α}\underline{β}\underline{γ}}w\underline{α}λ\underline{β}w∗ ; \underline{γ} where c_{\underline{α}\underline{β}\underline{γ}} ∈ 𝕂, and the sum runs over all finite multiindices \underline{α}, \underline{γ} with entries in I and \underline{β} with entries in I′ such that w\underline{α}λ\underline{β}w∗ ; \underline{γ} is a standard monomial. Elements written in this way (as linear combinations of distinct standard monomials) are said to be in a standard form. For the remainder of the paper, we will restrict our attention to the subspace 𝔞̂_{0} consisting of those elements ∑_{\underline{α},\underline{β},\underline{γ}} c_{\underline{α}\underline{β}\underline{γ}}w\underline{α}λ\underline{β}w∗ ; \underline{γ} ∈ 𝔞̂ with the following property:

(P1) For each \underline{γ}, there are only finitely many pairs (\underline{α}, \underline{β}) such that c_{\underline{α}\underline{β}\underline{γ}} ≠ 0.

Lemma 2.2 Term-by-term multiplication gives 𝔞̂_{0} the structure of an associative algebra.

Proof It is enough to show that this multiplication is well-defined and the resulting products are in 𝔞̂_{0}. Associativity then follows immediately from the associativity of 𝔞. Fix a multiindex \underline{ζ} = (ζ_{1}, …, ζ_{t}) of elements from I.

Let T = ∑_{\underline{α},\underline{β},\underline{γ}} c_{\underline{α}\underline{β}\underline{γ}}w\underline{α}λ\underline{β}w∗ ; \underline{γ} and T′ = ∑_{\underline{α′},\underline{β′},\underline{γ′}} d\underline{α′}\underline{β′}\underline{γ′}w\underline{α′}λ\underline{β′}w∗ ; \underline{γ′} be elements of

𝔞̂_{0} written in a standard form.

We consider the products P = P\underline{α}_{\underline{β}}\underline{γ}\underline{α′}\underline{β′}\underline{γ′} = c_{\underline{α}\underline{β}\underline{γ}}w\underline{α}λ\underline{β}w∗ ; \underline{γ}d\underline{α′}\underline{β′}\underline{γ′}w\underline{α′}λ\underline{β′}w∗ ; \underline{γ′}

which, when written in a standard form, contain a nonzero multiple of a standard monomial of the form w\underline{η}λ\underline{μ}w∗ ; \underline{ζ}. Note that w_{γ′_{i}}^{∗} ∈ \{w_{ζ_{j}}^{∗} | 1 ≤ j ≤ t\} for all w_{γ′_{i}}^{∗} occurring in the expression for P. Thus, there are only finitely many possible w∗ ; \underline{γ′} that may occur. For each such \underline{γ′}, there are only finitely many \underline{α′} = (α′_{1}, …, α′_{r}), \underline{β′} = (β′_{1}, …, β′_{s}) for which d\underline{α′}\underline{β′}\underline{γ′} ≠ 0, since T′ ∈ 𝔞̂_{0}. For each of these (finitely many) possible triples (\underline{α′}, \underline{β′}, \underline{γ′}) occurring in the expression for P, we see that every w_{γ_{i}}^{∗} ∈ {w_{α′_{j}}^{∗}, w_{ζ_{k}}^{∗} | 1 ≤ j ≤ r, 1 ≤ k ≤ t\}. (Any w_{γ_{i}}^{∗} ∉ \{w_{α′_{j}}^{∗} | 1 ≤ j ≤ r\} would commute with w\underline{α′}λ\underline{β′} and thus contribute a w_{γ_{i}}^{∗} term to the expression w∗ ; \underline{ζ}, which is impossible unless w_{γ_{i}}^{∗} ∈ \{w_{ζ_{k}}^{∗} | 1 ≤ k ≤ t\}.) For each of these finitely many possible \underline{γ}, there are only finitely many (\underline{α}, \underline{β}) such that c_{\underline{α}\underline{β}\underline{γ}} ≠ 0, so there are only finitely many (\underline{α}, \underline{β}, \underline{γ}, \underline{α′}, \underline{β′}, \underline{γ′}) such that a standard form expression for P contains a term of the form kw\underline{η}λ\underline{μ}w∗ ; \underline{ζ}, where k is a nonzero scalar. That is, the product T T′ is well-defined, and in 𝔞̂_{0}. □

For each x ∈ 𝔤 and α ∈ I, write x.w_{α} = ∑_{γ∈I} x_{γ}^{α}w_{γ} with x_{γ}^{α} ∈ 𝕂. Note

that x_{γ}^{α} = 0 for all but finitely many γ. Define the normal ordering : ∑ c_{αβ}w_{α}w_{β}^{∗} := ∑ c_{αβ} : w_{α}w_{β}^{∗} : where c_{αβ} ∈ 𝕂 and

: w_{α}w_{β}^{∗} := \{ \begin{matrix}−ρw_{β}^{∗}w_{α} & if α = β ∈ J \\ w_{α}w_{β}^{∗} & otherwise. \end{matrix}  (2.3)

Let

f_{x} = ∑_{α∈I} : (x.w_{α})w_{α}^{∗} : .  (2.4)

Note that f_{x} = ∑_{α,γ∈I} x_{γ}^{α} : w_{γ}w_{α}^{∗} : is a well-defined member of 𝔞̂_{0} for each x ∈ 𝔤. Thus, we may conduct the multiplications of the following two lemmas within 𝔞̂_{0}.

Lemma 2.5 For every x ∈ 𝔤, η ∈ I, and λ ∈ W′,

(i) [f_{x}, w_{η}] = ∑_{γ∈I} x_{γ}^{η}w_{γ} = x.w_{η}

(ii) [f_{x}, λ] = − ∑_{α,γ∈I} x_{γ}^{α}λ(w_{γ})w_{α}^{∗} = x.λ.

Proof If γ = α ∈ J, then : w_{γ}w_{α}^{∗} : = −ρw_{α}^{∗}w_{γ} = w_{γ}w_{α}^{∗} + 1. Otherwise, : w_{γ}w_{α}^{∗} : = w_{γ}w_{α}^{∗}. Therefore ad : w_{γ}w_{α}^{∗} : = ad w_{γ}w_{α}^{∗}, so

[f_{x}, w_{η}] = ∑_{α,γ∈I} x_{γ}^{α}[: w_{γ}w_{α}^{∗} :, w_{η}]
= ∑_{α,γ∈I} x_{γ}^{α}[w_{γ}w_{α}^{∗}, w_{η}]
= ∑_{α,γ∈I} x_{γ}^{α}w_{γ}{w_{α}^{∗}, w_{η}}_{ρ}
= ∑_{γ∈I} x_{γ}^{η}w_{γ} = x.w_{η}.

Similarly,

\begin{aligned}[f_{x}, λ] & = −ρ ∑_{α,γ∈I} x_{γ}^{α}\{w_{γ}, λ\}_{ρ}w_{α}^{∗} \\ & = − ∑_{α,γ∈I} x_{γ}^{α}λ(w_{γ})w_{α}^{∗}, \end{aligned}

and

\begin{aligned}(x.λ)(w_{α}) & = −λ(x.w_{α}) \\ & = − ∑_{γ∈I} x_{γ}^{α}λ(w_{γ}). \end{aligned}

Thus x.λ = − ∑_{α,γ∈I} x_{γ}^{α}λ(w_{γ})w_{α}^{∗} = [f_{x}, λ]. □

Lemma 2.6 [f_{x}, f_{y}] = f_{[x,y]} + Ω_{x,y}^{ρ} for some Ω_{x,y}^{ρ} ∈ 𝔞̂_{0} such that [Ω_{x,y}^{ρ}, 𝔞] = 0.

Proof By the Jacobi identity (on 𝔞̂_{0}, viewed as a Lie algebra), if u ∈ W or W′, then [[f_{x}, f_{y}], u] = [f_{x}, [f_{y}, u]] − [f_{y}, [f_{x}, u]] = x.(y.u) − y.(x.u) = [x, y].u = [f_{[x,y]}, u]. Thus [f_{x}, f_{y}] − f_{[x,y]} commutes with the generators of 𝔞, and hence centralizes 𝔞. □

3 Fock Space

Define an augmentation map ϵ : 𝔞^{+} → 𝕂 by the rule that

ϵ(w\underline{α}λ\underline{β}w∗ ; \underline{γ}) = \{ \begin{matrix}1 & if \underline{α} = \underline{β} = \underline{γ} = ∅ \\ 0 & otherwise. \end{matrix}

Let 𝕂v_{0} be the one-dimensional left 𝔞^{+}-module given by

μ.v_{0} = ϵ(μ)v_{0} for μ ∈ 𝔞^{+}.

Inducing to 𝔞 gives the Fock space V = V (𝔤, W, J , I, I′, ρ) = 𝔞 ⊗_{𝔞^{+}} 𝕂v_{0}.

Analogously, we define the left 𝔞_{res}-module V_{res} = V (𝔤, W, J , I, ρ) = 𝔞_{res} ⊗_{𝔞_{res}^{+}} 𝕂v_{0}, where 𝔞_{res}^{+} acts on 𝕂v_{0} by restriction of the 𝔞^{+}-action. We write av_{0} to denote a ⊗ v_{0} in either of V or V_{res}. Which module av_{0} inhabits should be clear from context.

By the relations of 𝔞, each v ∈ V (resp. V_{res}) can be written in the form av_{0} for some a ∈ 𝔞^{−} (resp. 𝔞_{res}^{−}). From the relations of 𝔞, we have the following useful formulas:

Lemma 3.1 Let w_{α}, w_{γ}, w_{α}^{∗}, w_{γ}^{∗} ∈ 𝔞_{res}. Then

(i) w_{α}w_{γ} = \{ \begin{matrix}w_{γ}w_{α} & if ρ = −1 \\ −w_{γ}w_{α} & if ρ = 1; \end{matrix}

(ii) w_{α}w_{γ}^{∗} = \{ \begin{matrix}w_{γ}^{∗}w_{α} − δ_{α,γ}1 & if ρ = −1 \\ −w_{γ}^{∗}w_{α} + δ_{α,γ}1 & if ρ = 1; \end{matrix}

(iii) w_{α}^{∗}w_{γ} = \{ \begin{matrix}w_{γ}w_{α}^{∗} + δ_{α,γ}1 & if ρ = −1 \\ −w_{γ}w_{α}^{∗} + δ_{α,γ}1 & if ρ = 1; \end{matrix}

(iv) w_{α}^{∗}w_{γ}^{∗} = \{ \begin{matrix}w_{γ}^{∗}w_{α}^{∗} & if ρ = −1 \\ −w_{γ}^{∗}w_{α}^{∗} & if ρ = 1. \end{matrix}

□

Iterating Lemma 3.1 gives

Lemma 3.2 Suppose w_{α}, w_{α}^{∗} ∈ 𝔞_{res}, and let

0 ≠ w\underline{γ}w∗ ; \underline{μ} = w_{γ_{1}}^{m_{1}}w_{γ_{2}}^{m_{2}} ⋯ w_{γ_{r}}^{m_{r}}w_{μ_{1}}^{∗n_{1}}w_{μ_{2}}^{∗n_{2}} ⋯ w_{μ_{s}}^{∗n_{s}} ∈ 𝔞_{res}

where each m_{i}, n_{j} is a positive integer,^{3} the γ_{i} are all pairwise distinct, and the μ_{j} are all pairwise distinct. Then

(i) w_{α}w\underline{γ}w∗ ; \underline{μ} = (−ρ)^{ℓ(\underline{γ})+ℓ(\underline{μ})}w\underline{γ}w∗ ; \underline{μ}w_{α}

− w\underline{γ} ∑_{j=1}^{s}(−ρ)^{ℓ(\underline{γ})+j+1}n_{j}δ_{α,μ_{j}}w_{μ_{1}}^{∗n_{1}} ⋯ w_{μ_{j−1}}^{∗n_{j−1}}w_{μ_{j}}^{∗n_{j}−1}w_{μ_{j+1}}^{∗n_{j+1}} ⋯ w_{μ_{s}}^{∗n_{s}}

(ii) w_{α}^{∗}w\underline{γ}w∗ ; \underline{μ} = (−ρ)^{ℓ(\underline{γ})+ℓ(\underline{μ})}w\underline{γ}w∗ ; \underline{μ}w_{α}^{∗}

+ ∑_{i=1}^{r}(−ρ)^{i}m_{i}δ_{α,γ_{i}}w_{γ_{1}}^{m_{1}} ⋯ w_{γ_{i−1}}^{m_{i−1}}w_{γ_{i}}^{m_{i}−1}w_{γ_{i+1}}^{m_{i+1}} ⋯ w_{γ_{r}}^{m_{r}}w∗ ; \underline{μ}.

□

Thus if w\underline{γ}w∗ ; \underline{μ} ∈ 𝔞^{−}, the action of w_{α} (resp. w_{β}^{∗})∈ 𝔞^{+} on w\underline{γ}w∗ ; \underline{μ}v_{0} is, up to a factor of ±1, \frac{∂}{∂w_{α}^{∗}} (resp. \frac{∂}{∂w_{β}}):

Lemma 3.3 Let w_{α}, w_{β}^{∗} ∈ 𝔞^{+}, and suppose w\underline{γ}w∗ ; \underline{μ} is as in Lemma 3.2. Then

(i) w_{α}w\underline{γ}w∗ ; \underline{μ}v_{0}

= −∑_{j=1}^{s}(−ρ)^{ℓ(\underline{γ})+j+1}n_{j}δ_{α,μ_{j}}w\underline{γ}w_{μ_{1}}^{∗n_{1}} ⋯ w_{μ_{j−1}}^{∗n_{j−1}}w_{μ_{j}}^{∗n_{j}−1}w_{μ_{j+1}}^{∗n_{j+1}} ⋯ w_{μ_{s}}^{∗n_{s}}v_{0}

(ii) w_{α}^{∗}w\underline{γ}w∗ ; \underline{μ}v_{0} = ∑_{i=1}^{r}(−ρ)^{i}m_{i}δ_{α,γ_{i}}w_{γ_{1}}^{m_{1}} ⋯ w_{γ_{i−1}}^{m_{i−1}}w_{γ_{i}}^{m_{i}−1}w_{γ_{i+1}}^{m_{i+1}} ⋯ w_{γ_{r}}^{m_{r}}w∗ ; \underline{μ}v_{0}.

□

3Note that if ρ = 1, then each m_{i} and n_{j} is equal to 1.

Corollary 3.4 Suppose ℓ(\underline{α}) + ℓ(\underline{β}) ≥ ℓ(\underline{γ}) + ℓ(\underline{μ}), w\underline{α}w∗ ; \underline{β} ∈ 𝔞^{∗}, and w\underline{γ}w∗ ; \underline{μ} is as in Lemma 3.2. Then

w\underline{α}w∗ ; \underline{β}w\underline{γ}w∗ ; \underline{μ}v_{0} = k(\underline{γ}, \underline{μ})δ_{\underline{α},\underline{μ}}δ_{\underline{β},\underline{γ}}m_{1}!m_{2}! ⋯ m_{r}!n_{1}!n_{2}! ⋯ n_{s}!v_{0}

where k(\underline{γ}, \underline{μ}) = ±1. □

Proposition 3.5 V_{res} is a simple 𝔞_{res}-module with 𝕂-basis

B = \{w\underline{α}w∗ ; \underline{β}v_{0} | w\underline{α}w∗ ; \underline{β} ∈ 𝔞^{−} is a standard monomial\}.

Proof By the observations before Lemma 3.1, the elements of B span V_{res}. Let v ∈ V_{res}. Write v = ∑_{\underline{α},\underline{β}} c_{\underline{α}\underline{β}}w\underline{α}w∗ ; \underline{β}v_{0} = 0 where each w\underline{α}w∗ ; \underline{β} is a standard monomial in 𝔞^{−}. Choose multiindices \underline{α}′ and \underline{β}′ so that w\underline{α}′w\underline{β}′ ∈ 𝔞^{−} and ℓ(\underline{α}′) + ℓ(\underline{β}′) = max\{ℓ(\underline{α}) + ℓ(\underline{β}) | c_{\underline{α}\underline{β}} ≠ 0\}. Then by Corollary 3.4,

w\underline{β}′w_{\underline{α}′}^{∗}v = kc\underline{α}_{′}\underline{β}′v_{0}  (3.6)

for some k ∈ 𝕂 \ 0. If v ≠ 0, then c\underline{α}_{′}\underline{β}′ can be taken to be nonzero. Hence every nonzero submodule V′ contains v_{0}. But since v_{0} generates V_{res}, V′ must be V_{res}, so V_{res} is simple.

If v = 0, then (3.6) reads 0 = w\underline{β}′w_{\underline{α}′}^{∗}v = kc\underline{α}_{′}\underline{β}′v_{0}. Hence c\underline{α}_{′}\underline{β}′ = 0, and by the maximality of ℓ(\underline{α}′) + ℓ(\underline{β}′), we see that every c_{\underline{α}\underline{β}} is zero. Therefore the elements of B are linearly independent and form a 𝕂-basis for V_{res}. □

Proposition 3.7 Let v ∈ V_{res}. Then v ∈ 𝕂v_{0} if and only if ((1 − ϵ)(𝔞_{res}^{+})).v = 0.

Proof If v ∈ 𝕂v_{0}, then ((1 − ϵ)(𝔞_{res}^{+}))v = 0, by the definition of V_{res}. Conversely, assume that ((1 − ϵ)(𝔞_{res}^{+}))v = 0. Write v = ∑_{\underline{α},\underline{β}} c_{\underline{α}\underline{β}}w\underline{α}w∗ ; \underline{β}v_{0} = 0 where each c_{\underline{α}\underline{β}} ∈ 𝕂, and w\underline{α}w∗ ; \underline{β} is an element of 𝔞_{res}^{−} in standard form. Choose multiindices \underline{α}′, \underline{β}′ such that ℓ(\underline{α}′) + ℓ(\underline{β}′) = max\{ℓ(\underline{α}) + ℓ(\underline{β}) | c_{\underline{α}\underline{β}} ≠ 0\} and c\underline{α}_{′}\underline{β}′ ≠ 0. By Corollary 3.4, w\underline{β}′w_{\underline{α}′}^{∗}v = kc\underline{α}_{′}\underline{β}′v_{0} for some k ∈ 𝕂 \ 0. But w\underline{α}′w_{\underline{β}′}^{∗} ∈ 𝔞_{res}^{−}, so by the definition of 𝔞_{res}^{+}, w\underline{β}′w_{\underline{α}′}^{∗} ∈ 𝔞_{res}^{+}. If \underline{α}′ and \underline{β}′ are not both equal to ∅, then w\underline{β}′w_{\underline{α}′}^{∗}v = ((1−ϵ)(w\underline{β}′w_{\underline{α}′}^{∗}))v = 0, a contradiction. Hence \underline{α}′ = \underline{β}′ = ∅, so we are done by the maximality of ℓ(\underline{α}′) + ℓ(\underline{β}′). □

Modules like V play an important role in statistical mechanics, where they represent the “space of states” for a given system. The cyclic vector v_{0} is viewed as a vacuum, and the element

w_{α_{1}}^{m_{1}}w_{α_{2}}^{m_{2}} ⋯ w_{α_{r}}^{m_{r}}λ_{β_{1}}^{n_{1}} ⋯ λ_{β_{s}}^{n_{s}}w_{γ_{1}}^{∗q_{1}} ⋯ w_{γ_{t}}^{∗q_{t}}v_{0}

(with every m_{i}, n_{j}, q_{k} a nonnegative integer, and w_{α_{i}}, λ_{β_{j}}, w_{γ_{k}}^{∗} ∈ 𝔞^{−}) corresponds to a state with m_{i} particles in state w_{α_{i}}, n_{j} particles in state λ_{β_{j}}, and q_{k} particles in state w_{γ_{k}}^{∗}. When ρ = −1 (resp. +1), V is called bosonic (resp. fermionic) Fock space, since the “particles” in V satisfy Bose-Einstein (resp. Fermi-Dirac) occupancy statistics.

Due to the normal ordering and the “finiteness condition” in our definition of a realization, the elements f_{x} defined in §2 have a well-defined left-multiplication action on V. This follows easily from Lemma 2.5 and the fact that all but finitely many monomials in any f_{x} act as 0 on the vacuum vector v_{0}. We can therefore interpret the elements f_{x} as operators on the Fock space V.

Moreover, for any α, β, γ, η ∈ I, the bracket [: w_{α}w_{β}^{∗} : ,: w_{γ}w_{η}^{∗} :] is an element of Span_{𝕂}{: w_{μ}w_{ν}^{∗} : | μ, ν ∈ I}⊕𝕂1, so [f_{x}, f_{y}] may be written in the form ∑_{μ,ν∈I} c_{μν} : w_{μ}w_{ν}^{∗} : for some c_{μν} ∈ 𝕂. Since f_{x} and f_{y} are given by a realization (𝔤, W, J , I, I′), the “finiteness condition” ensures that there are only finitely many (μ, ν) ∈ (I \ J) × J such that c_{μν} ≠ 0. Thus [f_{x}, f_{y}]v_{0} ∈ V_{res}. Now by Lemma 2.6, Ω_{x,y}^{ρ}v_{0} ∈ V_{res}. But since Ω_{x,y}^{ρ} commutes with the elements of 𝔞, we see that

((1 − ϵ)(𝔞_{res}^{+}))Ω_{x,y}^{ρ}v_{0} = Ω_{x,y}^{ρ}((1 − ϵ)(𝔞_{res}^{+}))v_{0} = 0

Hence by Proposition 3.7, Ω_{x,y}^{ρ}v_{0} ∈ 𝕂v_{0}. Writing Ω_{x,y}^{ρ}v_{0} = c_{x,y}^{ρ}v_{0} with c_{x,y}^{ρ} ∈ 𝕂 gives

Corollary 3.8 Span_{𝕂}{f_{x} | x ∈ 𝔤}⊕𝕂e is a Lie algebra with bracket defined by [f_{x}, f_{y}] = f_{[x,y]} + c_{x,y}^{ρ}e and [e, f_{x}] = 0 for all x, y ∈ 𝔤. □

4 Central Extensions

A central extension of 𝔤 is a Lie algebra 𝔤̃ and an epimorphism ϕ : 𝔤̃ → 𝔤 with ker ϕ contained in the center Z(𝔤̃) of 𝔤̃. Given two central extensions (𝔤̂, π) and (𝔤̃, ϕ) of 𝔤, a morphism (from (𝔤̂, π) to (𝔤̃, ϕ)) in the category of central extensions is a Lie algebra homomorphism μ : 𝔤̂ → 𝔤̃ such that ϕμ = π. We say that (𝔤̂, π) and (𝔤̃, ϕ) are isomorphic if the morphism μ is a bijection. The central extension (𝔤̂, π) is universal if there is a unique morphism from it to every other central extension of 𝔤.

Isomorphism classes of one-dimensional central extensions are in bijective correspondence with cohomology classes in H^{2}(𝔤; 𝕂). In particular, each class [c] ∈ H^{2}(𝔤; 𝕂) determines a central extension 𝔤̃ = 𝔤 ⊕ 𝕂c with the bracket [·, ·] in 𝔤̃ given by [x, y] = [x, y] + c(x, y)c for x, y ∈ 𝔤, where c is central and c is a representative of the class [c]. Conversely, if c : 𝔤 × 𝔤 → 𝕂 is 𝕂-bilinear and 𝔤̃ = 𝔤 ⊕ 𝕂c is a Lie algebra under the bracket [·, ·] defined above, then c is a representative of a cohomology class [c] in H^{2}(𝔤; 𝕂). Direct computation from the chain complex defining H^{2}(𝔤; 𝕂) shows that [c] ∈ H^{2}(𝔤; 𝕂) if and only if c satisfies the 2-cocycle identities:

(i) c(x, y) = −c(y, x) and

(ii) c(x, [y, z]) + c(y, [z, x]) + c(z, [x, y]) = 0.

Theorem 4.1 For every x, y ∈ 𝔤, let c(x, y) = ρc_{x,y}^{ρ}. Then [c] ∈ H^{2}(𝔤; 𝕂).

Proof Since x ↦ f_{x} is a linear map, the space S = Span_{𝕂}{f_{x} | x ∈ 𝔤} is a Lie algebra under the bracket [f_{x}, f_{y}]_{S} = f_{[x,y]}. Then by Corollary 3.8, the map (f_{x}, f_{y}) ↦ c_{x,y}^{ρ} is a 2-cocycle for S. Finally, [c] ∈ H^{2}(𝔤; 𝕂) since x ↦ f_{x} is a homomorphism from 𝔤 to S, and c(x, y) is a constant multiple of c_{x,y}^{ρ}.□

Corollary 4.2 Let 𝔤̃ = 𝔤 ⊕ 𝕂c with [x, y] = [x, y] + c(x, y)c where c is central, and let π : 𝔤̃ ↠ 𝔤 be the canonical projection. Then (𝔤̃, π) is a central extension of 𝔤. □

Theorem 4.3 The action c.v = ρv, x.v = f_{x}v for x ∈ 𝔤, v ∈ V gives a representation of 𝔤̃ on V.

Proof The map x ↦ f_{x}, c ↦ ρ1 extends to a linear transformation on 𝔤̃, so it suffices to check that

[x, y] .v = ([x, y] + c(x, y)c).v
= f_{[x,y]}v + ρc(x, y)v
= ([f_{x}, f_{y}] − Ω_{x,y}^{ρ})v + ρc(x, y)v
= [f_{x}, f_{y}]v
= f_{x}f_{y}v − f_{y}f_{x}v
= x.(y.v) − y.(x.v).  □

The relation c_{x,y}^{ρ}v_{0} = [f_{x}, f_{y}]v_{0} − f_{[x,y]}v_{0} (Lemma 2.6 and the discussion after Proposition 3.7) gives a way to calculate c(x, y) explicitly. In the notation of §2, x.w_{α} = ∑_{γ∈I} x_{γ}^{α}w_{γ} for every x ∈ 𝔤 and α ∈ I, and

Theorem 4.4 c(x, y) = ∑_{α∈J ,γ∈I\J} x_{α}^{γ}y_{γ}^{α} − ∑_{α∈I\J ,γ∈J} x_{α}^{γ}y_{γ}^{α}.

Proof Let κ : V_{res} → 𝕂 be the linear map defined by

κ(w\underline{α}w∗ ; \underline{β}v_{0}) = \{ \begin{matrix}1 & if \underline{α} = \underline{β} = ∅ \\ 0 & otherwise. \end{matrix}

for w\underline{α}w∗ ; \underline{β} ∈ 𝔞^{−}. Since c_{x,y}^{ρ} is a scalar, c_{x,y}^{ρ}v_{0} = [f_{x}, f_{y}]v_{0} − f_{[x,y]}v_{0} =

κ([f_{x}, f_{y}]v_{0})v_{0} − κ(f_{[x,y]}v_{0})v_{0}. For any z ∈ 𝔤, we have

κ(f_{z}v_{0}) = κ( ∑_{α,γ∈I} z_{γ}^{α} : w_{γ}w_{α}^{∗} : v_{0})
= κ( ∑_{α∈I} z_{α}^{α} : w_{α}w_{α}^{∗} : v_{0}) = 0,

by normal ordering since w_{α} ∈ 𝔞^{+} or w_{α}^{∗} ∈ 𝔞^{+}. Thus

c_{x,y}^{ρ} = κ([f_{x}, f_{y}]v_{0})
= κ( ∑_{α,γ∈I} y_{γ}^{α}[f_{x},: w_{γ}w_{α}^{∗} :]v_{0})
= κ( ∑_{α,γ∈I} y_{γ}^{α}([f_{x}, w_{γ}]w_{α}^{∗} + w_{γ}[f_{x}, w_{α}^{∗}])v_{0})
= κ( ∑_{α,γ∈I} x_{α}^{γ}y_{γ}^{α}(w_{α}w_{α}^{∗} − w_{γ}w_{γ}^{∗})v_{0}).

Let χ : I → {0, 1} with

χ(η) = \{ \begin{matrix}1 & if η ∈ J \\ 0 & otherwise. \end{matrix}

Note that w_{α} ∈ 𝔞^{+} or w_{α}^{∗} ∈ 𝔞^{+} (and likewise for w_{γ} and w_{γ}^{∗}). Therefore

w_{α}w_{α}^{∗}v_{0} = \{ \begin{matrix}0 & if w_{α}^{∗} ∈ 𝔞^{+} \\ −ρw_{α}^{∗}w_{α}v_{0} + ρv_{0} = ρv_{0} & if w_{α}^{∗} ∉ 𝔞^{+}. \end{matrix}

Hence (w_{α}w_{α}^{∗} − w_{γ}w_{γ}^{∗})v_{0} = ρ(χ(α) − χ(γ))v_{0}, so

c(x, y) = ρc_{x,y}^{ρ} = ρκ ( ∑_{α,γ∈I} x_{α}^{γ}y_{γ}^{α}ρ(χ(α) − χ(γ))v_{0})
= ∑_{α,γ∈I} x_{α}^{γ}y_{γ}^{α}(χ(α) − χ(γ))
= ∑_{α∈J ,γ∈I\J} x_{α}^{γ}y_{γ}^{α} − ∑_{α∈I\J ,γ∈J} x_{α}^{γ}y_{γ}^{α},

and both of these sums are finite by the finiteness condition on realizations. □

5 Embeddings into Lie Algebras of Infinite Matrices

Date-Jimbo-Kashiwara-Miwa [3] and Kac-Peterson [12] have introduced the Lie algebra \overline{𝔄}_{∞} consisting of infinite matrices (a_{ij})_{i,j∈ℤ} with a_{ij} ∈ 𝕂 and only a finite number of nonzero diagonals. The Lie algebra \overline{𝔄}_{∞} has a central extension 𝔄_{∞} = \overline{𝔄}_{∞} ⊕ 𝕂c given by the 2-cocycle α(E_{ij}, E_{ji}) = −α(E_{ji}, E_{ij}) = 1 for i ≤ 0, j > 0, and α(E_{ij}, E_{mn}) = 0 otherwise (cf. [13]).

By analogy, for any realization (𝔤, W, J , I, I′, ρ), we may view the space

\overline{𝔄} := \{ ∑_{α,β∈I} c_{αβ} : w_{α}w_{β}^{∗} :∈ 𝔞̂_{0} | c_{αβ} ∈ 𝕂\}

as a Lie algebra of (card I)×(card I) “matrices” ∑_{α,β∈I} c_{αβ}E_{αβ}, where E_{αβ} is

the matrix with 1 in the (α, β) position and 0 elsewhere. The Lie bracket in \overline{𝔄} is obtained by extending the ordinary “matrix Lie bracket” [E_{αβ}, E_{γη}] = δ_{β,γ}E_{αη} − δ_{α,η}E_{γβ} to \overline{𝔄}. Although the matrices in \overline{𝔄} may be of arbitrary dimension and need not have only finitely many nonzero “diagonals,” this bracket is well-defined, since it is simply the result of restricting the Lie bracket on 𝔞̂_{0} to \overline{𝔄} ⊕ 𝕂1 ⊂ 𝔞̂_{0} and then projecting onto \overline{𝔄}.

It is clear that the space 𝔄 = \overline{𝔄} ⊕ 𝕂1, under the restriction of the Lie bracket of 𝔞̂_{0}, is a central extension of \overline{𝔄}, and the resulting 2-cocycle is simply the “constant term” that occurs in a given bracket—that is, c(E_{αβ}, E_{γη}) = ϵ([: w_{α}w_{β}^{∗} :,: w_{γ}w_{η}^{∗} :]) = δ_{α,η}δ_{β,γ}(χ(α) − χ(γ)), in the notation of §3 and §4. The restriction of c to the Lie algebra of Corollary 4.2 is the 2-cocycle of §4. If 𝔤 = \overline{𝔄}_{∞}, W is the natural representation on (doubly infinite) column vectors (with canonical basis indexed by I = ℤ), I′ = ∅, and J is the nonpositive integers, then we get the 2-cocycle α and the Lie algebra 𝔄_{∞} of [3] and [12].

In particular, if V is faithful, the identification of the operator : w_{α}w_{β}^{∗} : with the matrix E_{αβ} and c with ρ1 gives an embedding of 𝔤̃ = 𝔤 ⊕ 𝕂c into 𝔄. The following proposition gives an easy criterion for faithfulness.

Proposition 5.1 The 𝔤̃-module V = V (𝔤, W, J , I, I′, ρ) is faithful if and only if W is a faithful 𝔤-module.

Proof Suppose W is not faithful. Then there is a nonzero x ∈ 𝔤 such that x.w = 0 for all w ∈ W. Thus f_{x} = ∑_{α∈I} : (x.w_{α})w_{α}^{∗} := 0, so for every v ∈ V,

x.v = f_{x}v = 0. Hence V is not faithful.

Conversely, suppose W is faithful and kc + x acts as zero on V for some k ∈ 𝕂 and x ∈ 𝔤. Then for every a ∈ 𝔞,

0 = (kc + x).(av_{0}) = (kρ + f_{x})av_{0}
= [kρ + f_{x}, a]v_{0} + a(kρ + f_{x})v_{0}
= [f_{x}, a]v_{0} + a(kc + x).v_{0}
= [f_{x}, a]v_{0}.

Therefore by Lemma 2.5, (x.w_{α})v_{0} = 0 = (x.w_{α}^{∗})v_{0} for all α ∈ I. By Proposition 3.5, it now follows that x.w_{α} = ∑_{γ∈J} x_{γ}^{α}w_{γ}. Then for any α, β ∈

I,

0 = (kc + x).(w_{α}w_{β}^{∗}v_{0}) = [f_{x}, w_{α}w_{β}^{∗}]v_{0}
= [f_{x}, w_{α}]w_{β}^{∗}v_{0} + w_{α}[f_{x}, w_{β}^{∗}]v_{0}
= (x.w_{α})w_{β}^{∗}v_{0} + w_{α}(x.w_{β}^{∗})v_{0}
= (x.w_{α})w_{β}^{∗}v_{0}
= ∑_{γ∈J} x_{γ}^{α}w_{γ}w_{β}^{∗}v_{0}
= −ρ ∑_{γ∈J} x_{γ}^{α}w_{β}^{∗}w_{γ}v_{0} + ∑_{γ∈J} x_{γ}^{α}{w_{γ}, w_{β}^{∗}}_{ρ}v_{0}
= ρx_{β}^{α}χ(β)v_{0},

in the notation of the proof of Theorem 4.4. Hence x_{β}^{α} = 0 for all α ∈ I and β ∈ J. But x.w_{α} = ∑_{γ∈J} x_{γ}^{α}w_{γ}, so this gives x.w_{α} = 0 for all α ∈ I. Since

W is faithful, x = 0, so kc = kc + x acts as zero on V. Hence k = 0, and V is faithful. □

We can use the “identity matrix” J := ∑_{γ∈I} : w_{γ}w_{γ}^{∗} : to decompose V into

submodules:

Proposition 5.2 The 𝔤̃-module V has a ℤ-grading V = ⨁_{r∈ℤ} V_{r}, where V_{r} is the J-eigenspace with eigenvalue r. Each V_{r} is a 𝔤̃-submodule of V.

Explicitly, V_{r} = Span_{𝕂}\{w\underline{α}λ\underline{β}w∗ ; \underline{γ}v_{0} | ℓ(\underline{α}) − ℓ(\underline{β}) − ℓ(\underline{γ}) = r\}.

Proof For any λ ∈ W′, [J, λ] = ∑_{γ∈I}[: w_{γ}w_{γ}^{∗} :, λ] = −ρ∑_{γ∈I}{w_{γ}, λ}_{ρ}w_{γ}^{∗} =

− ∑_{γ∈I} λ(w_{γ})w_{γ}^{∗}, and for any α ∈ I, − ∑_{γ∈I} λ(w_{γ})w_{γ}^{∗}(w_{α}) = −λ(w_{α}), so [J, λ] =

−λ. Likewise, for w ∈ W, [J, w] = w, and Jv_{0} = 0. Thus

Jw\underline{α}λ\underline{β}w∗ ; \underline{γ}v_{0} = [J, w\underline{α}λ\underline{β}w∗ ; \underline{γ}]v_{0} + w\underline{α}λ\underline{β}w∗ ; \underline{γ}Jv_{0}
= [J, w\underline{α}λ\underline{β}w∗ ; \underline{γ}]v_{0}
= (ℓ(\underline{α}) − ℓ(\underline{β}) − ℓ(\underline{γ}))w\underline{α}λ\underline{β}w∗ ; \underline{γ}v_{0}.

To see that V_{r} is a 𝔤̃-submodule, it is enough to observe that J commutes with f_{x} and c for all x ∈ 𝔤. But this is trivial, since [J,: w_{α}w_{β}^{∗} :] = [J, w_{α}]w_{β}^{∗} + w_{α}[J, w_{β}^{∗}] = w_{α}w_{β}^{∗} − w_{α}w_{β}^{∗} = 0 for all α, β ∈ I. □

As we will see in Example 6.1, the modules V_{r} need not be irreducible, even if W is irreducible.

6 Examples

In this section, we show that our notion of bosonic and fermionic realizations includes representations on symmetric and skew-symmetric tensors, as well as generalizes the Fock space constructions given by Feingold and Frenkel [5] for types A_{ℓ} and A_{ℓ}^{(1)} and by Gao [10] for { \atop 𝔤𝔩_{N}(ℂ_{q})}, where ℂ_{q} is the quantum torus in two variables. Also, Theorem 4.4 may be used to compute interesting central extensions. We illustrate this with the Virasoro Lie algebra and a central extension of 𝔤𝔩_{N}(A), where A is the ring of differential operators on the punctured plane ℂ^{×}. We anticipate that the techniques of this paper may be used to produce nontrivial representations of toroidal and other interesting Lie algebras, and we plan to investigate these and further applications in a later work.

Example 6.1 (Symmetric and skew-symmetric tensors)

Let 𝔤 be any Lie algebra with a module W. Consider a realization (𝔤, W, J , I, I′) with J = ∅. Then 𝔞^{−} is generated by {w_{α} | α ∈ I} and f_{x}v_{0} = 0 for every x ∈ 𝔤. The defining relations of 𝔞 make it clear that V is the module S(W) of symmetric tensors in the tensor algebra T (W) if ρ = −1, and is the module of skew-symmetric tensors ⨁_{r≥1} ⋀^{r} W, if ρ = 1. The central extension 𝔤̃ = 𝔤 ⊕ 𝕂c is obviously split, since the 2-cocycle c is trivial. Moreover, the submodules

V_{r} ≅ \{ \begin{matrix}S^{r}(W), the rth symmetric power of W & if ρ = −1 \\ ⋀^{r} W & if ρ = 1 \end{matrix}

are seldom irreducible, even if W is irreducible.

For instance, take 𝔤 = 𝔰𝔩_{2}(ℂ), W = S^{m}(ℂ^{2}), J = ∅, and ρ = −1, where ℂ^{2} is the natural module and m > 1. Then W is irreducible, but the Clebsch-Gordan rule (see [9], for instance) gives V_{2} = S^{2}(S^{m}(ℂ^{2})) ≅ ⨁_{0≤n≤ \frac{m}{2}} S^{2m−4n}(ℂ^{2}), so V_{2} is \underline{not} irreducible. Similar arguments can be made if ρ = 1.

Example 6.2 (Oscillator and spinor representations for 𝔤𝔩_{N}(𝕂))

The natural representation W = 𝕂^{N} of the Lie algebra 𝔤 = 𝔤𝔩_{N}(𝕂), with I = {1, 2, …, N} and J = ∅ gives the usual oscillator and spinor representations (as described in [5], for example) for ρ = −1 and ρ = 1, respectively. These are representations that come from the natural isomorphism 𝔤𝔩_{N}(𝕂) ≅ W ⊗W^{∗}, and can be thought of as the model for all bosonic and fermionic representations described in this paper (as discussed in §5).

Example 6.3 (Matrices over associative algebras)

Let A be an associative algebra over the field 𝕂, and let M be an Amodule. The Lie algebra 𝔤𝔩_{N}(A) of N×N matrices over A has a natural “leftmultiplication” action on the space M^{N} of N ×1 column vectors with entries in M. Specifically, let x(a) = (x_{ij}a)_{1≤i,j≤N} where x = (x_{ij})_{1≤i,j≤N} ∈ 𝔤𝔩_{N}(𝕂)

and a ∈ A, and let v(m) = ( v_{1}m ; v_{2}m ; ⋮ ; v_{N}m ) where v = ( v_{1} ; v_{2} ; ⋮ ; v_{N} ) ∈ 𝕂^{N} and

m ∈ M. Then x(a).v(m) = xv(a.m). Realizations of (𝔤𝔩_{N}(A), M^{N}, ρ) give representations for a variety of interesting Lie algebras, including Examples 6.4, 6.5, and 6.7 below.

Example 6.4 (Affine Lie algebra A_{N−1}^{(1)})

In the notation of Example 6.3, let A = 𝕂[t, t^{−1}], and let M = 𝕂[t, t^{−1}] be the (left) regular A-module. Then W = M^{N} is a module for 𝔤 = 𝔤𝔩_{N}(A). Let I = {1, 2 …, N} × ℤ with w_{(i,n)} = e_{i}(n) and J = {1, 2, …, N} × ℕ, where e_{i}(n) = e_{i}(t^{n}) and ℕ denotes the nonnegative integers. Then I′ = ∅, and the realization (𝔤, W, J , I, I′, ρ) is (with the exception of a small change to the normal order)^{4} the Fock space construction given by Feingold and Frenkel for the affine Lie algebra 𝔤̃ = 𝔤⊕𝕂c in [5]. If J = {1, 2, …, N}×ℤ^{+} where ℤ^{+} is the positive integers, then we recover the representation given by Gao ([10], 2.25) for his “vertical Lie algebra” L_{v} ≅ 𝔤̃.

Example 6.5 ({ \atop 𝔤𝔩_{N}(ℂ_{q})})

Let ℂ_{q} = ℂ[x^{±}, y^{±}]_{n.c.}/(yx−qxy) be the quantum torus in two variables.^{5} The associative algebra ℂ_{q} admits a representation on M = ℂ[t, t^{−1}] by x.t^{ℓ} = t^{ℓ+1} and y.t^{ℓ} = q^{ℓ}t^{ℓ}. Taking w_{(i,n)}, I, I′, and W as in Example 6.4, we let J = {1, 2, …, N} × ℤ^{+}. This realization gives Gao’s representation E_{ij}(q^{m}p^{n}) ↦ f_{ij}(m, n) = ∑_{s∈ℤ} q^{ns} : e_{i}(s + m)e_{j}(s)^{∗} :. In order to have the desired commutation relations between the operators f_{ij}(m, n), Gao adjusts his basis of the central extension spanned by f_{ij}(m, n) and 1 by replacing the f_{ij}(m, n) with F_{ij}(m, n) = f_{ij}(m, n) + k(i, j, m, n)1 for some constants k(i, j, m, n) ∈ ℂ. Note that such a change of basis is included in the boundary map ∂ : C^{1} → C^{2} of the Cartan-Eilenberg complex for H^{∗}(𝔤𝔩_{N}(ℂ_{q}); ℂ), so does not alter the 2-cocycle or the central extension

= \frac{1}{2} e_{i}(m)e_{j}(n)^{∗} − \frac{ρ}{2} e_{j}(n)^{∗}e_{i}(m) = \{ \begin{matrix}−e_{j}(n)^{∗}e_{i}(m) − \frac{ρ}{2} & if i = j and m = n \\ e_{i}(m)e_{j}(n)^{∗} & otherwise. \end{matrix}

4The normal ordering used by Feingold and Frenkel is : e_{i}(m)e_{j}(n)^{∗} :

Their change in normal order amounts only to replacing E_{ii}(0) with E_{ii}(0) − \frac{1}{2} c in our representation. They (and Gao [10]) use the notation a_{j}^{∗}(−n) to denote what we call e_{j}(n)^{∗}.

5Here q ∈ ℂ^{×}, and ℂ[x^{±}, y^{±}]_{n.c.} is the space of Laurent polynomials in two noncommuting variables x and y.

described above.^{6}

Example 6.6 (Virasoro Lie algebra)

Let 𝔤 be the Witt algebra Span_{𝕂}{L_{m} | m ∈ ℤ} with commutation relations [L_{m}, L_{n}] = (m − n)L_{m+n} for all m, n ∈ ℤ. The Witt algebra has a natural (and faithful) representation as the derivations L_{m} = −t^{m+1} \frac{d}{dt} on the space of Laurent polynomials, W = 𝕂[t, t^{−1}]. Taking I = ℤ, I′ = ∅, and J = ℤ^{+}, we obtain the 2-cocycle

c(L_{m}, L_{n}) = ∑_{i≥0,j<0}(L_{m})_{i}^{j}(L_{n})_{j}^{i} − ∑_{i<0,j≥0}(L_{m})_{i}^{j}(L_{n})_{j}^{i}.

Clearly L_{m}.t^{j} = −jt^{m+j}, so (L_{m})_{i}^{j} = −jδ_{i,m+j} and

c(L_{m}, L_{n}) = ∑_{i≥0,j<0} ijδ_{i,m+j}δ_{j,n+i} − ∑_{i<0,j≥0} ijδ_{i,m+j}δ_{j,n+i}

= ∑_{i=0}^{m−1} δ_{m,−n}i(i − m) − ∑_{i=m}^{−1} δ_{m,−n}i(i − m)
= δ_{m,−n} (\frac{m − m^{3}}{6} ) .

Adjusting our 2-cocycle c by a constant factor of −\frac{1}{2} gives the usual normalization of the central extension 𝔤̃ = 𝔤 ⊕ 𝕂c, called the Virasoro Lie algebra. It is well-known that this central extension is universal (cf. [1], [13], or [17], for instance).

Example 6.7 (𝔤̂𝔩_{N}(W_{1+∞}))

Let 𝕂[p, q^{±1}]_{n.c.} be the (unital) associative algebra of noncommuting polynomials in two variables p and q, localized at the ideal (q). Let A = 𝕂[p, q^{±1}]_{n.c.}/([pq]−1). The algebra A is the first Weyl algebra A_{1} localized on the multiplicative set {q^{m} | m ∈ ℕ}, and thus has a natural representation on 𝕂[t, t^{−1}] given by p.t^{ℓ} = ℓt^{ℓ−1}, q.t^{ℓ} = t^{ℓ+1}, and q^{−1}.t^{ℓ} = t^{ℓ−1}. If 𝕂 is algebraically closed, A is also the ring of differential operators on the “circle” {(x, y) ∈ 𝕂^{2} | x^{2} + y^{2} = 1} or the scheme 𝕂 \ {0} (cf. [2], [11]).^{7} Viewed as a Lie algebra in the usual way, the algebra A arises as a limit of objects called W_{n}-algebras (see [19], for instance), and is usually denoted W_{1+∞}.

6Gao’s central elements c(n) all act as our central element c and his c_{y} acts as 0.

7which is the punctured plane in the case 𝕂 = ℂ.

In the notation of Example 6.3, W = (𝕂[t, t^{−1}])^{N} is a representation for the Lie algebra 𝔤 = 𝔤𝔩_{N}(A), and we obtain a realization by taking w_{(i,n)} = e_{i}(t^{n}) ∈ W, I = {1, 2, …, N} × ℤ, I′ = ∅, and J = {1, 2, …, N} × ℕ. Let x(k, ℓ) = x(q^{k}p^{ℓ}) where x ∈ 𝔤𝔩_{N}(𝕂), k ∈ ℤ, and ℓ ∈ ℕ. Since the module action is E_{ij}(k, ℓ).w_{(m,n)} = δ_{j,m}(n)_{ℓ}w_{(i,n+k−ℓ)} where (n)_{ℓ} = n(n − 1) ⋯ (n − (ℓ − 1)), we have E_{ij}(k, ℓ)_{(r,s)}^{(m,n)} = δ_{i,r}δ_{j,m}δ_{n+k−ℓ,s}(n)_{ℓ}. Thus the Fock space representation is

E_{ij}(k, ℓ) ↦ ∑_{n∈ℤ}(n)_{ℓ} : w_{(i,n+k−ℓ)}w_{(j,n)}^{∗} :,

and by Theorem 4.4, the resulting 2-cocycle is

c(E_{ij}(k, ℓ), E_{mn}(r, s))

= ∑_{(a,b)∈J ,(u,v)∈I\J} E_{ij}(k, ℓ)_{(a,b)}^{(u,v)}E_{mn}(r, s)_{(u,v)}^{(a,b)}

− ∑_{(a,b)∈I\J ,(u,v)∈J} E_{ij}(k, ℓ)_{(a,b)}^{(u,v)}E_{mn}(r, s)_{(u,v)}^{(a,b)}

= δ_{i,n}δ_{j,m}( ∑_{b≥0,v<0} δ_{v+k−ℓ,b}δ_{b+r−s,v}(v)_{ℓ}(b)_{s}

− ∑_{b<0,v≥0} δ_{v+k−ℓ,b}δ_{b+r−s,v}(v)_{ℓ}(b)_{s})

= δ_{k+r,ℓ+s}δ_{j,m}δ_{i,n}( ∑_{b=0}^{s−r−1}(b + r − s)_{ℓ}(b)_{s} − ∑_{b=s−r}^{−1}(b + r − s)_{ℓ}(b)_{s})

with the convention that empty sums are 0.

By the lemma and identities in the Appendix, if s ≥ r, then

∑_{b=0}^{s−r−1}(b + r − s)_{ℓ}(b)_{s} = (−1)^{ℓ}ℓ!s!\binom{s − r + ℓ}{−r − 1}
= (−1)^{ℓ}ℓ!s!\binom{s − r + ℓ}{s + ℓ + 1}
= (−1)^{s+1}ℓ!s!\binom{r}{s + ℓ + 1}.

Likewise, if s < r, then r > 0, so

∑_{b=s−r}^{−1}(b + r − s)_{ℓ}(b)_{s} = (−1)^{s}ℓ!s!\binom{r}{r − s − ℓ − 1}
= (−1)^{s}ℓ!s!\binom{r}{s + ℓ + 1}.

Hence c(E_{ij}(k, ℓ), E_{mn}(r, s))

\begin{matrix}= δ_{k+r,ℓ+s}δ_{j,m}δ_{i,n}(−1)^{s+1}ℓ!s!\binom{r}{s+ℓ+1} \\ = tr(E_{ij}E_{mn})δ_{k+r,ℓ+s}(−1)^{s+1}ℓ!s!\binom{r}{s+ℓ+1}, \end{matrix}

so by the linearity of the trace,

c(x(k, ℓ), y(r, s)) = tr(xy)δ_{k+r,ℓ+s}(−1)^{s+1}ℓ!s!\binom{r}{ℓ + s + 1}

for any x, y ∈ 𝔤𝔩_{N}(𝕂).

The 2-cocycle c was first computed by Kac-Peterson [12], and the corresponding central extension { \atop 𝔤𝔩_{N}(A)} is universal ([4], [15], [16]). With a slight abuse of notation, we denote the central extension as 𝔤̂𝔩_{N}(W_{1+∞}). By the 2-cocycle formula obtained above, 𝔤̂𝔩_{N}(W_{1+∞}) contains full copies of the affine and Virasoro Lie algebras:

{ \atop 𝔤𝔩_{N}(𝕂)} ≅ Span_{𝕂}\{x(k, 0) | x ∈ 𝔤𝔩_{N}(𝕂), k ∈ ℤ\} ⊕ 𝕂c

V ir ≅ Span_{𝕂}{I(k, 1) | k ∈ ℤ} ⊕ 𝕂c

where I ∈ 𝔤𝔩_{N}(𝕂) is the identity matrix.

7 Appendix: A Combinatorial Lemma

In the statements below, we assume that all variables represent integers, and we follow the standard conventions that empty sums are 0, empty products are 1, and \binom{a}{b} = \frac{(a)_{b}}{b!} where

(a)_{b} = \{ \begin{matrix}a(a − 1) ⋯ (a − (b − 1)) & if b ≥ 0 \\ 0 & b < 0. \end{matrix}

We will use the following well-known identities without explicit mention:

(I1) \binom{a}{b} = (−1)^{b}\binom{b − a − 1}{b}
(I2) ∑_{n=0}^{c} \binom{a}{n}\binom{b}{c − n} = \binom{a + b}{c}

(I3) \binom{d}{b} = \binom{d}{d − b}

for any a, b, c ∈ ℤ and d ∈ ℕ. (See [6], for instance.)

Lemma 7.1 Let m, ℓ, s ≥ 0. Then

(i) ∑_{b=0}^{m−1}(b − m)_{ℓ}(b)_{s} = (−1)^{ℓ}ℓ!s!\binom{m + ℓ}{m − s − 1}

(ii) ∑_{b=−m}^{−1}(b + m)_{ℓ}(b)_{s} = (−1)^{s}ℓ!s!\binom{m + s}{m − ℓ − 1}.

Proof Note that (ii) will follow from (i) by interchanging ℓ and s and making the change of variables b ↦ b + m in (i). Thus we may restrict our attention to the first identity:

∑_{b=0}^{m−1}(b − m)_{ℓ}(b)_{s} = ∑_{b=0}^{m−1}(−1)^{ℓ}(m − b + ℓ − 1)_{ℓ}(b)_{s}
= ∑_{b=s}^{m−1}(−1)^{ℓ}ℓ!s!\binom{m − b + ℓ − 1}{ℓ}\binom{b}{s}
= (−1)^{ℓ}ℓ!s! ∑_{b=s}^{m−1} \binom{m − b + ℓ − 1}{m − b − 1}\binom{b}{b − s}
= (−1)^{ℓ+m−s−1}ℓ!s! ∑_{b=s}^{m−1} \binom{−ℓ − 1}{m − b − 1}\binom{−s − 1}{b − s}.

Changing the index of summation from b to b − s now gives

∑_{b=0}^{m−1}(b − m)_{ℓ}(b)_{s} = (−1)^{ℓ+m−s−1}ℓ!s! ∑_{b=0}^{m−1−s} \binom{−ℓ − 1}{m − b − s − 1}\binom{−s − 1}{b}
= (−1)^{ℓ+m−s−1}ℓ!s!\binom{−ℓ − s − 2}{m − s − 1}
= (−1)^{ℓ}ℓ!s!\binom{m + ℓ}{m − s − 1}.

□

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