Contemporary Mathematics

Two Realizations of Toroidal 𝔰𝔩_{2}(ℂ)

Ben L. Cox

Abstract. In this paper we present free field realizations of the universal central extension of the Lie algebra of trace zero two-by-two matrices with entries in the Laurent polynomials. One of these realizations generically gives a realization of imaginary Verma modules and the other gives Wakimoto type modules.

1. Introduction

In this paper we present two types of free field realizations of the universal central extension of the Lie algebra of trace zero two-by-two matrices with entries in the Laurent polynomials. One of these realizations generically gives a realization of imaginary Verma modules (see [Fut94]) and the other gives Wakimoto type modules. There are so many contributors to this line of work that the author feels incompetent in listing them all and is unable to give them the proper discussion they deserve. A sampling of some of these authors is A. Borel, R. Bott, A. Weil, B. Kostant, J. Bernstein, I. Gelfand and S. Gelfand. Articles more closely related to ours is that of H. Jakobsen and V. Kac [JK85] and then presumably Wakimoto [Wak86] and more generally B. Feigin and E. Frenkel [FF88, FF90b, FF90a]. These mathematicians and mathematical physicists in particular gave geometric constructions of certain representations of affine Kac-Moody algebras in terms of infinite sums of partial differential operators acting on a Fock space. The first realization of toroidal 𝔰𝔩(2) in 2-variables, mentioned in the abstract above, extends Wakimoto’s construction in [Wak86] and the second realization expands on the last section of the paper of D. Bernard and G. Felder (see [BF90]) to toroidal 𝔰𝔩(2) in n-variables n ≥ 2.

We review the Wakimoto type construction as motivation for the realization given in first main result, Theorem 1.9. Let H denote the Heisenberg Lie algebra with generators b_{n}, n ∈ ℤ, 1, subject to the relations

[b_{m}, b_{a}] = mδ_{m,−n}1 [b_{m}, 1] = 0.

If λ, κ ∈ ℂ, we let π_{λ}^{κ} be the H-module generated by a vector |λ⟩, subject to the relations

b_{n}|λ⟩ = 0, n > 0, b_{0}|λ⟩ = λ|λ⟩, 1|λ⟩ = κ|λ⟩.
\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

1991 Mathematics Subject Classification. Primary 17B67, 81R10.

We let A denote the Weyl algebra with generators a_{n} and a_{n}^{∗}, n ∈ ℤ, and relations

[a_{m}, a_{n}^{∗}] = δ_{m,−n}, [a_{m}, a_{n}] = [a_{m}^{∗}, a_{n}^{∗}] = 0.

There is an induced module M with generator |0⟩ that satisfies

a_{n}|0⟩ = 0, n ≥ 0, a_{n}^{∗}|0⟩ = 0, n > 0.

Introduce now the currents

b(z) = ∑_{n∈ℤ} b_{n}z^{−n−1},  h(z) = ∑_{n∈ℤ} h_{n}z^{−n−1},
e(z) = ∑_{n∈ℤ} e_{n}z^{−n−1},  f(z) = ∑_{n∈ℤ} f_{n}z^{−n−1},
a^{∗}(z) = ∑_{n∈ℤ} a_{n}^{∗}z^{−n},  a(z) = ∑_{n∈ℤ} a_{n}z^{−n−1},

where e_{n} = e ⊗ t^{n}, h_{n} = h ⊗ t^{n} and f_{n} = f ⊗ t^{n} are elements in the loop algebra for 𝔰𝔩(2). Then one has

Theorem 1.1 (Wakimoto, see [FBZ01]). Let k ∈ ℂ. The assignment

c ↦ k
e(z) ↦ a(z)
h(z) ↦ −2 : a^{∗}(z)a(z) : +b(z)
f(z) ↦ − : a^{∗}(z)^{2}a(z) : +k∂_{z}a^{∗}(z) + a^{∗}(z)b(z)

where the operators on the right act on M ⊗ π_{λ}^{2(k+2)}, defines a representation of the affine Lie algebra 𝔰̂𝔩(2).

One should compare this with our Theorem 1.9. In fact Theorem 1.9 was constructed using the above result as a starting point.

We briefly review the construction of Bernard and Felder [BF90] and Jakobsen and Kac [JK85] of a Fock space representation of { \atop 𝔰𝔩(n, C)} acting on the algebra ℂ[x_{m}|m ∈ ℤ] ⊗ ℂ[y_{k}|k ∈ ℕ^{∗}]: Jakobsen and Kac give a general realization of the action of 𝔰𝔩(2, R), R a commutative ring, on an induced highest weight module M, that in the case of R = ℂ[t, t^{−1}], t an indeterminate, and M = ℂ[x_{m}|m ∈ ℤ] is given by

f_{m} ↦ x_{m}
e_{m} ↦ − (∑_{i} λ_{i+m} \frac{∂}{∂x_{i}} + ∑_{i,j∈ℤ} x_{i+j+m} \frac{∂}{∂x_{i}} \frac{∂}{∂x_{j}} )

h_{m} ↦ − (λ_{m} + 2 ∑_{j∈ℤ} x_{j+m} \frac{∂}{∂x_{j}} ) .

Here λ_{m} = ∫_{S^{1}} z^{m} dμ where μ denotes a finite measure on the circle S^{1}, not concen-

TWO REALIZATIONS OF TOROIDAL 𝔰𝔩_{2}(ℂ) 3

trated in a finite number of points. One has λ_{m} = δ_{m,0} for μ the Lebesgue measure on S^{1}. (see [JK85]). Note in the above, that although the sums on the right are over ℤ, they collapse to a finite sum when acting on a polynomial in ℂ[x_{m}|m ∈ ℤ].

On the other hand D. Bernard and G. Felder (using a different Borel subalgebra than Feigin and Frenkel in [FF88]) defined an action through the use of a Borel-Weil like construction. The Borel subalgebra they choose is spanned by e_{m}, m ∈ ℤ, and h_{k}, k > 0 which is what H. Jakobsen and V. Kac call a natural Borel subalgebra. They then stipulate that elements g_{+} in the Borel subgroup B̂_{+} of the loop group ŜL_{2} can be written in the form

g_{+} = exp ( ∑_{m∈ℤ} x_{m}e_{m}) exp ( ∑_{m∈ℕ^{∗}} y_{m}h_{m})

where x_{m} and y_{m} are coordinate functions. Similarly one can define the Borel subalgebra B̂_{−}.

The Cartan subgroup H is generated by the exponential of h_{0} and c. A character χ_{K,J} of H is determined by specifying the value of c and h_{0}/2, by two numbers, the central charge K and the spin J respectively. Each character defines a one dimensional representation ℂ_{K;J} of B̂_{−} and from this one can construct the line bundle over ŜL_{2}/B̂_{−};

L_{K;J} := ŜL_{2} ×B̂_{−} ℂ_{K;J}.

Sections of this line bundle can be identified with functions f on ŜL_{2} satisfying

f(gg_{−}) = f(g) and f(gh) = f(g)χ_{K;J}(h)

for g ∈ ŜL_{2}, g_{−} ∈ B̂_{−} and h ∈ H. The loop group ŜL_{2} then acts on sections of L_{K;J} by

(g_{1} · f)(g_{2}) = f(g_{1}^{−1}g_{2}) for g_{1}, g_{2} ∈ ŜL_{2}.

This action can be differentiated to an action ϕ of the affine Lie algebra 𝔰̂𝔩_{2} on sections:

ϕ(x)f := (x · f) = \frac{d}{dt}(e^{tx} · f)|_{t=0} for x ∈ 𝔰̂𝔩_{2}.

In terms of the coordinate functions x_{m} and y_{k}, this action can be explicitly written in the form of differential operators

ϕ(e_{m}) = −\frac{∂}{∂x_{m}} ,
ϕ(h_{m})) = −2 ∑_{j} x_{j−m} \frac{∂}{∂x_{j}} − θ(m)\frac{∂}{∂y_{m}} + θ(−m)2mKy_{−m} + 2δ_{m,0}J,

ϕ(f_{m}) = ∑_{l,j} x_{l}x_{j−l−m} \frac{∂}{∂x_{j}} + ∑_{j>0} x_{j−m} \frac{∂}{∂y_{j}}

+ 2K ∑_{j>0} jy_{j}x_{−j−m} + Kmx_{−m} − 2Jx_{−m}.

Here θ(m) is one if m > 0 and zero otherwise. Note that if one views these operators as acting on the tensor product R = ℂ[x_{m}| m ∈ ℤ] ⊗ ℂ[y_{k}| k ∈ ℕ^{∗}] then the first and third summations for f_{m} can take values in some type of formal completion of R, rather than in R itself. As was noted in D. Bernard and G. Felder’s paper this realization is the same as that of H. Jakobsen and V. Kac’s after twisting the action by an automorphism, specializing to the case where K = 0 and quotienting out ℂ[y_{k}| k ∈ ℕ^{∗}]. More precisely the maps

ρ_{1}(e_{m}) = −f_{−m}, ρ_{1}(h_{m}) = h_{−m}, ρ_{1}(f_{m}) = −e_{−m}, ρ_{1}(c) = c,
ρ_{2}(x_{−m}) = \frac{∂}{∂x_{m}} , ρ_{2} (\frac{∂}{∂x_{m}} ) = x_{−m}, ρ_{2}(y_{k}) = −\frac{∂}{∂y_{k}} , ρ_{2} (\frac{∂}{∂y_{k}} ) = −y_{k}

define anti-automorphisms of ŝl(2, ℂ) and of the algebra of differential operators ℂ⟨x_{m}, ∂/∂x_{m}, y_{k}, ∂/∂y_{k}|m ∈ ℤ, k ∈ ℕ^{∗}⟩ respectively. Applying these two antiautomorphism to the above representation ϕ of D. Bernard and G. Felder lead to the representation ρ = ρ_{2} ∘ ϕ ∘ ρ_{1} so that

ρ(f_{m}) = x_{m},
ρ(h_{m}) = −2 ∑_{j} x_{j+m} \frac{∂}{∂x_{j}} + θ(−m)y_{−m} + θ(m)2mK \frac{∂}{∂y_{m}} + 2δ_{m,0}J,

ρ(e_{m}) = − ∑_{l,j} x_{j+l+m} \frac{∂}{∂x_{l}} \frac{∂}{∂x_{j}} + ∑_{j>0} y_{j} \frac{∂}{∂x_{−j−m}} + 2K ∑_{j>0} j \frac{∂}{∂y_{j}} \frac{∂}{∂x_{j−m}}

+ (Km + 2J)\frac{∂}{∂x_{−m}} .

Note that this action is well defined on the polynomial ring ℂ[x_{m}, y_{k}|m ∈ ℤ, k ∈ ℕ^{∗}].

Appropriate generalizations of the above steps are essentially the ones we used to construct the representation given in our second main result, Theorem 3.1. We have however omitted the tedious and non-rigorous calculations used in finding the explicit form of the realization.

The author would like to thank the organizers S. Berman, P. Fendley, Y. Huang, K. Misra, and B. Parshall for inviting him to the Conference on Infinite-dimensional Lie Theory. A special thanks goes to the referee who pointed out an error in an earlier version of this paper.

1.1. Toroidal Lie algebras with n = 2 variables. Much of the following notation comes from [MRY90] and [FBZ01]. Let

A = (A_{ij})_{i,j=0}^{1} = (\begin{matrix}2 & −2 \\ −2 & 2 \end{matrix}) ,  Â = (Â_{ij})_{i,j=0}^{2} = (\begin{matrix}2 & −2 & −2 \\ −2 & 2 & −2 \\ −2 & −2 & 2 \end{matrix})

and let Q = ℤα_{0} ⊕ ℤα_{1} be the free ℤ-module with basis Π = {α_{0}, α_{1}}. The set Π forms a base for the affine root system Δ sitting inside of Q. Let δ = α_{0} + α_{1} be the null root, ( |) denote the symmetric bilinear form defined by the matrix A and let

Δ_{±} = {±(n + 1)δ, ±α_{0} ± nδ, ±α_{1} ± nδ | n ∈ ℕ}

be the set of positive/negative roots. Let 𝔱 = 𝔱(A) denote the Lie algebra defined over ℂ given by generators

c, α̌_{0,k}, α̌_{1,k} x_{±α_{0},k}, x_{±α_{1},k}, k ∈ ℤ

subject to the relations

(1.1)  [c, α̌_{i,k}] = [c, x_{±α_{i},k}] = 0,
(1.2)  [α̌_{i,m}, α̌_{j,n}] = mA_{ij}δ_{m+n,0}c,
(1.3)  [α̌_{i,m}, x_{±α_{j},n}] = ±A_{ij}x_{±α_{j},m+n},
(1.4)  [x_{α_{i},m}, x_{−α_{j},n}] = −δ_{ij} (α̌_{i,m+n} + mδ_{m+n,0}c) ,
(1.5)  [x_{α_{i},m}, x_{α_{i},n}] = [x_{−α_{i},m}, x_{−α_{i},n}] = 0,
(1.6)  (ad x_{α_{i},0})^{3}x_{α_{j},n} = (ad x_{−α_{i},0})^{3}x_{−α_{j},n} = 0 for i ≠ j

for all m, n ∈ ℤ.

Lemma 1.2. The map ω ∈ End𝔱(2, ℂ)) defined by

ω(x_{±α_{i},m}) = x_{∓α_{j},m},  ω(α̌_{i,m}) = α̌_{j,m}, i ≠ j, ω(c) = −c

is an anti-automorphism.

Proof. Let us check just relations (2)-(4) are preserved by ω:

ω[α̌_{i,m}, α̌_{j,n}] = −mA_{ij}δ_{m+n,0}c = −[α̌_{σi,m}, α̌_{σj,n}] = [ω(α̌_{j,n}), ω(α̌_{i,m})]

ω[α̌_{i,m}, x_{±α_{j},n}] = ±A_{ij}x_{∓α_{σ(j)},m+n} = −[α̌_{σ(i),m}, x_{∓α_{σ(j)},n}]
= −[ω(α̌_{i,m}), ω(x_{±α_{j},n})],
ω[α̌_{i,m}, x_{±α_{1},n}] = ±A_{i1}x_{∓α_{0},m+n} = −[α̌_{j,m}, x_{∓α_{0},n}]
= [ω(α̌_{i,m}), ω(x_{±α_{1},n})],

and finally

ω[x_{α_{i},m}, x_{−α_{i},n}] = − (α̌_{j,m+n} + mδ_{m+n,0}c)
= −[x_{−α_{j},m}, x_{α_{j},n}]
= [ω(x_{α_{i},m}), ω(x_{−α_{i},n})].
□

1.2. Oscillator algebras. Set Γ = Q⊕ℤd and let ( |) be the ℤ-valued bilinear form on Γ given by

( | ) = (\begin{matrix}2 & −2 & 0 \\ −2 & 2 & 1 \\ 0 & 1 & 0 \end{matrix}) .

Set 𝔊 = ℂ ⊗_{ℤ} Γ, 𝔥 = ℂ ⊗_{ℤ} Q and define

𝔟̂ = ⨁_{k∈ℤ} 𝔊_{k} ⊕ ℂκ κ κ  𝔞̂ = ⨁_{k∈ℤ} 𝔥_{k} ⊕ ℂκ κ κ

where κ κ κ is a vector and 𝔊_{k} (resp. 𝔥_{k}) is an isomorphic copy of 𝔊 (resp. 𝔥) with the isomorphism being implemented by γ_{k} ↦ γ. Note that there is a slightly conflicting notation: The set {α_{0}, α_{1}} forms a base of the affine root system Δ, while α_{n}, n ∈ ℤ, α ∈ Δ form part of the a basis of the oscillator or Heisenberg Lie algebra 𝔟̂. It should be clear from the context which notation we are using. We can make 𝔞̂ and 𝔟̂ into infinite dimensional Heisenberg algebras by defining

(1.7) [α_{m}, β_{n}] = m(α|β)δ_{m+n,0}κ κ κ, α, β ∈ 𝔊, m, n ∈ ℤ and ℂκ κ κ central.

Set 𝔟̂_{±} = ⨁_{k∈ℕ^{∗}} 𝔊_{±k}. For fixed κ ∈ ℂ and λ ∈ 𝔊^{∗}, define the 𝔟̂-module, π_{λ}^{κ}(𝔟̂), with generator |λ⟩, such that

β_{m}|λ⟩ = 0, m > 0,  β_{0}|λ⟩ = λ(β)|λ⟩,  κ κ κ|λ⟩ = κ|λ⟩.

In other words since 𝔟̂_{+} ⊕ ℂκ κ κ is a subalgebra of 𝔟̂, the one dimensional 𝔟̂_{+} ⊕ ℂκ κ κmodule ℂ|λ⟩, can be used to construct via induction the 𝔟̂-module

π_{λ}^{κ}(𝔟̂) = U(𝔟̂) ⊗_{𝔟̂_{+}⊕ℂκ κ κ} ℂ|λ⟩.

If we take λ ∈ 𝔥^{∗}, we can define in a similar manner a representation of 𝔞̂ denoted by π_{λ}^{κ} .

1.3. The βγ-system. Let A be the Lie algebra with generators a_{α,m}, a_{β,n}^{∗}, 1, α, β ∈ {α_{0}, α_{1}}, m, n ∈ ℤ and relations

[a_{α,m}, a_{β,n}] = [a_{α,m}^{∗}, a_{β,n}^{∗}] = 0
[a_{α,m}, a_{β,n}^{∗}] = δ_{α,β}δ_{m+n,0}1
[a_{α,m}, 1] = [a_{α,m}^{∗}, 1] = 0.

Lemma 1.3. Consider the involution ¯ : Δ_{+} → Δ_{+} induced by \overline{α_{i}} = α_{j} for i ≠ j ∈ {0, 1}. It preserves the form ( , ) on Δ and it induces the anti-automorphism ω ∈ End(A ⊕ 𝔟̂) given by

ω(a_{α,m}) = a_{ᾱ,m+1}^{∗}, ω(a_{α,m}^{∗}) = a_{ᾱ,m−1},
ω(α_{m}) = ᾱ_{m}, ω(κ) = −κ ω(1) = 1.

Proof. This follows as

ω([a_{α,m}, a_{β,n}^{∗}]) = δ_{α,β}δ_{m+n,0}1 = −[a_{ᾱ,m+1}^{∗}, a_{β̄,n−1}]
ω([α_{m}, β_{n}]) = −m(α|β)δ_{m+n,0}κ = −[ᾱ_{m}, β̄_{n}]
□

Suppose M is a A-module generated by |0⟩, where

a_{α,m}|0⟩ = 0, m ≥ 0, a_{α,m}^{∗}|0⟩ = 0, m > 0

and 1 acts by the identity map. We could take M to be the commutative polynomial algebra

M = ℂ[z_{α_{i},m} | m ∈ ℤ, i = 0, 1].

Then we would have

a_{α,m} = \begin{cases}z_{α,m} & m < 0 \\ ∂/∂_{z_{α,m}} & m ≥ 0 \end{cases}

a_{α,m}^{∗} = \begin{cases}z_{α,−m} & m ≤ 0 \\ −∂/∂_{z_{α,−m}} & m > 0 \end{cases}

and 1 acts as multiplication by 1.

1.4. Formal Distributions and Wick’s Theorem. We need some more notation that will simplify some of the arguments later. This notation follows roughly [Kac98]: A formal distribution is an expression of the form

a(z, w, …) = ∑_{m,n,···∈ℤ} a_{m,n,...}z^{m}w^{n}

where the a_{m,n,...} lie in some fixed vector space V. We define ∂a(z) = ∂_{z}a(z) = ∑_{n} na_{n}z^{n−1}. We also have expansion about zero: There are two canonical embeddings of fields ι_{z,w} : ℂ(z − w) → ℂ[[z, w]] and ι_{w,z} : ℂ(z − w) → ℂ[[z, w]] where ι_{z,w}(a(z, w)) is formal Laurent series expansion in z^{−1} and −ι_{w,z}(a(z, w)) is formal Laurent series expansion in z. The formal delta function δ(z − w) is the formal distribution

δ(z − w) = z^{−1} ∑_{n∈ℤ} (\frac{z}{w} )^{n} = ι_{z,w} (\frac{1}{z − w} ) − ι_{w,z} (\frac{1}{z − w} ) .

For any sequence of elements {A_{(m)}}_{m∈ℤ} in the ring End(V), V a vector space, the formal distribution

A(z) : = ∑_{m∈ℤ} A_{(m)}z^{−m−1}

is called a field if for any v ∈ V, A_{(m)}v = 0 for m ≫ 0. We set

A^{−}(z) : = ∑_{m≥0} A_{(m)}z^{−m−1}, and A^{+}(z) := ∑_{m<0} A_{(m)}z^{−m−1}.

The normal ordered product of two distributions A(z) and B(w) is defined by

: A(z)B(w) := A^{+}(z)B(w) + B(w)A^{−}(z).

In particular

δ^{−}(z − w) = ι_{z,w} (\frac{1}{z − w} ) , δ^{+}(z − w) = −ι_{w,z} (\frac{1}{z − w} ) .

Note that

−∂_{z}δ(z − w) = ∂_{w}δ(z − w) = ι_{z,w} (\frac{1}{(z − w)^{2}} ) − ι_{w,z} (\frac{1}{(z − w)^{2}} ) .

For any positive affine root α ∈ Δ_{+}, we define

a_{α}^{∗}(z) = ∑_{n∈ℤ} a_{α,n}^{∗}z^{−n}, a_{α}(z) = ∑_{n∈ℤ} a_{α,n}z^{−n−1}

and

α(z) = ∑_{n∈ℤ} α_{n}z^{−n−1}.

We also define

x_{±α_{i}}(z) = ∑_{n∈ℤ} x_{±α_{i},n}z^{−n−1}, α̌_{i}(z) = ∑_{n∈ℤ} α̌_{i,n}z^{−n−1}, i = 0, 1.

In this case

[α(z), β(w)] = (α, β)κ∂_{w}δ(z − w),
[a_{α}(z), a_{β}^{∗}(w)] = δ_{α,β}1δ(z − w).

Set

H_{α_{i}}(z) : = ∑_{k=0,1}(α_{i}|α_{k}) : a_{α_{k}}(z)a_{α_{k}}^{∗}(z) :

Note that H_{α_{i}}(z) is well defined by the conditions imposed on M and

H_{α_{i}}(z) : = ∑_{k=0,1}(α_{i}|α_{k}) : a_{α_{k}}(z)a_{α_{k}}^{∗}(z) :

= 2(−1)^{i} ∑_{k,m∈ℤ} (: a_{α_{0},m−k}a_{α_{0},k}^{∗} : − : a_{α_{1},m−k}a_{α_{1},k}^{∗} :) z^{−m−1}

Corollary 1.4. For i ≠ j ∈ {0, 1}, (i). ω(α̌_{i}(z)) = α̌_{j}(z), and ω(α_{i}(z)) = α_{j}(z), (ii). ω(x_{±α_{i}}(z)) = x_{∓α_{j}}(z), (iii). ω(a_{α_{i}}^{∗±}(z)) = a_{α_{j}}^{±}(z), ω(a_{α_{i}}^{±}(z)) = a_{α_{j}}^{∗±}(z), (iv). ω(∂_{z}a_{α_{i}}^{∗±}(z)) = ∂_{z}a_{α_{j}}^{±}(z), (v). ω(H_{α_{i}}(z)) = H_{α_{j}}(z).

Proof. Since

a_{α_{i}}^{∗}(z) = ∑_{n∈ℤ} a_{α_{i},n}^{∗}z^{−n}

we get a_{α_{i},(n)}^{∗} = a_{α_{i},n+1}^{∗} and thus

ω(a_{α_{i}}^{∗+}(z)) = ∑_{n<0} ω(a_{α_{i},n+1}^{∗})z^{−n−1} = ∑_{n<0} a_{α_{j},n}z^{−n−1} = a_{α_{j}}^{+}(z)

Moreover

ω(∂_{z}a_{α_{i}}^{∗}(z)) = ∑_{n∈ℤ} ω(−na_{α_{i},n}^{∗})z^{−n−1} = ∑_{n∈ℤ} −na_{α_{j},n−1}z^{−n−1} = ∂_{z}a_{α_{j}}(z).

As a consequence of this we get

ω(: a_{α_{k}}^{∗}(z)a_{α_{k}}(z) :) = a_{α_{l}}^{+}(z)a_{α_{l}}^{∗}(z) + a_{α_{l}}^{∗}(z)a_{α_{l}}^{−}(z) =: a_{α_{l}}^{∗}(z)a_{α_{l}}(z) :

where l ≠ k. These three results prove the last three items. □

Let

⌊ab⌋ = a(z)b(w)− : a(z)b(w) := [a^{−}(z), b(w)]

(half of [a(z), b(w)]) denote the contraction of any two formal distributions with values in the Lie algebra A ⊕ 𝔟̂. For example

(1.8) ⌊a_{α}a_{β}^{∗}⌋ = ∑_{m≥0} δ_{α,β}z^{−m−1}w^{m} = δ_{α,β}δ^{−}(z − w) = δ_{α,β} ι_{z,w} (\frac{1}{z − w} )

(1.9) ⌊a_{β}^{∗}a_{α}⌋ = − ∑_{n<0} δ_{α,β}z^{n}w^{−n−1} = −δ_{α,β}δ^{+}(w − z) = δ_{α,β} ι_{z,w} (\frac{1}{w − z} ) .

Theorem 1.5 (Wick’s Theorem, see [BS83], [Hua98], or [Kac98] ). Let a_{i}(z) and b_{j}(z) be formal distributions with coefficients in the Lie algebra A⊕𝔟̂, satisfying

(1) [⌊a_{i}b_{j}⌋, c_{k}(z)] = 0, for all i, j, k and c = a or c = b,

(2) [a_{i}^{±}(z), b_{j}^{±}(w)] = 0 for all i and j.

Then

: a_{1}(z)a_{2}(z) ⋯ a_{n}(z) : b(w) = ∑_{i=1}^{n} : a_{1}(z) ⋯ ⌊a_{i}b⌋ ⋯ a_{n}(z) :

and

:a_{1}(z) ⋯ a_{m}(z) :: b_{1}(w) ⋯ b_{n}(w) :=

∑_{s=0}^{min(m,n)} ∑_{{i_{1}<···<is \atop j_{1}≠···≠js}} ⌊a_{i_{1}}b_{j_{1}}⌋ ⋯ ⌊a_{i_{s}}b_{i_{s}}⌋ : a_{1}(z) ⋯ a_{m}(z)b_{1}(w) ⋯ b_{n}(w) :_{(i_{1},...,i_{s};j_{1},...,j_{s})}

where the subscript (i_{1}, …, i_{s}; j_{1}, …, j_{s}) means that those factors a_{i}(z), b_{j}(w) with indices i ∈ {i_{1}, …, i_{s}}, j ∈ {j_{1}, …, j_{s}} are to be omitted from the product : a_{1}(z) ⋯ a_{m}(z)b_{1}(w) ⋯ b_{n}(w) :.

We will need the following two results, the first of which is Taylor’s Theorem.

Theorem 1.6 ([Kac98]). Let a(z) be a formal distribution. Then in the region |z − w| < |w|,

(1.10)  a(z) = ∑_{j=0}^{∞} ∂^{(j)}a(w)(z − w)^{j}.

Theorem 1.7 ([Kac98]). The following are equivalent

(i) [a(z), b(w)] = ∑_{j=0}^{N−1} ∂_{w}^{(j)}δ(z − w)c^{j}(w), where c^{j}(w) ∈ M[[w, w^{−1}]].

(ii) ⌊ab⌋ = ∑_{j=0}^{N−1} ι_{z,w} (\frac{1}{(z − w)^{j+1}} ) c^{j}(w).

In other words the singular part of the operator product expansion

⌊ab⌋ = ∑_{j=0}^{N−1} ι_{z,w} (\frac{1}{(z − w)^{j+1}} ) c^{j}(w)

completely determines the bracket of mutually local formal distributions a(z) and b(w). One writes

a(z)b(w) ∼ ∑_{j=0}^{N−1} \frac{c^{j}(w)}{(z − w)^{j+1}} .

For example

α_{i}(z)α_{j}(w) ∼ \frac{(α_{i}|α_{j})κ κ κ}{(z − w)^{2}}

and the defining relations for 𝔱(2, ℂ) become

(1.11)  α̌_{i}(z)α̌_{j}(w) ∼ (α_{i}|α_{j})c (z − w)^{−2}
(1.12)  α̌_{i}(z)x_{±α_{j}}(w) ∼ ±(α_{i}|α_{j})x_{±α_{j}}(w)(z − w)^{−1}
(1.13)  x_{α_{i}}(z)x_{−α_{j}}(w) ∼ −δ_{i,j}(α̌_{i}(z)(z − w)^{−1} + c(z − w)^{−2})
(1.14)  x_{±α_{i}}(z)x_{±α_{i}}(w) ∼ 0

and the last relation will follow from

(1.15)  x_{±α_{i}}(z_{1})[x_{±α_{i}}(z_{2}), x_{±α_{i}}(z_{3}), x_{±α_{j}}(w)] ∼ 0

10 BEN L. COX

Corollary 1.8. For i, j = 0, 1, α, β, γ ∈ {α_{0}, α_{1}},

H_{α_{i}}(z)a_{α_{l}}(w) ∼ −(α_{i}|α_{l})a_{α_{l}}(z)/ (z − w) ,
H_{α_{i}}(z)a_{α_{l}}^{∗}(w) ∼ (α_{i}|α_{l})a_{α_{l}}^{∗}(z)/(z − w),
H_{α_{i}}(z)∂_{w}a_{α_{l}}^{∗}(w) ∼ (α_{i}|α_{l})a_{α_{l}}^{∗}(z)/(z − w)^{2},
[H_{α_{i}}(z), H_{α_{j}}(w)] = − ( ∑_{k=0,1}(α_{i}|α_{k})(α_{j}|α_{k})) ∂_{w}δ(z − w),
[H_{α_{i}}(z),: a_{α}(w)a_{β}^{∗}(w)a_{γ}^{∗}(w) :] = (α_{i}|β + γ − α) : a_{α}(w)a_{β}^{∗}(w)a_{γ}^{∗}(w) : δ(z − w)

− (α_{i}|α) (δ_{α,β}a_{γ}^{∗}(w) + δ_{α,γ}a_{β}^{∗}(w)) ∂_{w}δ(z − w),

[H_{α_{i}}(z),: a_{α}(w)a_{β}(w)a_{γ}^{∗}(w) :] = (α_{i}|γ − α − β) : a_{α}(w)a_{β}(w)a_{γ}^{∗}(w) : δ(z − w)

− (α_{i}|γ) (δ_{γ,β}a_{γ}(w) + δ_{α,γ}a_{β}(w)) ∂_{w}δ(z − w).

Proof. Now by (1.8) and (1.9) and by Wick’s Theorem

H_{α_{i}}(z)a_{α_{l}}(w) = ∑_{k=0,1}(α_{i}|α_{k}) : a_{α_{k}}(z)a_{α_{k}}^{∗}(z) : a_{α_{l}}(w)

∼ (α_{i}|α_{l})a_{α_{l}}(z) (\frac{1}{w − z} )

and

H_{α_{i}}(z)a_{α_{l}}^{∗}(w) = ∑_{k=0,1}(α_{i}|α_{k}) : a_{α_{k}}(z)a_{α_{k}}^{∗}(z) : a_{α_{l}}^{∗}(w)

∼ (α_{i}|α_{l})a_{α_{l}}^{∗}(z) (\frac{1}{z − w} ) .

The third identity comes from differentiating the second with respect to w.

On the other hand by Wick’s Theorem

: a_{ν}(z)a_{μ}^{∗}(z) :: a_{α}(w)a_{β}^{∗}(w) :
=: a_{α}(w)a_{β}^{∗}(w)a_{ν}(z)a_{μ}^{∗}(z) : −δ_{α,μ}δ^{+}(w − z) : a_{ν}(z)a_{β}^{∗}(w) :
+ δ_{β,ν}δ^{−}(z − w) : a_{α}(w)a_{μ}^{∗}(z) : −δ_{α,μ}δ_{β,ν}ι_{z,w}(z − w)^{−2}.

Thus

H_{α_{i}}(z)H_{α_{j}}(w) = ∑_{k,l=0,1}(α_{i}|α_{k})(α_{j}|α_{l}) : a_{α_{k}}(z)a_{α_{k}}^{∗}(z) :: a_{α_{l}}(w)a_{α_{l}}^{∗}(w) :

= ∑_{k,l=0,1}(α_{i}|α_{k})(α_{j}|α_{l}) : a_{α_{j}}(z)a_{α_{k}}(w)a_{α_{j}}^{∗}(z)a_{α_{k}}^{∗}(w) :

+ ∑_{l=0,1}(α_{i}|α_{l})(α_{j}|α_{l}) : a_{α_{l}}(w)a_{α_{l}}^{∗}(z) : ι_{z,w} \frac{1}{(z − w)}

+ ∑_{k=0,1}(α_{i}|α_{k})(α_{j}|α_{k}) : a_{α_{k}}(z)a_{α_{k}}^{∗}(w) : ι_{z,w} \frac{1}{(w − z)}

− ((α_{i}|α_{0})(α_{j}|α_{0}) + (α_{i}|α_{1})(α_{j}|α_{1})) ι_{z,w} \frac{1}{(z − w)^{2}} ,

which can be rewritten as

[H_{α_{i}}(z), H_{α_{j}}(w)] = ∑_{k=0,1}(α_{i}|α_{k})(α_{j}|α_{k}) (ι_{w,z} \frac{1}{(w − z)^{2}} − ι_{z,w} \frac{1}{(z − w)^{2}} )

= − ∑_{k=0,1}(α_{i}|α_{k})(α_{j}|α_{k})∂_{w}δ(z − w).

Again by (1.8) and (1.9) and by Wick’s Theorem

: a_{α_{k}}(z)a_{α_{k}}^{∗}(z) :: a_{α}(w)a_{β}^{∗}(w)a_{γ}^{∗}(w) :

=: a_{α_{k}}(z)a_{α_{k}}^{∗}(z)a_{α}(w)a_{β}^{∗}(w)a_{γ}^{∗}(w) :

+ δ_{α_{k},α} : a_{α_{k}}(z)a_{β}^{∗}(w)a_{γ}^{∗}(w) : ι_{z,w}(\frac{1}{w − z} )

+ δ_{α_{k},β} : a_{α}(w)a_{α_{k}}^{∗}(z)a_{γ}^{∗}(w) : ι_{z,w}(\frac{1}{z − w} )

+ δ_{α_{k},γ} : a_{α}(w)a_{β}^{∗}(w)a_{α_{k}}^{∗}(z) : ι_{z,w}(\frac{1}{z − w} )

− (δ_{α_{k},α}δ_{α_{k},β}a_{γ}^{∗}(w) + δ_{α_{k},α}δ_{α_{k},γ}a_{β}^{∗}(w)) ι_{z,w}(z − w)^{−2}.

Hence the last identity follows from

H_{α_{i}}(z) : a_{α}(w)a_{β}^{∗}(w)a_{γ}^{∗}(w) := ∑_{k=0,1}(α_{i}|α_{k}) : a_{α_{k}}(z)a_{α_{k}}^{∗}(z) :: a_{α}(w)a_{β}^{∗}(w)a_{γ}^{∗}(w) :

∼ ∑_{k=0,1}(α_{i}|α_{k})(( − δ_{α_{k},α} : a_{α_{k}}(z)a_{β}^{∗}(w)a_{γ}^{∗}(w) :

+δ_{α_{k},β} : a_{α}(w)a_{α_{k}}^{∗}(z)a_{γ}^{∗}(w) : +δ_{α_{k},γ} : a_{α}(w)a_{β}^{∗}(z)a_{α_{k}}^{∗}(z) : )ι_{z,w}(z − w)^{−1} − (δ_{α_{k},α}δ_{α_{k},β}a_{γ}^{∗}(w) + δ_{α_{k},α}δ_{α_{k},γ}a_{β}^{∗}(w)) ι_{z,w}(z − w)^{−2})

∼ ( − (α_{i}|α) : a_{α}(z)a_{β}^{∗}(w)a_{γ}^{∗}(w) : +(α_{i}|β) : a_{α}(w)a_{β}^{∗}(z)a_{γ}^{∗}(w) : +(α_{i}|γ) : a_{α}(z)a_{β}^{∗}(z)a_{μ}^{∗}(w) : )ι_{z,w}(z − w)^{−1} −(α_{i}|α) (δ_{α,β}a_{γ}^{∗}(w) + δ_{α,γ}a_{β}^{∗}(w)) ι_{z,w}(z − w)^{−2}.

□

Theorem 1.9. For any fixed κ ∈ ℂ and λ ∈ 𝔥^{∗}, the vector space M ⊗ π_{λ}^{κ} becomes a representation for 𝔱(2, ℂ) where the action is defined by

π_{λ,κ}(c) = κ − 4,
π_{λ,κ}(x_{−α_{0}}(z)) = a_{α_{0}}^{∗}(z), π_{λ,κ}(x_{α_{1}}(z)) = a_{α_{1}}(z),
π_{λ,κ}(α̌_{i}(z)) = −H_{α_{i}}(z) + α_{i}(z),

π_{λ,κ}(x_{α_{0}}(z)) = : a_{α_{0}}(z)^{2}(z)a_{α_{0}}^{∗}(z) : +(α_{0}|α_{1}) : a_{α_{0}}(z)a_{α_{1}}(z)a_{α_{1}}^{∗}(z) :
−α_{0}(z)a_{α_{0}}(z) + (κ − 4)∂_{z}a_{α_{0}}(z),
π_{λ,κ}(x_{−α_{1}}(z)) = : a_{α_{1}}(z)a_{α_{1}}^{∗}(z)^{2} : +(α_{0}|α_{1}) : a_{α_{0}}(z)a_{α_{0}}^{∗}(z)a_{α_{1}}^{∗}(z) :
−α_{1}(z)a_{α_{1}}^{∗}(z) + (κ − 4)∂_{z}a_{α_{1}}^{∗}(z).

Note that the signs in the description of π_{λ,κ}(x_{α_{0}}(z)) and π_{λ,κ}(x_{−α_{1}}(z)) are minus what appears in Theorem 1.1, due to the minus in the relation (1.4) of the toroidal Lie algebra.

Observe that if λ(α_{0}) = λ(α_{1}), then ωπ_{λ,κ} = π_{λ,κ}ω.

Proof. Let us check that the relations are satisfied. By Wick’s Theorem

π_{λ,κ}(α̌_{i}(z))π_{λ,κ}(α̌_{j}(w)) = (−H_{α_{i}}(z) + α_{i}(z))(−H_{α_{j}}(w) + α_{j}(w))

∼ (−(α_{i}|α_{0})(α_{j}|α_{0}) − (α_{i}|α_{1})(α_{j}|α_{1}) + (α_{i}|α_{j})κ) /(z − w)^{2}

∼ π_{λ,κ} ((α_{i}|α_{j})c ∂_{w}δ(z − w))

By Lemma 1.8 we have

π_{λ,κ}(α̌_{i}(z))π_{λ,κ}(x_{−α_{0}}(w)) = (−H_{α_{i}}(z) + α_{i}(z))a_{α_{0}}^{∗}(w) ∼ −(α_{i}|α_{0})a_{α_{0}}^{∗}(z)/(z − w)
∼ π_{λ,κ}(−A_{i0}x_{−α_{0}}(w)δ(z − w)),

and

π_{λ,κ}(α̌_{i}(z))π_{λ,κ}(x_{α_{1}}(w)) = (−H_{α_{i}}(z) + α_{i}(z))a_{α_{1}}(w) ∼ (α_{i}|α_{1})a_{α_{1}}(z)/(z − w)
∼ π_{λ,κ}(A_{i1}x_{α_{1}}(w)δ(z − w)),

Now using Taylor’s Theorem, Wick’s Theorem, Theorem 1.7, together with Corollary 1.8, we get

π_{λ,κ}(α̌_{i}(z))π_{λ,κ}(x_{α_{0}}(w)) = (−H_{α_{i}}(z) + α_{i}(z))
( : a_{α_{0}}(w)^{2}a_{α_{0}}^{∗}(w) : +(α_{0}|α_{1}) : a_{α_{0}}(w)a_{α_{1}}(w)a_{α_{1}}^{∗}(w) :
− α_{0}(w)a_{α_{0}}(w) + (κ − 4)∂_{w}a_{α_{0}}(w))

∼ ((α_{i}|α_{0}) : a_{α_{0}}(w)^{2}a_{α_{0}}^{∗}(w) : ι_{z,w}(1/(z − w))

+ (α_{0}|α_{1})(α_{0}|α_{i}) : a_{α_{0}}(w)a_{α_{1}}(w)a_{α_{1}}^{∗}(w) : ι_{z,w}(1/(z − w))

+ 2(α_{i}|α_{0})a_{α_{0}}(w)ι_{z,w}(1/(z − w)^{2}) + (α_{i}|α_{1})(α_{0}|α_{1})a_{α_{0}}(w)ι_{z,w}(1/(z − w)^{2})

− (α_{0}|α_{i})κa_{α_{0}}(w)ι_{z,w}(1/(z − w)^{2}) + (α_{0}|α_{i})(κ − 4)a_{α_{0}}(z)ι_{z,w}(1/(z − w)^{2})

− (α_{i}|α_{0})α_{0}(w)a_{α_{0}}(z)/(z − w))

TWO REALIZATIONS OF TOROIDAL 𝔰𝔩_{2}(ℂ) 13

∼ ((α_{i}|α_{0}) : a_{α_{0}}(w)^{2}a_{α_{0}}^{∗}(w) : ι_{z,w}(1/(z − w)) + (α_{0}|α_{1})(α_{0}|α_{i}) : a_{α_{0}}(w)a_{α_{1}}(w)a_{α_{1}}^{∗}(w) : ι_{z,w}(1/(z − w)) + 2(α_{i}|α_{0})a_{α_{0}}(w)ι_{z,w}(1/(z − w)^{2}) + (α_{i}|α_{1})(α_{0}|α_{1})a_{α_{0}}(w)ι_{z,w}(1/(z − w)^{2}) − (α_{0}|α_{i})κa_{α_{0}}(w)ι_{z,w}(1/(z − w)^{2}) + (α_{0}|α_{i})(κ − 4)(a_{α_{0}}(w) + ∂_{w}a_{α_{0}}(w)(z − w))ι_{z,w}(1/(z − w)^{2}) − (α_{i}|α_{0})α_{0}(w)a_{α_{0}}(z)/(z − w))

∼ π_{λ,κ}(A_{i0}x_{α_{0}}(w)δ(z − w)). Next we have

: π_{λ,κ}(x_{α_{0}}(z)) : : π_{λ,κ}(x_{α_{0}}(w)) := ( : a_{α_{0}}(z)^{2}a_{α_{0}}^{∗}(z) : +(α_{0}|α_{1}) : a_{α_{0}}(z)a_{α_{1}}(z)a_{α_{1}}^{∗}(z) :
− α_{0}(z)a_{α_{0}}(z) + (κ − 4)∂_{z}a_{α_{0}}(z) : )
( : a_{α_{0}}(w)^{2}a_{α_{0}}^{∗}(w) : +(α_{0}|α_{1}) : a_{α_{0}}(w)a_{α_{1}}(w)a_{α_{1}}^{∗}(w) :
− α_{0}(w)a_{α_{0}}(w) + (κ − 4)∂_{w}a_{α_{0}}(w) : )

∼ 2 : a_{α_{0}}(z)^{2}a_{α_{0}}(w)a_{α_{0}}^{∗}(w) : ι_{z,w} (1/(w − z)) + 2 : a_{α_{0}}(z)a_{α_{0}}(w)^{2}a_{α_{0}}^{∗}(z) : ι_{z,w} (1/(z − w)) − 4 : a_{α_{0}}(z)a_{α_{0}}(w) : ι_{z,w} (1/(z − w)^{2}) − 2 : a_{α_{0}}(w)^{2}a_{α_{1}}(z)a_{α_{1}}^{∗}(z) : ι_{z,w} (1/(z − w)) − α_{0}(z) : a_{α_{0}}(w)^{2} : ι_{z,w} (1/(z − w)) − (κ − 4) : a_{α_{0}}(w)^{2} : ι_{z,w} (1/(z − w)^{2}) + 2 : a_{α_{0}}(z)^{2}a_{α_{1}}(w)a_{α_{1}}^{∗}(w) : ι_{z,w} (1/(z − w)) + 4( : a_{α_{0}}(z)a_{α_{0}}(w)a_{α_{1}}(z)a_{α_{1}}^{∗}(w) : ι_{z,w} (1/(z − w)) + : a_{α_{0}}(z)a_{α_{0}}(w)a_{α_{1}}^{∗}(z)a_{α_{1}}(w) : ι_{z,w} (1/(w − z)) − : a_{α_{0}}(z)a_{α_{0}}(w) : ι_{z,w} (1/(z − w))^{2} )

+ : α_{0}(w)a_{α_{0}}(z)^{2} : ι_{z,w} (1/(z − w)) + 2κ : a_{α_{0}}(z)a_{α_{0}}(w) : ι_{z,w} (1/(z − w))^{2} − (κ − 4) : a_{α_{0}}(z)^{2} : ι_{z,w} (1/(z − w))^{2}

Under the assumption λ(α_{0}) = λ(α_{1}) we have for i ≠ j ∈ {0, 1},

[π_{λ,κ}(α̌_{i}(z)), π_{λ,κ}(x_{−α_{1}}(w))] = [ωπ_{λ,κ}(ωα̌_{i}(z)), ωπ_{λ,κ}(ωx_{−α_{1}}(w))]
= −ω[π_{λ,κ}α̌_{j}(z)), π_{λ,κ}(x_{α_{0}}(w))]
= −π_{λ,κ}(A_{j0}x_{−α_{1}}(z))δ(z − w))
= −π_{λ,κ}(A_{i1}x_{−α_{1}}(z)δ(z − w)).

All this really means is that the calculation for (a) above holds when applying ω and replacing λ(α_{0}) by λ(α_{1}).

: π_{λ,κ}(x_{α_{0}}(z)) : : π_{λ,κ}(x_{−α_{0}}(w)) :=: a_{α_{0}}(z)^{2}a_{α_{0}}^{∗}(z) : +(α_{0}|α_{1}) : a_{α_{0}}(z)a_{α_{1}}(z)a_{α_{1}}^{∗}(z) :
− α_{0}(z)a_{α_{0}}(z) + (κ − 4)∂_{z}a_{α_{0}}(z) : : a_{α_{0}}^{∗}(w) :

∼ (2 : a_{α_{0}}(z)a_{α_{0}}^{∗}(z) : +(α_{0}|α_{1}) : a_{α_{1}}(z)a_{α_{1}}^{∗}(z) : −α_{0}(z))/(z − w)

− (κ − 4)/(z − w)^{2}
∼ π_{λ,κ}(−(α̌_{0}(z)δ(z − w) + ∂_{w}δ(z − w)c)).

Again under the assumption λ(α_{0}) = λ(α_{1}), we get

[π_{λ,κ}(x_{α_{1}}(z)), π_{λ,κ}(x_{−α_{1}}(w))] = [ωπ_{λ,κ}(ω(x_{α_{1}}(z))), ωπ_{λ,κ}(ω(x_{−α_{1}}(w)))]
= −ωπ_{λ,κ}[x_{−α_{0}}(z), x_{α_{0}}(w))]
= −ωπ_{λ,κ}(α̌_{0}(w)δ(w − z) − ∂_{z}δ(w − z)c)
= π_{λ,κ}(−(α̌_{1}(z)δ(z − w) + ∂_{w}δ(z − w)c)).

Recall Engel’s bracket notation [z_{1}, z_{2}, …, z_{n}] = [z_{1}, [z_{2}, …, [z_{n−1}, z_{n}] ⋯]. Since each bracket with a a_{α_{0}}(z) (resp. a_{α_{1}}(z)) removes the appearance of an a_{α_{0}}^{∗}(z) (resp. a_{α_{1}}^{∗}(z)), it is clear from the defining relations that

[π_{λ,κ}(x_{α_{1}}(z_{1})), π_{λ,κ}(x_{α_{1}}(z_{2})), π_{λ,κ}(x_{α_{1}}(z_{3})), π_{λ,κ}(x_{α_{0}}(w))] = 0,
[π_{λ,κ}(x_{−α_{0}}(z_{1})), π_{λ,κ}(x_{−α_{0}}(z_{2})), π_{λ,κ}(x_{−α_{0}}(z_{3})), π_{λ,κ}(x_{−α_{1}}(w))] = 0.

This proves half of the relations (1.6). For the other two we have

π_{λ,κ}(x_{α_{0}}(z_{3}))π_{λ,κ}(x_{α_{1}}(w))

= ( : a_{α_{0}}(z_{3})a_{α_{0}}(z_{3})a_{α_{0}}^{∗}(z_{3}) : +(α_{0}|α_{1}) : a_{α_{0}}(z_{3})a_{α_{1}}(z_{3})a_{α_{1}}^{∗}(z_{3}) :

− α_{0}(z_{3})a_{α_{0}}(z_{3}) + (κ − 4)∂_{z}a_{α_{0}}(z_{3}))a_{α_{1}}(w)
∼ (α_{0}|α_{1}) : a_{α_{0}}(z_{3})a_{α_{1}}(z_{3}) : /(w − z_{3})

and thus [π_{λ,κ}(x_{α_{0}}(z_{3})), π_{λ,κ}(x_{α_{1}}(w))] = (α_{0}|α_{1}) : a_{α_{0}}(z_{3})a_{α_{1}}(z_{3}) : δ(w − z_{3}). As a consequence, in order to compute [π_{λ,κ}(x_{α_{0}}(z_{2})), π_{λ,κ}(x_{α_{0}}(z_{3})), π_{λ,κ}(x_{α_{1}}(w))], we calculate

π_{λ,κ}(x_{α_{0}}(z_{2}))[π_{λ,κ}(x_{α_{0}}(z_{3})), π_{λ,κ}(x_{α_{1}}(w))] =
( : a_{α_{0}}(z_{2})a_{α_{0}}(z_{2})a_{α_{0}}^{∗}(z_{2}) : +(α_{0}|α_{1}) : a_{α_{0}}(z_{2})a_{α_{1}}(z_{2})a_{α_{1}}^{∗}(z_{2}) :
− α_{0}(z_{2})a_{α_{0}}(z_{2}) + (κ − 4)∂_{z_{2}}a_{α_{0}}(z_{2}))
· (α_{0}|α_{1}) : a_{α_{0}}(z_{3})a_{α_{1}}(z_{2}) : δ(w − z_{3})
∼ (α_{0}|α_{1}) : a_{α_{0}}(z_{2})^{2}a_{α_{1}}(z_{2}) : δ(z_{2} − z_{3})δ(w − z_{3})

Thus

[π_{λ,κ}(x_{α_{0}}(z_{2})),π_{λ,κ}(x_{α_{0}}(z_{3})), π_{λ,κ}(x_{α_{1}}(w))]
= (α_{0}|α_{1}) : a_{α_{0}}(z_{2})^{2}a_{α_{1}}(z_{2}) : δ(z_{2} − z_{3})δ(w − z_{3})

Finally we have

π_{λ,κ}(x_{α_{0}}(z_{1}))[π_{λ,κ}(x_{α_{0}}(z_{2})), π_{λ,κ}(x_{α_{0}}(z_{3})), π_{λ,κ}(x_{α_{1}}(w))] =
( : a_{α_{0}}(z_{1})a_{α_{0}}(z_{1})a_{α_{0}}^{∗}(z_{1}) : +(α_{0}|α_{1}) : a_{α_{0}}(z_{1})a_{α_{1}}(z_{1})a_{α_{1}}^{∗}(z_{1}) :
− α_{0}(z_{1})a_{α_{0}}(z_{1}) + (κ − 4)∂_{z_{1}}a_{α_{0}}(z_{1}))
· (α_{0}|α_{1}) : a_{α_{0}}(z_{2})^{2}a_{α_{1}}(z_{2}) : δ(z_{2} − z_{3})δ(w − z_{3})
∼ 0.

Applying ω as we have done above in the other cases, provides a proof that the last Serre relation is satisfied. □

2. Toroidal 𝔰𝔩(2, ℂ) in n-variables

Fix a decomposition of ℤ^{p+1} = ℤ_{−}^{p+1} ∪ {0} ∪ ℤ_{+}^{p+1} into three subsets such that ℤ_{±}^{p+1} are sets closed under vector addition i.e. for example, if j, k ∈ ℤ_{+}^{p+1}, then j + k ∈ ℤ_{+}^{p+1}. Declare m > 0 if m ∈ ℤ_{+}^{p+1} and m < 0 if m ∈ ℤ_{−}^{p+1}. Define the function θ by

θ(m) = \begin{cases}1 if m > 0 \\ 0 otherwise. \end{cases}

The standard basis for 𝔰𝔩(2, ℂ) is given by

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

We denote the Killing form by (x|y) = tr (xy) and set x(m) = t^{m} ⊗ x for x, y ∈ 𝔰𝔩(2, ℂ) and m ∈ ℤ^{p+1}. In this case the toroidal 𝔱 = 𝔱(2, ℂ) has generators e_{m}, f_{m}, h_{m}, K_{m,i} with m ∈ ℤ^{p+1}, 0 ≤ i ≤ p and defining relations written in the form

(R1)  [h_{m}, h_{n}] = 2 ∑_{i=0}^{p} m_{i}K_{m+n,i}
(R2)  [h_{m}, e_{n}] = 2e_{m+n}
(R3)  [h_{m}, f_{n}] = −2f_{m+n}

(R4)  [e_{m}, f_{n}] = h_{m+n} + ∑_{i=0}^{p} m_{i}K_{m+n,i}
(R5)  [e_{m}, e_{n}] = [f_{m}, f_{n}] = 0.

(R6)  K_{m,i} is central for all i and m and ∑_{i=0}^{p} m_{i}K_{m,i} = 0

16 BEN L. COX

3. Imaginary Verma Type Modules

Theorem 3.1. Fix κ_{m,i} and λ in ℂ subject to the two conditions

∑_{i=0}^{p} m_{i}κ_{m,i} = 0 for all m and ∑_{i=0}^{p} m_{i}κ_{−m−n,i} = 0 for all m > 0, n > 0. The assignment

ϕ_{λ,κ κ κ}(f_{j}) = x_{j},
ϕ_{λ,κ κ κ}(h_{j}) = −2 ∑_{k} x_{j+k}∂_{x_{k}} − 2θ(j) ∑_{l>0} ∂_{y_{l}}( ∑_{i=0}^{p} j_{i}κ_{−j+l,i})

+ θ(−j)y_{−j} − θ(−j) ∑_{l>0} ∂_{y_{l}}( ∑_{i=0}^{p} j_{i}κ_{−j+l,i}) − λδ_{j,0},

ϕ_{λ,κ κ κ}(e_{j}) = − ∑_{k,l} x_{j+k+l}∂_{x_{k}}∂_{x_{l}} + ∑_{k;j+k<0} y_{−j−k}∂_{x_{k}} + ∑_{k} ∂_{x_{k}} ∑_{i=0}^{p} k_{i}κ_{−j−k,i}

− 2 ∑_{k,l;l>0,j+k≥0} ∂_{x_{k}}∂_{y_{l}}( ∑_{i=0}^{p} l_{i}κ_{−j−k+l,i})

− ∑_{k,l;l>0,j+k<0} ∂_{x_{k}}∂_{y_{l}}( ∑_{i=0}^{p} l_{i}κ_{−j−k+l,i}) − λ∂_{x_{−j}}

ϕ_{λ,κ κ κ}(K_{m,i}) = −κ_{−m,i}.

defines a representation of 𝔱 on the space of polynomials ℂ[x_{j}, y_{k}|k < 0, j, k ∈ ℤ^{p+1}].

Observe that all of the infinite sums above collapse to finite sums when applied to a polynomial in ℂ[x_{j}, y_{k}| k < 0, j, k ∈ ℤ^{p+1}].

TWO REALIZATIONS OF TOROIDAL 𝔰𝔩_{2}(ℂ) 17

Proof.

[ϕ_{λ,κ κ κ}(h_{m}), ϕ_{λ,κ κ κ}(h_{n})] = [ − 2 ∑_{k} x_{m+k}∂_{x_{k}}, −2 ∑_{k} x_{n+k}∂_{x_{k}}]

−θ(m)θ(−n)[2 ∑_{l>0} ∂_{y_{l}} ∑_{i=0}^{p} m_{i}κ_{−m+l,i}, y_{−n}]

−θ(−m)θ(−n)[ ∑_{l>0} ∂_{y_{l}} ∑_{i=0}^{p} m_{i}κ_{−m+l,i}, y_{−n}]

−θ(−m)θ(n)[y_{−m}, 2 ∑_{l>0} ∂_{y_{l}} ∑_{i=0}^{p} n_{i}κ_{−n+l,i}]

−θ(−m)θ(−n)[y_{−m}, ∑_{l>0} ∂_{y_{l}} ∑_{i=0}^{p} n_{i}κ_{−n+l,i}]

= −2θ(m)θ(−n) ∑_{i=0}^{p} m_{i}κ_{−m−n,i}, +2θ(−m)θ(n) ∑_{i=0}^{p} n_{i}κ_{−m−n,i}

+ 2θ(−m)θ(−n) ∑_{i=0}^{p} n_{i}κ_{−m−n,i}

= ϕ_{λ,κ κ κ} (∑_{i=0}^{p} m_{i}K_{m+n,i})

[ϕ_{λ,κ κ κ}(h_{m}), ϕ_{λ,κ κ κ}(e_{n})] = [ − 2 ∑_{k} x_{m+k}∂_{x_{k}} − 2θ(m) ∑_{l>0} ∂_{y_{l}} (∑_{i=0}^{p} m_{i}κ_{−m+l,i})

+ θ(−m)y_{−m} − θ(−m) ∑_{l>0} ∂_{y_{l}} (∑_{i=0}^{p} m_{i}κ_{−m+l,i}) ,

− ∑_{k,l} x_{n+k+l}∂_{x_{k}}∂_{x_{l}} + ∑_{k;n+k<0} y_{−n−k}∂_{x_{k}} + ∑_{k} ∂_{x_{k}} ∑_{i=0}^{p} k_{i}κ_{−n−k,i}

− 2 ∑_{\begin{matrix}k,l;l>0, \\ n+k≥0 \end{matrix}}∂_{x_{k}}∂_{y_{l}}(∑_{i=0}^{p} l_{i}κ_{−n−k+l,i}) − ∑_{\begin{matrix}k,l;l>0, \\ n+k<0 \end{matrix}}∂_{x_{k}}∂_{y_{l}}(∑_{i=0}^{p} l_{i}κ_{−n−k+l,i}) − λ∂_{x_{−n}}]

= [ − 2 ∑_{k} x_{m+k}∂_{x_{k}}, − ∑_{k,l} x_{n+k+l}∂_{x_{k}}∂_{x_{l}} + ∑_{k;n+k<0} y_{−n−k}∂_{x_{k}}

+ ∑_{k} ∂_{x_{k}} ∑_{i=0}^{p} k_{i}κ_{−n−k,i} − 2 ∑_{\begin{matrix}k,l;l>0, \\ n+k≥0 \end{matrix}}∂_{x_{k}}∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−n−k+l,i})

− ∑_{\begin{matrix}k,l;l>0, \\ n+k<0 \end{matrix}}∂_{x_{k}}∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−n−k+l,i}) − λ∂_{x_{−n}}]

+ [ − 2θ(m) ∑_{l>0} ∂_{y_{l}} (∑_{i=0}^{p} m_{i}κ_{−m+l,i}) + θ(−m)y_{−m}

− θ(−m) ∑_{l>0} ∂_{y_{l}} (∑_{i=0}^{p} m_{i}κ_{−m+l,i}) , ∑_{k;n+k<0} y_{−n−k}∂_{x_{k}}

−2 ∑_{\begin{matrix}k,l;l>0, \\ n+k≥0 \end{matrix}}∂_{x_{k}}∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−n−k+l,i}) − ∑_{\begin{matrix}k,l;l>0, \\ n+k<0 \end{matrix}}∂_{x_{k}}∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−n−k+l,i}) ]

= −2 ∑_{k,l} x_{m+n+k+l}∂_{x_{k}}∂_{x_{l}} + 2 ∑_{k;m+n+k<0} y_{−m−n−k}∂_{x_{k}}

+ 2 ∑_{k} ∂_{x_{k}} ∑_{i=0}^{p}(m_{i} + k_{i})κ_{−m−n−k,i} − 4 ∑_{\begin{matrix}k,l;l>0, \\ m+n+k≥0 \end{matrix}}∂_{x_{k}}∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−m−n−k+l,i})

− 2 ∑_{\begin{matrix}k,l;l>0, \\ m+n+k<0 \end{matrix}}∂_{x_{k}}∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−m−n−k+l,i}) − 2λ∂_{x_{−m−n}}

−2θ(m) ∑_{k;n+k<0} ∂_{x_{k}} (∑_{i=0}^{p} m_{i}κ_{−m−n−k,i})

− θ(−m) ∑_{k;n+k<0} ∂_{x_{k}} (∑_{i=0}^{p} m_{i}κ_{−m−n−k,i})

− 2θ(−m) ∑_{k;n+k≥0} ∂_{x_{k}} (∑_{i=0}^{p} m_{i}κ_{−n−k−m,i})

− θ(−m) ∑_{k;n+k<0} ∂_{x_{k}} (∑_{i=0}^{p} m_{i}κ_{−n−k−m,i})

= 2ϕ_{λ,κ κ κ} (e_{m+n}) .

[ϕ_{λ,κ κ κ}(e_{m}), ϕ_{λ,κ κ κ}(f_{n})]

= −[ ∑_{k,l} x_{m+k+l}∂_{x_{k}}∂_{x_{l}} − ∑_{k;m+k<0} y_{−m−k}∂_{x_{k}} − ∑_{k} ∂_{x_{k}} ∑_{i=0}^{p} k_{i}κ_{−m−k,i}

+ 2 ∑_{\begin{matrix}k,l;l>0, \\ m+k≥0 \end{matrix}}∂_{x_{k}}∂_{y_{l}}(∑_{i=0}^{p} l_{i}κ_{−m−k+l,i})

+ ∑_{\begin{matrix}k,l;l>0, \\ m+k<0 \end{matrix}}∂_{x_{k}}∂_{y_{l}}(∑_{i=0}^{p} l_{i}κ_{−m−k+l,i}) + λ∂_{x_{−m}}, x_{n}]

= −2 ∑_{k} x_{m+n+k}∂_{x_{k}} + θ(m + n)y_{−m−n} + ∑_{i=0}^{p} n_{i}κ_{−m−n,i}

− 2θ(m + n) ∑_{l>0} ∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−m−n+l,i})

− θ(−m − n) ∑_{l>0} ∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−m−n+l,i}) − λδ_{m+n,0}

= ϕ_{λ,κ κ κ} (h_{m+n} + ∑_{i=0}^{p} m_{i}K_{m+n,i}) .

[ϕ_{λ,κ κ κ}(e_{m}), ϕ_{λ,κ κ κ}(e_{n})]

= [ ∑_{k,l} x_{m+k+l}∂_{x_{k}}∂_{x_{l}} − ∑_{k;m+k<0} y_{−m−k}∂_{x_{k}} − ∑_{k} ∂_{x_{k}} ∑_{i=0}^{p} k_{i}κ_{−m−k,i}

+ 2 ∑_{\begin{matrix}k,l;l>0, \\ m+k≥0 \end{matrix}}∂_{x_{k}}∂_{y_{l}}(∑_{i=0}^{p} l_{i}κ_{−m−k+l,i})+ ∑_{\begin{matrix}k,l;l>0, \\ m+k<0 \end{matrix}}∂_{x_{k}}∂_{y_{l}}(∑_{i=0}^{p} l_{i}κ_{−m−k+l,i}) + λ∂_{x_{−m}},

∑_{k,l} x_{n+k+l}∂_{x_{k}}∂_{x_{l}} − ∑_{k;n+k<0} y_{−n−k}∂_{x_{k}} − ∑_{k} ∂_{x_{k}} ∑_{i=0}^{p} k_{i}κ_{−n−k,i}

+ 2 ∑_{\begin{matrix}k,l;l>0, \\ n+k≥0 \end{matrix}}∂_{x_{k}}∂_{y_{l}}(∑_{i=0}^{p} l_{i}κ_{−n−k+l,i}) + ∑_{\begin{matrix}k,l;l>0, \\ n+k<0 \end{matrix}}∂_{x_{k}}∂_{y_{l}}(∑_{i=0}^{p} l_{i}κ_{−n−k+l,i}) + λ∂_{x_{−n}}]

= [ ∑_{k,l} x_{m+k+l}∂_{x_{k}}∂_{x_{l}}, − ∑_{r;n+r<0} y_{−n−r}∂_{x_{r}} − ∑_{r} ∂_{x_{r}} ∑_{i=0}^{p} r_{i}κ_{−n−r,i}

+ 2 ∑_{\begin{matrix}r,s;s>0, \\ n+r≥0 \end{matrix}}∂_{x_{r}}∂_{y_{s}}(∑_{i=0}^{p} s_{i}κ_{−n−r+s,i}) + ∑_{\begin{matrix}r,s;s>0, \\ n+r<0 \end{matrix}}∂_{x_{r}}∂_{y_{s}}(∑_{i=0}^{p} s_{i}κ_{−n−r+s,i})+ λ∂_{x_{−n}}]

− [ ∑_{k;m+k<0} y_{−m−k}∂_{x_{k}}, ∑_{r,s} x_{n+r+s}∂_{x_{r}}∂_{x_{s}}

+ 2 ∑_{\begin{matrix}r,s;s>0, \\ n+r≥0 \end{matrix}}∂_{x_{r}}∂_{y_{s}} (∑_{i=0}^{p} s_{i}κ_{−n−r+s,i}) + ∑_{\begin{matrix}r,s;s>0, \\ n+r<0 \end{matrix}}∂_{x_{r}}∂_{y_{s}} (∑_{i=0}^{p} s_{i}κ_{−n−r+s,i}) ]

− ∑_{r,s} ∂_{x_{r}}∂_{x_{s}} ∑_{i=0}^{p}(n_{i} + r_{i} + s_{i})κ_{−m−n−r−s,i}

+ 2[ ∑_{\begin{matrix}k,l;l>0, \\ m+k≥0 \end{matrix}}∂_{x_{k}}∂_{y_{l}}(∑_{i=0}^{p} l_{i}κ_{−m−k+l,i}) , ∑_{k,l} x_{n+k+l}∂_{x_{k}}∂_{x_{l}} − ∑_{k;n+k<0} y_{−n−k}∂_{x_{k}}]

+ [ ∑_{\begin{matrix}k,l;l>0, \\ m+k<0 \end{matrix}}∂_{x_{k}}∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−m−k+l,i}) + λ∂_{x_{−m}}, ∑_{k,l} x_{n+k+l}∂_{x_{k}}∂_{x_{l}}

− ∑_{k;n+k<0} y_{−n−k}∂_{x_{k}} + ∑_{k} ∂_{x_{k}} ∑_{i=0}^{p} k_{i}κ_{−n−k,i}

+ 2 ∑_{\begin{matrix}k,l;l>0, \\ n+k≥0 \end{matrix}}∂_{x_{k}}∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−n−k+l,i})

+ ∑_{\begin{matrix}k,l;l>0, \\ n+k<0 \end{matrix}}∂_{x_{k}}∂_{y_{l}} (∑_{i=0}^{p} l_{i}κ_{−n−k+l,i}) + λ∂_{x_{−n}}]

= ∑_{k,l;n+m+k+l<0} y_{−n−m−k−l}∂_{x_{k}}∂_{x_{l}} + ∑_{k,l} ∂_{x_{k}}∂_{x_{l}} ∑_{i=0}^{p}(m_{i} + k_{i} + l_{i})κ_{−n−m−k−l,i}

−2 ∑_{k,l,s;s>0, m+n+k+l≥0}∂_{x_{k}}∂_{x_{l}}∂_{y_{s}} (∑_{i=0}^{p} s_{i}κ_{−m−n−k−l+s,i})

− ∑_{k,l,s;s>0, m+n+k+l<0}∂_{x_{k}}∂_{x_{l}}∂_{y_{s}} (∑_{i=0}^{p} s_{i}κ_{−m−n−k−l+s,i}) − λ ∑_{k,l} δ_{m+n+k+l,0 0 0}∂_{x_{k}}∂_{x_{l}}

− ∑_{k,l;n+m+k+l<0} y_{−n−m−k−l}∂_{x_{k}}∂_{x_{l}}

− 2 ∑_{\begin{matrix}r,k;k+m<0, \\ n+r≥0 \end{matrix}}∂_{x_{r}}∂_{x_{k}} (∑_{i=0}^{p}(m_{i} + k_{i})κ_{−m−n−r−k,i})

− ∑_{\begin{matrix}r,k;k+m<0, \\ n+r<0 \end{matrix}}∂_{x_{r}}∂_{x_{k}} (∑_{i=0}^{p}(m_{i} + k_{i})κ_{−m−n−r−k,i})

− ∑_{r,s} ∂_{x_{r}}∂_{x_{s}} ∑_{i=0}^{p}(n_{i} + r_{i} + s_{i})κ_{−m−n−r−s,i}

+ 2 ∑_{k,l,s;s>0, m+n+k+l≥0}∂_{x_{k}}∂_{x_{l}}∂_{y_{s}} (∑_{i=0}^{p} s_{i}κ_{−m−n−k−l+s,i})

+ 2 ∑_{\begin{matrix}r,k;k+n<0, \\ m+r≥0 \end{matrix}}∂_{x_{r}}∂_{x_{k}} (∑_{i=0}^{p}(n_{i} + k_{i})κ_{−m−n−r−k,i})

+ ∑_{\begin{matrix}r,k;k+n<0, \\ m+r<0 \end{matrix}}∂_{x_{r}}∂_{x_{k}} (∑_{i=0}^{p}(n_{i} + k_{i})κ_{−m−n−r−k,i})

+ ∑_{k,l,s;s>0, m+n+k+l<0}∂_{x_{k}}∂_{x_{l}}∂_{y_{s}} (∑_{i=0}^{p} s_{i}κ_{−m−n−k−l+s,i}) + λ ∑_{k,l} δ_{m+n+k+l,0 0 0}∂_{x_{k}}∂_{x_{l}}

= − ∑_{k,l} ∂_{x_{k}}∂_{x_{l}} ∑_{i=0}^{p} n_{i}κ_{−n−m−k−l,i}

− ∑_{\begin{matrix}l,k;k+m<0, \\ n+l≥0 \end{matrix}}∂_{x_{l}}∂_{x_{k}} (∑_{i=0}^{p}(m_{i} + k_{i})κ_{−m−n−l−k,i})

− ∑_{l,k;k+m<0} ∂_{x_{l}}∂_{x_{k}} (∑_{i=0}^{p}(m_{i} + k_{i})κ_{−m−n−l−k,i})

+ ∑_{l,k} ∂_{x_{l}}∂_{x_{k}} ∑_{i=0}^{p} m_{i}κ_{−m−n−l−k,i}

+ ∑_{\begin{matrix}l,k;l+n<0, \\ m+k≥0 \end{matrix}}∂_{x_{l}}∂_{x_{k}} (∑_{i=0}^{p}(n_{i} + l_{i})κ_{−m−n−l−k,i})

+ ∑_{l,k;l+n<0} ∂_{x_{l}}∂_{x_{k}} (∑_{i=0}^{p}(n_{i} + l_{i})κ_{−m−n−l−k,i}) = 0.

□

Remark 3.2. In order to get a realization of the toroidal-Virasoro algebra (see [BB99] and [ERM94]) one needs to describe how certain derivations act on the representations above. There are at least two other realizations of 𝔱(2, ℂ), one a homogeneous realization and the other a principal realization, that are more closely related to those representations in [ERM94] and [BB99] and for which it is more appropriate to take into account these derivations. We will describe them and some of their properties in a future publication.

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Department of Mathematics, University of Charleston, 66 George Street, Charleston SC 29424

E-mail address: bcox@cofc.edu