Irreducible Modules for Extended Affine Lie Algebras

Yuly Billig^{1∗} and Michael Lau^{2}^{∗}

1Carleton University, School of Mathematics and Statistics, 1125 Colonel By Drive, Ottawa, Ontario, Canada K1S 5B6 Email: billig@math.carleton.ca

2Université Laval, Département de mathématiques et de statistique, 1045, av. de la Médecine, Québec, Québec, Canada G1V 0A6 Email: mlau@mat.ulaval.ca

Abstract. We construct irreducible modules for twisted toroidal Lie algebras and extended affine Lie algebras. This is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules. We illustrate our method with examples of extended affine Lie algebras of Clifford type.

MSC: 17B67, 17B69. Keywords: twisted toroidal Lie algebras, extended affine Lie algebras, thin coverings, twisted modules.

0 Introduction

Extended affine Lie algebras (EALAs) are natural generalizations of the affine Kac-Moody algebras. They come equipped with a nondegenerate symmetric invariant bilinear form, a finite-dimensional Cartan subalgebra, and a discrete root system. Originally introduced in the contexts of singularity theory and mathematical physics, their structure theory has been extensively studied for over 15 years. (See [2, 3, 19] and the references therein.)

Their representations are much less well understood. Early attempts to replicate the highest weight theory of the affine setting were stymied by the lack of a triangular decomposition; later work considered only the untwisted toroidal Lie algebras and a few other isolated examples.

∗Both authors gratefully acknowledge funding from the Natural Sciences and Engineering Research Council of Canada.

As a result of major breakthroughs announced in [3] and [19], it is now clear that, except for (extensions of) matrix algebras over non-cyclotomic quantum tori, every extended affine Lie algebra can be constructed as an extension of a twisted multiloop algebra. These results have inspired the present paper, in which we use a twisting procedure to explicitly obtain irreducible generalized highest weight modules for EALAs associated with every twisted multiloop algebra.

In more detail, let 𝔤 be a finite-dimensional simple Lie algebra over the complex numbers ℂ, with commuting automorphisms σ_{0}, σ_{1}, …, σ_{N} of orders m_{0}, m_{1}, …, m_{N}, respectively. Fix primitive m_{i}th roots of unity ξ_{i} ∈ ℂ for every i, and let each σ_{i} act as an automorphism of the (N +1)-torus 𝕋^{N+1} = (ℂ^{×})^{N+1}, by sending the point (x_{0}, x_{1}, …, x_{N}) ∈ 𝕋^{N+1} to the point (x_{0}, …, ξ_{i}x_{i} …, x_{N}). The twisted multiloop algebra L(𝔤; σ) consists of the σ_{0}, …, σ_{N}-equivariant 𝔤valued regular functions on 𝕋^{N+1}, under pointwise Lie bracket. Next, we take the universal central extension of L(𝔤; σ) and adjoin the Lie algebra of equivariant vector fields on the torus, possibly twisted with a 2-cocycle. This produces the full twisted toroidal Lie algebra

𝔤_{T}(σ) = L(𝔤; σ) ⊕ K_{Λ} ⊕ D_{Λ}.

This Lie algebra does not admit a nondegenerate invariant symmetric bilinear form, and is thus too large to be an extended affine Lie algebra. Indeed, the largest extended affine Lie algebra 𝔤_{E}(σ) associated with L(𝔤; σ) is the Lie algebra obtained by adjoining only the divergence zero vector fields to the universal central extension of L(𝔤; σ):

𝔤_{E}(σ) = L(𝔤; σ) ⊕ K_{Λ} ⊕ S_{Λ}.

See Section 1 for details.

After describing the twisted toroidal Lie algebras, we discuss σ-twists of vertex Lie algebras in Section 2. These structures may be thought of as the twisted analogues of the vertex Lie algebras of Dong, Li, and Mason [9]. They are examples of Γ-twisted formal distribution algebras, a more general construction appearing in the work of Kac [13]. We go on to prove that the twisted toroidal Lie algebra 𝔤_{T}(σ_{0}, 1, …, 1) is a σ_{0}-twist of the (untwisted) full toroidal Lie algebra 𝔤_{T}(1, …, 1).

Past work by one of the authors identifies a quotient V_{T} of the universal enveloping vertex operator algebra (VOA) of 𝔤_{T}(1, …, 1) as a tensor product of a lattice VOA, an affine VOA, and a VOA V_{𝔤𝔩𝔙𝔦𝔯} associated with the affine Lie algebra 𝔤̂𝔩_{N} and the Virasoro Lie algebra. This allows irreducible representations of 𝔤_{T}(1, …, 1) and 𝔤_{E}(1, …, 1) to be constructed from tensor products of modules for the tensor components of V_{T}. See [5, 6] for details.

In the present paper, we show that a tensor product V +_{Hyp} ⊗ W ⊗ L_{𝔤𝔩𝔙𝔦𝔯} of modules for the corresponding lattice VOA, twisted affine Lie algebra 𝔤̂(σ_{0}), and V_{𝔤𝔩𝔙𝔦𝔯} can be given the structure of a module for the Lie algebra 𝔤_{T}(σ_{0}, 1, …, 1). This is done by using general theorems about σ_{0}-twists of vertex Lie algebras to reduce most of the verifications to work previously done in the untwisted case.

To obtain irreducible modules for the Lie algebras 𝔤_{T}(σ_{0}, σ_{1}, …, σ_{N}) and 𝔤_{E}(σ_{0}, σ_{1}, …, σ_{N}), we use the technique of thin coverings introduced in a previous paper [7]. Thin coverings are a tool for constructing graded-simple modules from simple ungraded modules over a graded algebra. By taking a thin covering with respect to ⟨σ_{1}⟩ × ⋯ × ⟨σ_{N}⟩ of an irreducible highest weight module for a twisted affine algebra 𝔤̂(σ_{0}), we produce irreducible representations for the twisted toroidal Lie algebras 𝔤_{T}(σ) and 𝔤_{E}(σ). These lowest energy modules have weight decompositions into finite-dimensional weight spaces, and the action of the centres of 𝔤_{T}(σ) and 𝔤_{E}(σ) is given by a central character.

We illustrate our method by explicitly constructing irreducible representations for two of the more exotic extended affine Lie algebras. In the process, we give a detailed discussion of how to realize Jordan torus EALAs of Clifford type as extensions of twisted multiloop algebras, and how to find the thin coverings used in our construction. Vertex operator representations of some Clifford type EALAs were previously constructed in [23] and [18]. Unlike this earlier work, our construction yields irreducible modules.

1 Twisted Toroidal Lie Algebras

Let 𝔤 be a finite-dimensional simple Lie algebra over the complex numbers ℂ, with commuting automorphisms σ_{0}, σ_{1}, …, σ_{N} of (finite) orders m_{0}, m_{1}, …, m_{N}, respectively. Fix primitive m_{i}th roots of unity ξ_{i} ∈ ℂ for i = 0, 1, …, N. Define two sublattices Γ ⊆ ℤ^{N} and Λ ⊆ ℤ^{N+1}:

Γ = m_{1}ℤ × ⋯ × m_{N}ℤ, Λ = m_{0}ℤ × Γ.

For each s ∈ ℤ^{N+1}, we write s = (s_{0}, s), where s_{0} ∈ ℤ, s ∈ ℤ^{N}, and denote by \overline{s} = (\overline{s}_{0}, \overline{s}) its image under the canonical map ℤ^{N+1} → ℤ^{N+1}/Λ. Likewise, f(t) will denote a Laurent polynomial in the N + 1 variables t_{0}^{±1/m_{0}}, t_{1}^{±1}, …, t_{N}^{±1}. However, in recognition of the special role played by the first variable t_{0}, the multi-index exponential notation t^{r} = t_{1}^{r_{1}}t_{2}^{r_{2}} ⋯ t_{N}^{r_{N}} will be reserved for N-tuples r = (r_{1}, r_{2}, …, r_{N}) ∈ ℤ^{N}.

The Lie algebra 𝔤 has a common eigenspace decomposition

𝔤 = ⨁_{\overline{s}∈ℤ^{N+1}/Λ} 𝔤\overline{s},

where 𝔤\overline{s} = \{x ∈ 𝔤 | σ_{i}x = ξ_{i}^{s_{i}}x for i = 0, 1, …, N\}. The corresponding twisted multiloop algebra

L(𝔤; σ) = ∑_{s∈ℤ^{N+1}} t_{0}^{s_{0}/m_{0}}t^{s} ⊗ 𝔤\overline{s}  (1.1)
⊆ ℂ[t_{0}^{±1/m_{0}}, t_{1}^{±1}, …, t_{N}^{±1}] ⊗ 𝔤  (1.2)

has Lie bracket given by

[f_{1}(t)g_{1}, f_{2}(t)g_{2}] = f_{1}(t)f_{2}(t)[g_{1}, g_{2}].  (1.3)

For simplicity of notation, we sometimes drop the tensor product symbol ⊗, as in (1.3).

Let R = ℂ[t_{0}^{±1/m_{0}}, t_{1}^{±1}, …, t_{N}^{±1}] be the algebra of Laurent polynomials, and let R_{Λ} = ℂ[t_{0}^{±1}, t_{1}^{±m_{1}}, …, t_{N}^{±m_{N}}] ⊆ R.

We will write Ω_{R}^{1} (respectively, Ω_{R_{Λ}}^{1}) for the space of Kähler differentials of R (resp., R_{Λ}). As a left R-module, Ω_{R}^{1} has a natural basis consisting of the 1-forms k_{p} = t_{p}^{−1}dt_{p} for p = 0, …, N. Likewise,

Ω_{R_{Λ}}^{1} = ⨁_{p=0}^{N} R_{Λ}k_{p}.

For each f ∈ R, the differential map d : R → Ω_{R}^{1} is defined as

d(f) = ∑_{p=0}^{N} d_{p}(f)k_{p},

where d_{p} = t_{p} \frac{∂}{∂t_{p}} for p = 0, …, N.

Kassel [16] has shown that the centre K of the universal central extension (R ⊗ 𝔤) ⊕ K of R ⊗ 𝔤 can be realized as

K = Ω_{R}^{1}/d(R).

The multiplication in the universal central extension is given by

[f_{1}(t)x, f_{2}(t)y] = f_{1}(t)f_{2}(t)[x, y] + (x | y)f_{2}d(f_{1}),  (1.4)

for all f_{1}, f_{2} ∈ R and x, y ∈ 𝔤, where (x | y) is a symmetric invariant bilinear form on 𝔤. This form is normalized by the condition that the induced form on the dual of the Cartan subalgebra satisfies (α | α) = 2 for long roots α. Similarly, the Lie algebra L(𝔤; σ) can be centrally extended by K_{Λ} = Ω_{R_{Λ}}^{1}/d(R_{Λ}) using the Lie bracket (1.4). This central extension of the twisted multiloop algebra is also universal [20].

Let D = Der R be the Lie algebra of derivations of R, and let D_{Λ} = Der R_{Λ} ⊆ D. The space D (resp., D_{Λ}) acts on R ⊗ 𝔤 (resp., L(𝔤; σ)) by

[f_{1}(t)d_{a}, f_{2}(t)x] = f_{1}d_{a}(f_{2})x.  (1.5)

There is also a compatible action of D (resp., D_{Λ}) on K (resp., on K_{Λ}) via the Lie derivative:

[f_{1}(t)d_{a}, f_{2}(t)k_{b}] = f_{1}d_{a}(f_{2})k_{b} + δ_{ab}f_{2}d(f_{1}).  (1.6)

The multiplication in the semidirect product Lie algebra (R ⊗ 𝔤) ⊕ K ⊕ D can be twisted by any K-valued 2-cocycle τ ∈ H^{2}(D, K):

[f_{1}(t)d_{a}, f_{2}(t)d_{b}] = f_{1}d_{a}(f_{2})d_{b} − f_{2}d_{b}(f_{1})d_{a} + τ(f_{1}d_{a}, f_{2}d_{b}).  (1.7)

We will use cocycles

τ = μτ_{1} + ντ_{2}  (1.8)

parametrized by μ, ν ∈ ℂ. To define these cocycles, recall that the Jacobian v^{J} of a vector field v = ∑_{a} f_{a}(t)d_{a} is the matrix with (a, b)-entry d_{b}(f_{a}), for 0 ≤ a, b ≤ n. In this notation,

τ_{1}(v, w) = T r(v^{J}d(w^{J}))
τ_{2}(v, w) = T r(v^{J})d(T r(w^{J})),

where T r denotes the trace and the differential map d is defined element-wise on the matrix w^{J}.

The resulting Lie algebra

𝔤_{T} = (R ⊗ 𝔤) ⊕ K ⊕_{τ} D  (1.9)

is called the toroidal Lie algebra. When restricted to D_{Λ}, the cocycle τ restricts to a cocycle (also denoted by τ) in the space H^{2}(D_{Λ}, K_{Λ}). This gives the full twisted toroidal Lie algebra

𝔤_{T}(σ) = L(𝔤; σ) ⊕ K_{Λ} ⊕_{τ} D_{Λ}  (1.10)

with Lie bracket given by (1.4) – (1.7).

We will also consider the closely related (twisted) toroidal extended affine Lie algebra (EALA). A derivation v is called divergence zero (or skew-centroidal) if T r(v^{J}) = 0. We will denote the subalgebra of divergence zero derivations of R (resp., R_{Λ}) by S (resp., S_{Λ}). Note that the cocycle τ_{2} vanishes when restricted to the space S, so when working in the EALA setting, we can assume that τ = μτ_{1}. The toroidal EALA is the Lie algebra

𝔤_{E} = (R ⊗ 𝔤) ⊕ K ⊕_{τ} S ⊆ 𝔤_{T}.  (1.11)

Analogously, the twisted toroidal EALA is the subalgebra

𝔤_{E}(σ) = L(𝔤; σ) ⊕ K_{Λ} ⊕_{τ} S_{Λ} ⊆ 𝔤_{T}(σ).  (1.12)

The Lie algebras 𝔤_{E} and 𝔤_{E}(σ) possess non-degenerate invariant bilinear forms.

2 Vertex Lie Algebras and their σ-Twists

In this section, we describe a general construction that will reduce the work of verifying certain relations in the twisted toroidal setting to verifying the analogous relations in the untwisted setting. We begin by recalling the definition of vertex Lie algebra. In our exposition, we will follow the paper of Dong, Li, and Mason [9]. Similar constructions appear in the work of Kac [12], under the name Lie formal distribution algebra.

Let L be a Lie algebra with basis {u(n), c(−1) | u ∈ U, c ∈ C, n ∈ ℤ}, where U and C are some index sets. Define the corresponding formal fields in L[[z, z^{−1}]]:

u(z) = ∑_{n∈ℤ} u(n)z^{−n−1}  (2.1)
c(z) = c(−1)z^{0},  (2.2)

for each u ∈ U and c ∈ C. Let F be the subspace of L[[z, z^{−1}]] spanned by the fields u(z), c(z), and their derivatives of all orders.

The delta function is defined as

δ(z) = ∑_{j∈ℤ} z^{j}.

Definition 2.3 A Lie algebra L with basis as above is called a vertex Lie algebra if the following two conditions hold:

(VL1) For all u_{1}, u_{2} ∈ U, there exist n ≥ 0 and f_{0}(z), …, f_{n}(z) ∈ F such that

[u_{1}(z_{1}), u_{2}(z_{2})] = ∑_{j=0}^{n} f_{j}(z_{2}) [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{j} δ (\frac{z_{2}}{z_{1}} )] .

(VL2) The element c(−1) is central in L for all c ∈ C.

Let L^{(−)} be the subspace with basis {u(n), c(−1) | u ∈ U, c ∈ C, n < 0}, and let L^{(+)} be the subspace of L with basis {u(n) u ∈ U, n ≥ 0}. Then L = L^{(−)} ⊕ L^{(+)} and both L^{(−)} and L^{(+)} are in fact subalgebras of L.

The universal enveloping vertex algebra V_{L} of a vertex Lie algebra L is the induced module

V_{L} = Ind_{L^{(+)}}^{L}(ℂ𝟙) = U(L^{(−)}) ⊗_{ℂ} 𝟙,  (2.4)

where ℂ𝟙 is a trivial 1-dimensional L^{(+)}-module.

The following result appears as Theorem 4.8 in [9]. (See also [12].)

Theorem 2.5 Let L be a vertex Lie algebra. Then V_{L} has the structure of a vertex algebra with vacuum vector 𝟙. The infinitesimal translation operator T is the derivation of V_{L} given by T (u(n)) = −nu(n − 1) and T (c(−1)) = 0 for all u ∈ U, c ∈ C. The state-field correspondence is defined by the formula

Y (a_{1}(−n_{1} − 1) ⋯ a_{k−1}(−n_{k−1} − 1)a_{k}(−n_{k} − 1)𝟙, z)

=: (\frac{1}{n_{1}!} (\frac{∂}{∂z} )^{n_{1}} a_{1}(z)) ⋯

: (\frac{1}{n_{k−1}!} (\frac{∂}{∂z} )^{n_{k−1}} a_{k−1}(z)) (\frac{1}{n_{k}!} (\frac{∂}{∂z} )^{n_{k}} a_{k}(z)) : ⋯:,

where a_{j} ∈ U and n_{j} ≥ 0, or a_{j} ∈ C and n_{j} = 0.

Next we will define a twisted vertex Lie algebra (cf. [13, 15]).

We consider a vertex Lie algebra L graded by a cyclic group ℤ/mℤ for which the generating fields u(z), c(z) are homogeneous. That is, L = ⨁_{\overline{k}∈ℤ/mℤ} L\overline{k} is a ℤ/mℤ-graded Lie algebra, and there is a decomposition U = ⋃_{\overline{k}∈ℤ/mℤ} U\overline{k} for which u(n) ∈ L\overline{k} and c(−1) ∈ L\overline{0} for all c ∈ C, u ∈ U\overline{k}, and n ∈ ℤ. Let ξ be a primitive mth root of 1. This grading defines an automorphism σ : L → L of order m by σ(x) = ξ^{k}x for x ∈ L\overline{k}.

Let L(σ) be a space with the basis

⋃_{\overline{k}∈ℤ/mℤ} \{\overline{u}(n), \overline{c}(−1) | u ∈ U\overline{k}, n ∈ k/m + ℤ, c ∈ C \} .  (2.6)

We define fields

\overline{u}(z) = ∑_{n∈k/m+ℤ} \overline{u}(n)z^{−n−1} ∈ L(σ)[[z^{−1/m}, z^{1/m}]],  (2.7)
\overline{c}(z) = \overline{c}(−1)z^{0},  (2.8)

for all u ∈ U\overline{k} and c ∈ C. Let \overline{F} be the space spanned by the fields \overline{u}(z), \overline{c}(z), and their derivatives of all orders. The correspondence u(z) ↦ \overline{u}(z), c(z) ↦ \overline{c}(z) extends to a vector space isomorphism : F → \overline{F} commuting with the derivative \frac{d}{dz}.

We will use the twisted delta function δ_{k}(z) = z^{k/m}δ(z) when working with σ-twists of vertex Lie algebras. More precisely, the σ-twist of a vertex Lie algebra L is a vector space L(σ) equipped with a Lie bracket defined by the relations

(T1) [\overline{u}_{1}(z_{1}), \overline{u}_{2}(z_{2})] = ∑_{j=0}^{n} \overline{f}_{j}(z_{2}) [z_{1}^{−1} (\frac{∂}{∂z_{2}} )^{j} δ_{k} (\frac{z_{2}}{z_{1}} )],

where the f_{j} are as in (VL1) and u_{1} ∈ U\overline{k}, u_{2} ∈ U.

(T2) The elements \overline{c}(−1) are central in L(σ) for all c ∈ C,

It follows from [15] that L(σ) is indeed a Lie algebra.

Observe that the twisted affine Kac-Moody algebras are examples of twisted vertex Lie algebras. In this paper, our main example of a vertex Lie algebra will be the full toroidal Lie algebra 𝔤_{T}. Its σ_{0}-twist will be the twisted toroidal Lie algebra 𝔤_{T}(σ_{0}, 1, …, 1).

The next theorem (see e.g., [17]), will be very helpful in our construction of modules for twisted toroidal Lie algebras.

Theorem 2.9 Let L(σ) be a σ-twist of a vertex Lie algebra L, and let V_{L} be the universal enveloping vertex algebra of L. Then every σ-twisted V_{L}-module M is a module for the Lie algebra L(σ).

Let us recall the definition of a twisted module of a vertex algebra [17, 14]. Let σ be an automorphism of order m of a vertex algebra V. Consider the grading of V by the cyclic group (\frac{1}{m}ℤ)/ℤ, where for each coset \overline{α} = k/m + ℤ we define the component

V\overline{α} = \{v ∈ V | σ(v) = ξ^{k}v\} .

For each coset \overline{α} = k/m + ℤ, fix a representative α ∈ \overline{α}. If \overline{α} = ℤ, we set α = 0. Let M be a vector space with a map

Y_{M} : V → End (M)[[z^{1/m}, z^{−1/m}]].  (2.10)

Write Y_{M}(a, z) = ∑_{j∈\overline{α}} a_{(j)}^{M}z^{−j−1}, with each a_{(j)}^{M} ∈ End (M).

Definition 2.11 A vector space M together with a map Y_{M} as above is called a σ-twisted module for V if the following axioms hold:

Y_{M}(a, z) ∈ z^{−α}End (M)[[z, z^{−1}]],  (2.12)

a_{(α+n)}^{M}v = 0 for n ≫ 0,  (2.13)
Y_{M}(𝟙, z) = Id_{M}z^{0},  (2.14)
Y_{M}(T (a), z) = \frac{d}{dz} Y_{M}(a, z),  (2.15)

∑_{j=0}^{∞} \binom{m}{j}Y_{M}(a_{(n+j)}b, z)z^{m−j}

= ∑_{j=0}^{∞}(−1)^{j}\binom{n}{j} (a_{(m+n−j)}^{M}Y_{M}(b, z)z^{j} − (−1)^{n}Y_{M}(b, z)a_{(m+j)}^{M}z^{n−j}) (2.16)

for all a ∈ V\overline{α}, m ∈ α + ℤ, b ∈ V, v ∈ M, and n ∈ ℤ.

Letting n = 0 in the twisted Borcherds identity (2.16), one gets the commutator formula for a ∈ V\overline{α}, b ∈ V\overline{β}, m ∈ \overline{α}, and n ∈ \overline{β}:

[a_{(m)}^{M}, b_{(n)}^{M}] = ∑_{j=0}^{∞} \binom{m}{j} (a_{(j)}b)_{(m+n−j)}^{M} .  (2.17)

The twisted normally ordered product is defined as (see [15])

: Y_{M}(a, z)Y_{M}(b, z) : = Y_{M}(a, z)_{+}Y_{M}(b, z) + Y_{M}(b, z)Y_{M}(a, z)_{−} ,  (2.18)

where

Y_{M}(a, z)_{−} = ∑_{j=0}^{∞} a_{(α+j)}^{M}z^{−α−j−1}, Y_{M}(a, z)_{+} = ∑_{j=−1}^{−∞} a_{(α+j)}^{M}z^{−α−j−1}. (2.19)

Note that when a ∈ V\overline{0}, this coincides with the usual normal ordered product. Letting m = α and n = −1 in the twisted Borcherds identity, one gets:

∑_{j=0}^{∞} \binom{α}{j}Y_{M}(a_{(−1+j)}b, z)z^{−j} =: Y_{M}(a, z)Y_{M}(b, z) : .  (2.20)

3 Representations of Twisted Toroidal Lie Algebras

In this section, we use the representation theory of the full toroidal Lie algebras to construct irreducible representations for the full twisted toroidal Lie algebras. We begin by describing the toroidal vertex operator algebra (VOA) that controls the representation theory of the full toroidal Lie algebra 𝔤_{T}. We then show that twisted modules for this toroidal VOA yield representations of the twisted toroidal Lie algebra 𝔤_{T}(σ_{0}, 1, …, 1). Finally, we realize irreducible modules for 𝔤_{T}(σ_{0}, σ_{1}, …, σ_{N}) as subspaces of the irreducible modules for 𝔤_{T}(σ_{0}, 1, …, 1) using thin coverings.

3.1 The toroidal VOA V_{T}

The toroidal vertex operator algebra V_{T} that controls the representation theory of the full toroidal Lie algebra 𝔤_{T} is a quotient of the universal enveloping vertex algebra of 𝔤_{T} [5]. It is a tensor product of three VOAs:

V_{T} = V +_{Hyp} ⊗ V_{aff} ⊗ V_{𝔤𝔩𝔙𝔦𝔯}.  (3.1)

Here V +_{Hyp} is a sub-VOA of a lattice VOA, V_{aff} is an affine VOA, and V_{𝔤𝔩𝔙𝔦𝔯} is a twisted 𝔤̂𝔩_{N}-Virasoro VOA. We will give brief descriptions of these VOAs and refer to [5] for details.

The vertex operator algebra V +_{Hyp} is a sub-VOA of a lattice vertex algebra associated with a hyperbolic lattice. As a vector space, it is a tensor product of a Laurent polynomial algebra with a Fock space:

V +_{Hyp} = ℂ[q_{1}^{±1}, …, q_{N}^{±1}] ⊗ 𝔉,

𝔉 = ℂ[u_{pj}, v_{pj} | p = 1, 2, …, N, j = 1, 2, 3, …].

In the description of the action of 𝔤_{T} on V_{T}, we will use the following vertex operators:

K_{0}(r, z) = Y (q^{r}, z)
= q^{r} exp (∑_{p=1}^{N} r_{p} ∑_{j=1}^{∞} u_{pj}z^{j}) exp (− ∑_{p=1}^{N} r_{p} ∑_{j=1}^{∞} \frac{z^{−j}}{j} \frac{∂}{∂v_{pj}} ) ,

K_{a}(z) = Y (u_{a1}, z) = ∑_{j=1}^{∞} ju_{aj}z^{j−1} + ∑_{j=1}^{∞} \frac{∂}{∂v_{aj}} z^{−j−1},
K_{a}(r, z) = Y (u_{a1}q^{r}, z) = K_{a}(z)K_{0}(r, z),
D_{a}(z) = Y (v_{a1}, z) = ∑_{j=1}^{∞} jv_{aj}z^{j−1} + q_{a} \frac{∂}{∂q_{a}} z^{−1} + ∑_{j=1}^{∞} \frac{∂}{∂u_{aj}} z^{−j−1},

ω_{Hyp}(z) = Y (∑_{p=1}^{N} u_{p1}v_{p1}, z) = ∑_{p=1}^{N} : K_{p}(z)D_{p}(z) : ,

for a = 1, 2, …, N and r ∈ ℤ^{N}. The last expression is the Virasoro field of this VOA, and the rank of V +_{Hyp} is 2N.

Remark. The vertex algebra V +_{Hyp} has a family of modules

M_{Hyp}(α, β) = e^{βv}q^{α}ℂ[q_{1}^{±1}, …, q_{N}^{±1}] ⊗ 𝔉,  (3.2)

where α ∈ ℂ^{N}, β ∈ ℤ^{N}, and βv = β_{1}v_{1} + ⋯ + β_{N}v_{N}. See [5] for the description of the action of V +_{Hyp} on M_{Hyp}(α, β). All constructions of modules in this paper admit a straightforward generalization by shifting the algebra of Laurent polynomials by the factor e^{βv}q^{α} and replacing Y (q^{r}, z) with Y_{M_{Hyp}(α,β)}(q^{r}, z). For the sake of simplicity of exposition, we will not be using these modules in the present paper.

The second factor in (3.1) is the usual affine vertex operator algebra V_{aff} of noncritical level c ∈ ℂ associated with the affine Lie algebra

𝔤̂ = (𝔤 ⊗ ℂ[t_{0}, t_{0}^{−1}]) ⊕ ℂC_{aff}.

We denote its Virasoro field by ω_{aff}(z). The corresponding Virasoro algebra has central charge c dim 𝔤/(c + h^{∨}), where h^{∨} is the dual Coxeter number of 𝔤. This material may be found in any of the introductory books on vertex operator algebras. (See [12], for instance.)

The remaining VOA in the tensor product (3.1) is associated with the twisted 𝔤̂𝔩_{N}-Virasoro algebra 𝔤𝔩𝔙𝔦𝔯, which is the universal central extension of the Lie algebra

(ℂ[t_{0}, t_{0}^{−1}] ⊗ 𝔤𝔩_{N}(ℂ)) ⋊ Der ℂ[t_{0}, t_{0}^{−1}].  (3.3)

This central extension is obtained by adjoining a 4-dimensional space spanned by the basis {C_{𝔰𝔩_{N}} , C_{Heis}, C_{V ir}, C_{V H}}.

We fix the natural projections

ψ_{1} : 𝔤𝔩_{N}(ℂ) → 𝔰𝔩_{N}(ℂ)  (3.4)
ψ_{2} : 𝔤𝔩_{N}(ℂ) → ℂ  (3.5)

where ψ_{2}(u) = T r(u)/N, ψ_{1}(u) = u − ψ_{2}(u)I, and I is the N × N identity matrix. The multiplication in 𝔤𝔩𝔙𝔦𝔯 is given by

[L(n), L(m)] = (n − m)L(n + m) + \frac{n^{3} − n}{12}δ_{n+m,0}C_{V ir}
[L(n), u(m)] = −mu(n + m) − (n^{2} + n)δ_{n+m,0}ψ_{2}(u)C_{V H}
[u(n), v(m)] = [u, v](n + m)

+nδ_{n+m,0}(T r (ψ_{1}(u)ψ_{1}(v))C_{𝔰𝔩_{N}} + ψ_{2}(u)ψ_{2}(v)C_{Heis}),

where L(n) is the Virasoro operator −t_{0}^{n+1} \frac{∂}{∂t_{0}} and u(m) = t_{0}^{m}⊗u for u ∈ 𝔤𝔩_{N}(ℂ).

The twisted 𝔤̂𝔩_{N}-Virasoro algebra 𝔤𝔩𝔙𝔦𝔯 is a vertex Lie algebra [5, Prop 3.5], and let V_{𝔤𝔩𝔙𝔦𝔯} be its universal enveloping vertex algebra with central charge given by a central character γ. We write

c_{𝔰𝔩_{N}} = γ(C_{𝔰𝔩_{N}}),
c_{Heis} = γ(C_{Heis}),
c_{V ir} = γ(C_{V ir}),
c_{V H} = γ(C_{V H}).

The Virasoro field of V_{𝔤𝔩𝔙𝔦𝔯} is

ω_{𝔤𝔩𝔙𝔦𝔯}(z) = Y (L(−2)𝟙, z) = ∑_{j∈ℤ} L(j)z^{−j−2}.

For i, j = 1, …, N, let

E_{ij}(z) = Y (E_{ij}(−1)𝟙, z) = ∑_{k∈ℤ} E_{ij}(k)z^{−k−1},

where E_{ij} ∈ 𝔤𝔩_{N} is the matrix with 1 in the (i, j)-position, and zero elsewhere.

3.2 Representations of 𝔤_{T}(σ_{0}, 1, …, 1)

For a = 1, …, N and r = (r_{0}, r) ∈ ℤ^{N+1}, we now define fields (in a single variable z) whose Fourier coefficients span the Lie algebra 𝔤_{T}(σ_{0}, 1, …, 1):

k_{0}(r, z) = ∑_{j∈ℤ} t_{0}^{j}t^{r}k_{0}z^{−j},  (3.6)
k_{a}(r, z) = ∑_{j∈ℤ} t_{0}^{j}t^{r}k_{a}z^{−j−1},  (3.7)
x(r, z) = ∑_{j∈r_{0}/m_{0}+ℤ} t_{0}^{j}t^{r}xz^{−j−1} for each x ∈ 𝔤\overline{r},  (3.8)
d̃_{a}(r, z) = ∑_{j∈ℤ} (t_{0}^{j}t^{r}d_{a} − νr_{a}t_{0}^{j}t^{r}k_{0})z^{−j−1},  (3.9)
d̃_{0}(r, z) = − ∑_{j∈ℤ} (t_{0}^{j}t^{r}d_{0} − (μ + ν) (j + \frac{1}{2}) t_{0}^{j}t^{r}k_{0}) z^{−j−2}, (3.10)

where μ and ν are the parameters of the cocycle τ = μτ_{1}+ντ_{2}, as in (1.8). In the case where σ_{0} = 1 (and m_{0} = 1), these fields may be viewed as the generating fields of the (untwisted) full toroidal Lie algebra 𝔤_{T}. (See [5] for details.)

We consider the commutation relations between these fields. Most of these relations can be taken directly from work done for the untwisted case [5, Eqn 5.7]. The only exceptions are those relations involving fractional powers of z– namely, the relations involving the field x(r, z). Verifying these relations is a completely straightforward calculation. For x ∈ 𝔤\overline{r}, y ∈ 𝔤\overline{s}, a = 1, …, N, and i = 0, …, N, we see that

[x(r, z_{1}), y(s, z_{2})] = [x, y](r + s, z_{2}) [z_{1}^{−1}δ_{r_{0}} (\frac{z_{2}}{z_{1}} )]

+(x|y)k_{0}(r + s, z_{2}) [z_{1}^{−1} \frac{∂}{∂z_{2}} δ_{r_{0}} (\frac{z_{2}}{z_{1}} )]

+(x|y) ∑_{p=1}^{N} r_{p}k_{p}(r + s, z_{2}) [z_{1}^{−1}δ_{r_{0}} (\frac{z_{2}}{z_{1}} )] , (3.11)
[d̃_{a}(r, z_{1}), y(s, z_{2})] = s_{a}y(r + s, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )] ,  (3.12)
[d̃_{0}(r, z_{1}), y(s, z_{2})] = \frac{∂}{∂z_{2}} (y(r + s, z_{2}) [z_{1}^{−1}δ (\frac{z_{2}}{z_{1}} )]) ,  (3.13)
[x(r, z_{1}), k_{i}(s, z_{2})] = 0.  (3.14)

The above computations demonstrate that the twisted toroidal Lie algebra 𝔤_{T}(σ_{0}, 1, …, 1) is a σ_{0}-twist of the untwisted toroidal Lie algebra 𝔤_{T}. Indeed, the automorphism σ_{0} : 𝔤 → 𝔤 naturally lifts to an automorphism

σ_{0} : 𝔤_{T} → 𝔤_{T}  (3.15)

by setting σ_{0}(t_{0}^{r_{0}}t^{r}x) = t_{0}^{r_{0}}t^{r}σ_{0}(x) = ξ_{0}^{r_{0}}t_{0}^{r_{0}}t^{r}x for each x ∈ 𝔤\overline{r} and letting σ_{0} act trivially on D and K. Comparing (3.11)–(3.14) with the corresponding computations in the untwisted toroidal case [5, Eqn. 5.7], we have now verified the following proposition:

Proposition 3.16 The Lie algebra 𝔤_{T}(σ_{0}, 1, …, 1) is a σ_{0}-twist of the vertex Lie algebra 𝔤_{T}. □

It was shown in [5] that for certain central characters γ, the toroidal VOA V_{T} = V +_{Hyp} ⊗ V_{aff} ⊗ V_{𝔤𝔩𝔙𝔦𝔯} is a quotient of the universal enveloping vertex algebra V_{𝔤_{T}}. We will see that the automorphism σ_{0} induces an automorphism (again denoted by σ_{0}) of V_{T} that is compatible with the natural lift of σ_{0} to V_{𝔤_{T}}. Then every σ_{0}-twisted V_{T}-module is a σ_{0}-twisted V_{𝔤_{T}}-module, and also a 𝔤_{T}(σ_{0}, 1, …, 1)-module by Theorem 2.9.

Theorem 3.17 [5] (i) Let V_{aff} be the universal enveloping vertex algebra for 𝔤̂ at nonzero, non-critical level c. Let V_{𝔤𝔩𝔙𝔦𝔯} be the universal enveloping vertex algebra of the Lie algebra 𝔤𝔩𝔙𝔦𝔯 with the following central character:

c_{𝔰𝔩_{N}} = 1 − μc, c_{Heis} = N(1 − μc) − N^{2}νc,
c_{V H} = N(\frac{1}{2} − νc), c_{V ir} = 12c(μ + ν) − 2N − \frac{c dim 𝔤}{c + h^{∨}},  (3.18)

where μ and ν are as in (1.8). Then there exists a homomorphism of vertex algebras

ϕ : V_{𝔤_{T}} → V +_{Hyp} ⊗ V_{aff} ⊗ V_{𝔤𝔩𝔙𝔦𝔯},

defined by the correspondence of fields:

k_{0}(r, z) ↦ cK_{0}(r, z),  (3.19)
k_{a}(r, z) ↦ cK_{a}(r, z),  (3.20)
x(r, z) ↦ Y (x(−1)𝟙, z)K_{0}(r, z),  (3.21)
d̃_{a}(r, z) ↦ : D_{a}(z)K_{0}(r, z) : + ∑_{p=1}^{N} r_{p}E_{pa}(z)K_{0}(r, z),  (3.22)
d̃_{0}(r, z) ↦ : (ω_{Hyp}(z) + ω_{aff}(z) + ω_{𝔤𝔩𝔙𝔦𝔯}(z))K_{0}(r, z) :

+ ∑_{1≤i,j≤N} r_{i}K_{j}(z)E_{ij}(z)K_{0}(r, z)

+(μc − 1) ∑_{p=1}^{N} r_{p} (\frac{∂}{∂z} K_{p}(z)) K_{0}(r, z),  (3.23)

for all r ∈ ℤ^{N}, x ∈ 𝔤, and a = 1, …, N.

(ii) Let U be an irreducible V_{aff}-module of nonzero, non-critical level c and let L_{𝔤𝔩𝔙𝔦𝔯} be an irreducible V_{𝔤𝔩𝔙𝔦𝔯}-module with the above central character. Then (3.19)-(3.23) define the structure of an irreducible 𝔤_{T}-module on

ℂ[q_{1}^{±1}, …, q_{N}^{±1}] ⊗ 𝔉 ⊗ U ⊗ L_{𝔤𝔩𝔙𝔦𝔯}.
□

The automorphism σ_{0} : 𝔤 → 𝔤 lifts to an automorphism of 𝔤_{T} as in (3.15). It then lifts in the obvious way to a VOA automorphism σ_{0} : V_{𝔤_{T}} → V_{𝔤_{T}}. It can also be extended to an automorphism 𝔤̂ → 𝔤̂ and to σ_{0} : V_{aff} → V_{aff} by setting

σ_{0}(f(t_{0})x) = f(t_{0})σ_{0}(x), σ_{0}(C_{aff}) = C_{aff},

for all x ∈ 𝔤. This lets us identify σ_{0} with the map 1 ⊗ σ_{0} ⊗ 1 on the tensor product V_{T}:

1 ⊗ σ_{0} ⊗ 1 : V +_{Hyp} ⊗ V_{aff} ⊗ V_{𝔤𝔩𝔙𝔦𝔯} → V +_{Hyp} ⊗ V_{aff} ⊗ V_{𝔤𝔩𝔙𝔦𝔯}.

Lemma 3.24 The automorphisms σ_{0} on V_{𝔤_{T}} and V_{T} are compatible with the homomorphism ϕ : V_{𝔤_{T}} → V_{T} in the sense that

σ_{0} ∘ ϕ = ϕ ∘ σ_{0}.

Proof Note that the only field of (3.6)–(3.10) that is not fixed by σ_{0} is x(r, z):

σ_{0}(x(r, z)) = ξ_{0}^{r_{0}}x(r, z)

for all x ∈ 𝔤\overline{r}. The action of σ_{0} on the right-hand sides of (3.19)–(3.23) is clearly also trivial, with the exception of its action on (3.21) and (3.23). Note that the only term of (3.23) on which σ_{0} can act nontrivially is the affine Virasoro field

ω_{aff}(z) = \frac{1}{2(c + h^{∨})} ∑_{i=1}^{dim 𝔤} : x_{i}(z)x^{i}(z) :,

where {x_{i}} and {x^{i}} are dual bases of 𝔤, relative to the normalized invariant bilinear form. These bases can be chosen to consist of eigenvectors for σ_{0}. Since the invariant bilinear form is σ_{0}-invariant (Lemma A.1 in Appendix below), we see that the product of eigenvalues of x_{i} and x^{i} is 1, for each i. This means that ω_{aff}(z) is also fixed by σ_{0}. On the remaining term (3.21), it is clear that for homogeneous x ∈ 𝔤, the eigenvalues of σ_{0} agree on the left- and right-hand sides. Therefore, ϕ ∘ σ_{0} = σ_{0} ∘ ϕ. □

Let W be an irreducible σ_{0}-twisted module of the affine vertex operator algebra V_{aff}. The following lemma says that the tensor product V +_{Hyp}⊗W ⊗L_{𝔤𝔩𝔙𝔦𝔯} is a σ_{0}-twisted module of V_{T} = V +_{Hyp} ⊗ V_{aff} ⊗ V_{𝔤𝔩𝔙𝔦𝔯}.

Lemma 3.25 Let A and B be vertex operator algebras, and assume that A is equipped with a finite-order automorphism η. Extend η to A ⊗ B as η ⊗ 1 : A ⊗ B → A ⊗ B. If U is an η-twisted module for A and V is a module for B, then U ⊗ V is an η-twisted module for A ⊗ B.

Proof This lemma is a straightforward consequence of [17, Prop. 3.17]. □

Corollary 3.26 Let W be an irreducible σ_{0}-twisted V_{aff}-module. Then the tensor product V +_{Hyp} ⊗ W ⊗ L_{𝔤𝔩𝔙𝔦𝔯} is a module for the Lie algebra 𝔤_{T}(σ_{0}, 1, …, 1).

Proof By Lemma 3.25, V +_{Hyp} ⊗ W ⊗ L_{𝔤𝔩𝔙𝔦𝔯} is a σ_{0}-twisted module for V_{T}. By Lemma 3.24, every σ_{0}-twisted V_{T}-module is also a σ_{0}-twisted V_{𝔤_{T}}-module. Finally, since 𝔤_{T}(σ_{0}, 1, …, 1) is a σ_{0}-twist of 𝔤_{T}, we see that V +_{Hyp} ⊗ W ⊗ L_{𝔤𝔩𝔙𝔦𝔯} is a module for 𝔤_{T}(σ_{0}, 1, …, 1), by Theorem 2.9. □

Note that by Theorem 2.9, σ_{0}-twisted V_{aff}-modules are bounded modules for the twisted affine Lie algebra

𝔤̂(σ_{0}) = ∑_{j∈ℤ} t_{0}^{j/m_{0}}𝔤\overline{j} ⊕ ℂC_{aff}.

In order to explicitly describe the action of the Lie algebra 𝔤_{T}(σ_{0}, 1, …, 1), we need to modify formulas (3.21) and (3.23) in (3.19)–(3.23).

The twisted field x(r, z) = ∑_{j∈r_{0}/m_{0}+ℤ} t_{0}^{j}t^{r}xz^{−j−1} for each x ∈ 𝔤\overline{r}, is represented by the twisted vertex operator

Y_{W}(x(−1)𝟙, z)K_{0}(r, z),  (3.27)

where Y_{W}(x(−1)𝟙, z) represents the action of the twisted affine field x(z) = ∑_{j∈r_{0}/m_{0}+ℤ} t_{0}^{j}xz^{−j−1} on the module W.

In (3.23), the Virasoro field ω_{aff}(z) is replaced with the twisted vertex operator Y_{W}(ω_{aff}, z). The latter operator may be written down using (2.20) (cf. [15]):

Y_{W}(ω_{aff}, z) = \frac{1}{2(c + h^{∨})}( ∑_{i} : Y_{W}(x_{i}(−1)𝟙, z)Y_{W}(x^{i}(−1)𝟙, z) :

− ∑_{i} α_{i}z^{−1}Y_{W}([x_{i}, x^{i}](−1)𝟙, z) − c ∑_{i} \binom{α_{i}}{2}z^{−2}Id_{W}), (3.28)

where {x_{i}}, {x^{i}} are dual bases of 𝔤 that are homogeneous relative to the grading 𝔤 = ⨁_{\overline{r}} 𝔤\overline{r} and α_{i} is a representative of the coset r_{0}/m_{0} + ℤ for which x_{i} ∈ 𝔤\overline{r}.

3.3 Representations of 𝔤_{T}(σ_{0}, σ_{1}, …, σ_{N})

We are now ready to describe irreducible representations of the twisted toroidal Lie algebra 𝔤_{T}(σ) = 𝔤_{T}(σ_{0}, σ_{1}, …, σ_{N}). In this subsection, we describe how to construct these representations from the tensor product V +_{Hyp} ⊗ W ⊗ L_{𝔤𝔩𝔙𝔦𝔯} of Corollary 3.26. We will prove their irreducibility in Section 4.

In order to specify the spaces on which 𝔤_{T}(σ) acts, we recall the definition of thin covering of a module [7]. Let L = ⨁_{g∈G} L_{g} be a Lie algebra graded by a finite abelian group G, and let U be a (not necessarily graded) module for L. A covering of U is a collection of subspaces U_{g} (g ∈ G) satisfying the following axioms

(i) ∑_{g∈G} U_{g} = U

(ii) L_{g}U_{h} ⊆ U_{g+h} for all g, h ∈ G.

A covering {U_{g} | g ∈ G} is a thin covering if there is no other covering {U′_{g} | g ∈ G} of U with U′_{g} ⊆ U_{g} for all g ∈ G.

The automorphisms σ_{1}, …, σ_{N} extend to commuting automorphisms of the twisted affine Lie algebra 𝔤̂(σ_{0}). This gives a grading of 𝔤̂(σ_{0}) by the finite abelian group ℤ^{N}/Γ. Let \{W\overline{r} | \overline{r} ∈ ℤ^{N}/Γ)\} be a thin covering of the irreducible bounded 𝔤̂(σ_{0})-module W fixed in Corollary 3.26. The thin coverings of quasifinite modules like W were classified in [7].

Theorem 3.29 The space

M = ∑_{r∈ℤ^{N}} q^{r} ⊗ 𝔉 ⊗ W\overline{r} ⊗ L_{𝔤𝔩𝔙𝔦𝔯},  (3.30)

is a 𝔤_{T}(σ)-submodule in V +_{Hyp} ⊗ W ⊗ L_{𝔤𝔩𝔙𝔦𝔯}.

Proof We only need to verify that M is closed under the action of the twisted fields Y_{W}(x(−1)𝟙, z)K_{0}(r, z) for x ∈ 𝔤\overline{r} and Y_{W}(ω_{aff}, z). This, however, follows immediately from the definition of a covering. □

4 Irreducibility

We now state one of the main results of this paper.

Theorem 4.1 Let 𝔉 be the Fock space ℂ[u_{pj}, v_{pj} | p = 1, …, N, j = 1, 2, …]. Let L_{𝔤𝔩𝔙𝔦𝔯} be an irreducible highest weight module for the twisted 𝔤̂𝔩_{N}-Virasoro algebra with central character given by (3.18), and let W be an irreducible bounded module for the twisted affine algebra 𝔤̂(σ_{0}) at level c ≠ 0, −h^{∨}. Let \{W\overline{r} | \overline{r} ∈ ℤ^{N}/Γ\} be a thin covering relative to the automorphisms σ_{1}, …, σ_{N} of 𝔤̂(σ_{0}). Then the space

M = ∑_{r∈ℤ^{N}} q^{r} ⊗ 𝔉 ⊗ W\overline{r} ⊗ L_{𝔤𝔩𝔙𝔦𝔯}

is an irreducible module for the twisted toroidal Lie algebra 𝔤_{T}(σ) with the action given by (3.19), (3.20), (3.22), (3.23), (3.27), and (3.28).

The proof of this theorem will be split into a sequence of lemmas. Consider a nonzero submodule N ⊆ M. We need to show that N = M.

Lemma 4.2 Let {U^{A}} be the standard monomial basis of the Fock space 𝔉 = ℂ[u_{pj}, v_{pj} | p = 1, …, N, j = 1, 2, …]. Then w = ∑_{A} U^{A} ⊗ f_{A} ∈ N, where

f_{A} ∈ ∑_{r∈ℤ^{N}} q^{r} ⊗ W\overline{r} ⊗ L_{𝔤𝔩𝔙𝔦𝔯}

if and only if 1 ⊗ f_{A} ∈ N for all A.

Proof The Lie algebra 𝔤_{T}(σ) contains the components of the fields k_{a}(0, z), d̃_{a}(0, z), a = 1, …, N, which act as multiplication and differentiation operators on the Fock space ℂ[u_{pj}, v_{pj} | p = 1, …, N, j = 1, 2, …]. This Fock space is an irreducible module over the Heisenberg Lie algebra generated by these operators, which proves the claim of this lemma. □

The key technique for proving irreducibility under the action of some vertex operators is the following observation: any subspace stabilized by the moments of the (untwisted) vertex operators Y (a, z) and Y (b, z) is also (setwise) invariant under the moments of the vertex operators Y (a_{(k)}b, z) for all k ∈ ℤ. This is an immediate consequence of the Borcherds identity [12] with k, n ∈ ℤ:

(a_{(k)}b)_{(n)} = ∑_{j≥0}(−1)^{k+j+1}\binom{k}{j}b_{(n+k−j)}a_{(j)} + ∑_{j≥0}(−1)^{j}\binom{k}{j}a_{(k−j)}b_{(n+j)}.

The case of twisted modules requires a more delicate analysis.

Lemma 4.3 The space N is closed under the action of the vertex operator Y (ω_{Hyp}, z).

Proof The Lie algebra fields k_{a}(0, z) and d̃_{a}(0, z) act as the vertex operators Y (u_{a1}, z) and Y (v_{a1}, z), respectively. Since

ω_{Hyp} = ∑_{p=1}^{N} u_{p1}_{(−1)}v_{p1},

the lemma now follows from the Borcherds identity observation. □

Lemma 4.4 The space N is closed under the action of the vertex operators Y (E_{ab}(−1)𝟙, z), for all a, b.

Proof For b = 1, …, N and r ∈ Γ, the Lie algebra field d̃_{b}(r, z) is represented by the vertex operator Y (v_{b1}q^{r}, z) + ∑_{p=1}^{N} r_{p}Y (E_{pb}(−1)q^{r}, z). Taking r = 0, this becomes Y (v_{b1}, z). Combining this with the fact that k_{0}(r, z) is represented by cY (q^{r}, z), we see that N is invariant under the action of Y (v_{b1}q^{r}, z) =: Y (v_{b1}, z)Y (q^{r}, z) :, and hence also under the field ∑_{p=1}^{N} r_{p}Y (E_{pb}(−1)q^{r}, z). Since q_{(−1)}^{−r}(E_{pb}(−1)q^{r}) = E_{pb}(−1)𝟙, we obtain that the space N is invariant under ∑_{p=1}^{N} r_{p}Y (E_{pb}(−1)𝟙, z). Finally choosing r_{a} = m_{a} and r_{p} = 0 for p ≠ a, we see that N is invariant under the action of Y (E_{ab}(−1)𝟙, z). □

Lemma 4.5 Let r ∈ ℤ^{N+1} and x ∈ 𝔤\overline{r}. Then N is closed under the action of q^{r}Y_{W}(x(−1)𝟙, z).

Proof The submodule N is closed under the action of Y_{W}(x(−1)𝟙, z)Y (q^{r}, z) since this operator represents the action of the Lie algebra field x(r, z). Let w ∈ N. We would like to show that the coefficients of all powers of z in q^{r}Y_{W}(x(−1)𝟙, z)w belong to N. By Lemma 4.2, we may assume that w does not involve u_{pj}, v_{pj}. In this case,

Y_{W}(x(−1)𝟙, z)Y (q^{r}, z)w = q^{r}Y_{W}(x(−1)𝟙, z)w + (terms involving u_{pj}).

Applying Lemma 4.2 again, we conclude that N is closed under the action of q^{r}Y (x(−1)𝟙, z). □

Lemma 4.6 The space N is closed under the action of the twisted vertex operator Y_{W}(ω_{aff}, z).

Proof The action of Y_{W}(ω_{aff}, z) is given by (3.28). The dual bases {x_{i}}, {x^{i}} may be assumed to be homogeneous. For each i, let r^{(i)} ∈ ℤ^{N+1} so that x_{i} ∈ 𝔤\overline{r}_{(i)} and x^{i} ∈ 𝔤_{−\overline{r}^{(i)}}. Then

: Y_{W}(x_{i}(−1)𝟙, z)Y_{W}(x^{i}(−1)𝟙, z) :
= : (q^{r^{(i)}}Y_{W}(x_{i}(−1)𝟙, z)) (q^{−r^{(i)}}Y_{W}(x^{i}(−1)𝟙, z)) :, (4.7)

so Lemma 4.5 implies that N is invariant under the operator : Y_{W}(x_{i}(−1)𝟙, z)Y_{W}(x^{i}(−1)𝟙, z) :.

Note also that [x_{i}, x^{i}] ∈ 𝔤\overline{0}. Thus the components of the field ∑_{j∈ℤ} t_{0}^{j}[x_{i}, x^{i}]z^{−j−1} belong to 𝔤̂(σ_{0}). Since this field is represented by Y_{W}([x_{i}, x^{i}](−1)𝟙, z), we conclude that N is invariant under this operator. The last summand in (3.28) involves the identity operator, which leaves N invariant. This completes the proof of the lemma. □

Lemma 4.8 The space N is closed under the action of the vertex operator Y (ω_{𝔤𝔩𝔙𝔦𝔯}, z).

Proof The Lie algebra field d̃_{0}(0, z) is represented by the vertex operator

Y (ω_{Hyp}, z) + Y (ω_{𝔤𝔩𝔙𝔦𝔯}, z) + Y_{W}(ω_{aff}, z).

Since N is closed under d̃_{0}(0, z), Lemmas 4.3 and 4.6 imply that N is closed under Y (ω_{𝔤𝔩𝔙𝔦𝔯}, z). □

We are now ready to complete the proof of Theorem 4.1. The Fock space 𝔉 is an irreducible module for the Heisenberg subalgebra in 𝔤_{T}(σ) spanned by the components of the fields d̃_{a}(0, z), k_{a}(0, z), a = 1, …, N, together with the central element k_{0}. The space L_{𝔤𝔩𝔙𝔦𝔯} is an irreducible module for the twisted 𝔤̂𝔩_{N}-Virasoro algebra.

With respect to the commuting automorphisms σ_{1}, …, σ_{N} : 𝔤̂(σ_{0}) → 𝔤̂(σ_{0}), we can form the twisted multiloop Lie algebra

L(𝔤̂(σ_{0}); σ_{1}, …, σ_{N}) = ∑_{s∈ℤ^{N}} t^{s} ⊗ 𝔤̂(σ_{0})\overline{s}.

The twisted affine Lie algebra 𝔤̂(σ_{0}) is generated by the subspaces t_{0}^{r_{0}/m_{0}}𝔤\overline{r} for r = (r_{0}, r) ∈ ℤ^{N+1}. The corresponding operators q^{r}Y_{W}(x(−1)𝟙, z), with x ∈ 𝔤\overline{r}, thus generate the action of the Lie algebra L(𝔤̂(σ_{0}), σ_{1}, …, σ_{N}) on the module

∑_{r∈ℤ^{N}} q^{r} ⊗ W\overline{r}.

By [7, Section 5], we see that this space is a ℤ^{N+1}-graded-simple module for the twisted multiloop algebra L(𝔤̂(σ_{0}), σ_{1}, …, σ_{N}).

To complete the proof of the theorem, we will use the following fact about tensor products of modules.

Lemma 4.9 Let A and B be associative unital algebras graded by an abelian group G. Suppose that V and W are G-graded-simple modules for A and B, respectively, with V_{γ} finite-dimensional for all γ ∈ G. For α ∈ G, denote by V^{(α)} a G-graded A-module, obtained from V by a shift in grading: V_{γ}^{(α)} = V_{γ+α}. Assume that for all α ∈ G, α ≠ 0, there is no grading-preserving isomorphism of A-modules between V and V^{(α)}. Then V ⊗ W is a G-graded-simple A ⊗ Bmodule.

Proof We need to show that V ⊗ W can be generated by any homogeneous nonzero element u ∈ (V ⊗ W)_{γ}. Let us write

u = ∑_{α∈G} ∑_{i} v_{α}^{i} ⊗ w_{α}^{i} (finite sum),

where v_{α}^{i} ∈ V_{α}, w_{α}^{i} ∈ W_{γ−α}. Without loss of generality, we may assume that the set of vectors {v_{α}^{i}} is linearly independent, and the vectors {w_{α}^{i}} are nonzero. Let v ⊗ w be one of the terms in the sum above. The modules in {V^{(α)}|α ∈ G} do not have admit grading-preserving isomorphisms between pairs of distinct modules, so we may apply the quasifinite density theorem ([7, Thm. A.2]) to conclude that there exists a ∈ A_{0} such that av = v, while av_{α}^{i} = 0 for all other i and α. Thus the A ⊗ B submodule generated by u contains v ⊗ w = (a ⊗ 1)u, where both v and w are homogeneous. Acting on v ⊗ w with A ⊗ 1 and 1 ⊗ B, and taking into account that both modules V and W are graded-simple, we conclude that the submodule generated by u is V ⊗ W. □

The space 𝔉 ⊗ L_{𝔤𝔩𝔙𝔦𝔯} is an irreducible module for the universal enveloping algebra of the direct sum of the infinite-dimensional Heisenberg algebra and the twisted 𝔤̂𝔩_{N}-Virasoro algebra 𝔤𝔩𝔙𝔦𝔯. This module has a natural ℤ-grading, which we extend to a ℤ^{N+1} grading by setting (𝔉 ⊗ L_{𝔤𝔩𝔙𝔦𝔯})_{(r_{0},r)} = 0 whenever r ≠ 0. We immediately see by comparing the characters that there are no grading-preserving isomorphisms between this module and the modules obtained from it by shifts in the grading. By the result of Section 5 of [7], the space ∑_{r∈ℤ^{N}} q^{r} ⊗W\overline{r} is a ℤ^{N+1}-graded-simple module for the twisted multiloop algebra L(𝔤̂(σ_{0}), σ_{1}, …, σ_{N}). Since the Lie algebra 𝔤_{T}(σ) contains the derivations d_{0}, d_{1}, …, d_{N}, the space N is a ℤ^{N+1}-graded submodule of M, and every element of N can be reduced to a homogeneous element using d_{0}, d_{1}, …, d_{N}. By Lemmas 4.3–4.8, N is closed under the action of the Heisenberg Lie algebra, of 𝔤𝔩𝔙𝔦𝔯, and of L(𝔤̂(σ_{0}), σ_{1}, …, σ_{N}). Applying Lemma 4.9, we see that N = M, and thus M is irreducible.

5 Irreducible modules for twisted toroidal EALAs

Irreducible modules for untwisted toroidal extended affine algebras were constructed in [6]. The techniques developed in the previous sections can be used to extend this construction and obtain irreducible modules for the twisted toroidal EALAs.

The twisted toroidal EALA

𝔤_{E} = (R ⊗ 𝔤) ⊕ K ⊕_{τ} S

is spanned by elements d_{0}, d_{1}, …, d_{N} and by the moments of the fields

k_{0}(s, z) = ∑_{j∈ℤ} t_{0}^{j}t^{s}k_{0}z^{−j},
k_{a}(s, z) = ∑_{j∈ℤ} t_{0}^{j}t^{s}k_{a}z^{−j−1},
x(r, z) = ∑_{j∈r_{0}/m_{0}+ℤ} t_{0}^{j}t^{r}xz^{−j−1} for each x ∈ 𝔤\overline{r},
d̃_{ab}(s, z) = ∑_{j∈ℤ} (s_{b}t_{0}^{j}t^{s}d_{a} − s_{a}t_{0}^{j}t^{s}d_{b})z^{−j−1},
d̂_{a}(s, z) = ∑_{j∈ℤ} (jt_{0}^{j}t^{s}d_{a} + s_{a}t_{0}^{j}t^{s}d̃_{0} + \frac{s_{a}}{2cN} (N − 1 + μc)t_{0}^{j}t^{s}k_{0}) z^{−j−2},

where s ∈ Γ, r ∈ ℤ^{N+1}, and a, b = 1, …, N.

In the representation theory of 𝔤_{E}, the twisted 𝔤̂𝔩_{N}-Virasoro algebra 𝔤𝔩𝔙𝔦𝔯 is replaced with its subalgebra, the semidirect product 𝔰𝔩𝔙𝔦𝔯 of the Virasoro algebra with the affine algebra 𝔰̂𝔩_{N}:

𝔰𝔩𝔙𝔦𝔯 = (ℂ[t_{0}, t_{0}^{−1}] ⊗ 𝔰𝔩_{N} ⊕ ℂC_{𝔰𝔩_{N}} ) ⋊ (Der ℂ[t_{0}, t_{0}^{−1}] ⊕ ℂC_{V ir}) .

Now we can state the theorem describing irreducible modules for the twisted toroidal EALA 𝔤_{E}(σ).

Theorem 5.1 Let W be an irreducible bounded 𝔤̂(σ_{0})-module of level c ≠ 0, −h^{∨}, with a thin covering \{W\overline{r} | \overline{r} ∈ ℤ^{N}/Γ\} relative to automorphisms σ_{1}, …, σ_{N}. Let L_{𝔰𝔩𝔙𝔦𝔯} be an irreducible highest weight module for 𝔰𝔩𝔙𝔦𝔯 with a central character γ:

γ(C_{𝔰𝔩_{N}}) = 1 − μc, γ(C_{V ir}) = 12(1 − \frac{1}{N}) + 12μc(1 + \frac{1}{N}) − 2N − \frac{c dim(𝔤)}{c + h^{∨}}.

Then the space

∑_{r∈ℤ^{N}} q^{r} ⊗ 𝔉 ⊗ W\overline{r} ⊗ L_{𝔰𝔩𝔙𝔦𝔯}

has the structure of an irreducible module for the twisted toroidal extended affine Lie algebra 𝔤_{E}(σ_{0}, σ_{1}, …, σ_{N}) with the action given by

k_{0}(s, z) ↦ cK_{0}(s, z),  (5.2)
k_{a}(s, z) ↦ cK_{a}(s, z),  (5.3)
x(r, z) ↦ Y_{W}(x(−1)𝟙, z)K_{0}(s, z),  (5.4)
d̃_{ab}(s, z) ↦: (s_{b}D_{a}(z) − s_{a}D_{b}(z)) K_{0}(s, z) :

+ s_{b} ∑_{{p=1 \atop p≠a}} s_{p}E_{pa}(z)K_{0}(s, z) − s_{a} ∑_{{p=1 \atop p≠b}} s_{p}E_{pb}(z)K_{0}(s, z)

+ s_{a}s_{b}(E_{aa} − E_{bb})(z)K_{0}(s, z),  (5.5)
d̂_{a}(s, z) ↦ s_{a} : (ω_{Hyp}(z) + ω_{𝔰𝔩𝔙𝔦𝔯}(z) + Y_{W}(ω_{aff}, z)) K_{0}(s, z) :
+s_{a} ∑_{p,ℓ=1}^{N} s_{p}ψ_{1}(E_{pℓ})(z)K_{ℓ}(s, z) + s_{a}(μc − 1) ∑_{p=1}^{N} s_{p} (\frac{∂}{∂z} K_{p}(z)) K_{0}(s, z)

−(z^{−1} + \frac{∂}{∂z} ) (: D_{a}(z)K_{0}(s, z) : + ∑_{p=1}^{N} s_{p}ψ_{1}(E_{pa})(z)K_{0}(s, z)) , (5.6)

where s ∈ Γ, r ∈ ℤ^{N+1}, a, b = 1, …, N, and ψ_{1} is the natural projection (3.4) ψ_{1} : 𝔤𝔩_{N}(ℂ) → 𝔰𝔩_{N}(ℂ).

This theorem is based on its untwisted analogue [6, Thm. 5.5]. The proof is completely parallel to the proof of Theorems 3.29 and 4.1 and will be omitted.

6 Example: EALAs of Clifford Type

We now apply the general theory that we have developed to construct irreducible representations of EALAs coordinatized by Jordan tori of Clifford type.

6.1 Multiloop realization

Fix a positive integer m. Let 2ℤ^{m} ⊆ S ⊆ ℤ^{m}, where S is a union of some cosets of the subgroup 2ℤ^{m} ⊆ ℤ^{m}. We assume that ℤ^{m} is generated by S as a group. Let \overline{μ} be the image of μ ∈ ℤ^{m} under the map ℤ^{m} → ℤ^{m}/2ℤ^{m}, and let r be the cardinality of \overline{S} = S/2ℤ^{m}. We identify ℤ^{m}/2ℤ^{m} with the multiplicative group {−1, 1}^{m} = ℤ_{2}^{m}.

A Jordan torus of Clifford type is a Jordan algebra

J = ⨁_{μ∈S} ℂs^{μ}

with multiplication given by

s^{μ}s^{η} = \{ \begin{matrix}s^{μ+η} & if \overline{μ} = \overline{0}, \overline{η} = \overline{0}, or \overline{μ} = \overline{η}, \\ 0 & otherwise. \end{matrix}

Let L_{J} = {L_{a} | a ∈ J} be the set of left multiplication operators L_{a} : b ↦ ab on J. The Tits-Kantor-Koecher algebra associated with J is the Lie algebra

TKK(J) = (J ⊗ 𝔰𝔩_{2}(ℂ)) ⊕ [L_{J}, L_{J}],

where

[a ⊗ x, b ⊗ y] = ab ⊗ [x, y] + (x|y)[L_{a}, L_{b}],
[d, a ⊗ x] = da ⊗ x = −[a ⊗ x, d],
[d, d′] = dd′ − d′d

for all a, b ∈ J, x, y ∈ 𝔰𝔩_{2}(ℂ), and d, d′ ∈ [L_{J}, L_{J}].

We now introduce some notation which will be used to realize TKK(J) as a multiloop algebra. Let U be an (r + 2)-dimensional vector space with a basis \{v_{i} | i ∈ I\}, where I = \{1, 2, 3\} ∪ (\overline{S} \ \{\overline{0}\}). Define a symmetric bilinear form on U by declaring that this basis is orthonormal. If i ∈ {1, 2, 3}, let i_{p} = 1, and if i = \overline{μ} ∈ \overline{S} \ \{\overline{0}\}, let i_{p} = \overline{μ}_{p} ∈ \{−1, 1\} for all p ∈ \{1, …, m\}. For each i, j, k ∈ I, define e_{ij} ∈ 𝔰𝔬(U) by

e_{ij}(v_{k}) = δ_{jk}v_{i} − δ_{ik}v_{j}.

Let σ_{p} be an orthogonal transformation on U defined by

σ_{p}(v_{i}) = i_{p}v_{i}

for all i ∈ I and p ∈ {1, …, m}. We identify each σ_{p} with an automorphism of 𝔰𝔬(U) where σ_{p} acts on 𝔰𝔬(U) by conjugation. Each of these σ_{p} ∈ Aut (𝔰𝔬(U)) has order 2.

Remark. In our construction the index set I is obtained from \overline{S} by triplicating \overline{0} into \{1, 2, 3\}. This is done in order to create a 3-dimensional subalgebra 𝔰𝔬_{3}(ℂ) ≅ 𝔰𝔩_{2}(ℂ), fixed under all the involutions σ_{p}. Analogous gradings on 𝔰𝔬_{2^{m}}(ℂ) were considered in [22, 1, 4].

Theorem 6.1 The Tits-Kantor-Koecher algebra TKK(J) is isomorphic to the twisted multiloop algebra G = L(𝔰𝔬_{r+2}(ℂ); σ_{1}, …, σ_{m}) via the following map:

ϕ : TKK(J) → L(𝔰𝔬_{r+2}(ℂ); σ_{1}, …, σ_{m})
s^{μ} ⊗ X_{1} ↦ \{ \begin{matrix}T^{μ} ⊗ e_{32} & if \overline{μ} = \overline{0} \\ T^{μ} ⊗ e_{\overline{μ}1} & otherwise, \end{matrix}
s^{μ} ⊗ X_{2} ↦ \{ \begin{matrix}T^{μ} ⊗ e_{13} & if \overline{μ} = \overline{0} \\ T^{μ} ⊗ e_{\overline{μ}2} & otherwise, \end{matrix}
s^{μ} ⊗ X_{3} ↦ \{ \begin{matrix}T^{μ} ⊗ e_{21} & if \overline{μ} = \overline{0} \\ T^{μ} ⊗ e_{\overline{μ}3} & otherwise, \end{matrix}
[L_{s^{γ}} , L_{s^{η}}] ↦ T^{γ+η} ⊗ e_{\overline{γ}\overline{η}},

for all μ, γ, η ∈ S with \overline{γ}, \overline{η}, \overline{γ + η} ≠ \overline{0}. Here \{X_{1}, X_{2}, X_{3}\} is a basis of 𝔰𝔩_{2}(ℂ) with relations

[X_{1}, X_{2}] = X_{3}, [X_{2}, X_{3}] = X_{1}, [X_{3}, X_{1}] = X_{2}.

Proof Observe that σ_{p}(e_{ij}) = i_{p}j_{p}e_{ij} for all i, j ∈ I and p ∈ {1, …, m}. This implies that the image of ϕ is contained in the twisted multiloop algebra L(𝔰𝔬_{r+2}(ℂ); σ_{1}, …, σ_{m}). The verification that ϕ is a homomorphism is tedious but straightforward, and will be omitted. It is clear that ϕ is injective. To see that it is surjective, we note that TKK(J) and L(𝔰𝔬_{r+2}(ℂ); σ_{1}, …, σ_{m}) have natural ℤ^{m}-gradings given by

deg(s^{μ} ⊗ X_{i}) = μ,
deg[L_{s^{γ}} , L_{s^{η}}] = γ + η,
deg(T^{μ} ⊗ e_{ij}) = μ.

The map ϕ is then homogeneous of degree 0. It is now sufficient to verify that the dimensions of the corresponding graded components of TKK(J) and L(𝔰𝔬_{r+2}(ℂ); σ_{1}, …, σ_{m}) are the same.

It is easy to see that

dim TKK(J)_{μ} = dim TKK(J)_{γ},
dim G_{μ} = dim G_{γ}

whenever \overline{μ} = \overline{γ}. This allows us to define

a′ ; \overline{μ} = dim TKK(J)_{μ}
a′′ ; \overline{μ} = dim G_{μ}

for all μ ∈ ℤ^{m}. Instead of proving that a′ ; \overline{μ} = a′′ ; \overline{μ} for each \overline{μ} ∈ ℤ_{2}^{m}, we will show that

∑_{\overline{μ}∈ℤ_{2}^{m}} a′ ; \overline{μ} = ∑_{\overline{μ}∈ℤ_{2}^{m}} a′′ ; \overline{μ}.  (6.2)

Since the map ϕ is injective and homogeneous of degree 0, the latter equality will imply that a′ ; \overline{μ} = a′′ ; \overline{μ}, and ϕ is thus an isomorphism. We now verify (6.2).

The contribution of the space J⊗𝔰𝔩_{2}(ℂ) in the sum ∑_{\overline{μ}∈ℤ_{2}^{m}} a′ ; \overline{μ} is 3r, while the space [L_{J}, L_{J}] contributes \binom{r−1}{2}. Thus ∑_{\overline{μ}∈ℤ_{2}^{m}} a′ ; \overline{μ} = 3r +\binom{r−1}{2}. The right-hand side of (6.2) is simply the dimension \binom{r+2}{2} of 𝔰𝔬_{r+2}(ℂ). Since 3r+\binom{r−1}{2} = \binom{r+2}{2}, we are done. □

We are interested in EALAs associated with the universal central extension of TKK(J). The above multiloop realization of TKK(J) yields a description of such EALAs in the setup of Section 1 as 𝔤_{E}(σ_{1}, …, σ_{m}) with 𝔤 = 𝔰𝔬_{r+2}(ℂ).

We now consider the representation theory of these Jordan torus EALAs. To conform with the notation in the rest of the paper, we will set m = N + 1, and we number the variables of the Jordan torus from 0 to N. Likewise, the automorphisms of 𝔰𝔬_{r+2}(ℂ) under consideration become σ_{0}, σ_{1}, …, σ_{N} and the variables in the multiloop algebra are thus changed from T_{1}, …, T_{m} to t_{0}^{1/2}, t_{1}, …, t_{N}.

According to Theorem 3.29, the piece of the simple module M specific to the Jordan torus EALA is the irreducible highest weight module W for the twisted affine Lie algebra 𝔰̂𝔬_{r+2}(σ_{0}), and its thin covering \{W\overline{r}\} with respect to the automorphisms {σ_{1}, …, σ_{N}}.

The Lie algebra 𝔰̂𝔬_{r+2}(σ_{0}) is isomorphic to the untwisted or twisted affine Lie algebra, depending on whether or not σ_{0} is an inner automorphism of 𝔰𝔬_{r+2}(ℂ).

Lemma 6.3 Let U be a finite-dimensional vector space with an orthonormal basis {v_{i} | i ∈ I}. Let σ ∈ GL(U), with σ(v_{i}) = ±v_{i}. Then conjugation by σ is an inner automorphism of 𝔰𝔬(U) if and only if the matrix of σ in this basis has an even number of −1’s on the diagonal or an even number of +1’s.

Proof If σ has an even number of −1’s on the diagonal, then σ ∈ SO(U), and it is inner. Suppose that σ has an even number of +1’s on the diagonal. Since σ acts on 𝔰𝔬(U) by conjugation, −σ induces the same automorphism, and −σ is inner by the previous argument. Finally, suppose that σ has an odd number of +1’s and an odd number of −1’s on the diagonal. Then dim U is even, and 𝔰𝔬(U) is of type D. Then, multiplying σ by an appropriate diagonal matrix with an even number of −1’s on the diagonal, we can get a diagonal matrix τ with all +1’s except for the entry −1 in the last position. Choosing a Cartan subalgebra of 𝔰𝔬(U) and a basis of its root system as in Subsection 6.3 below, one can easily see that τ is the Dynkin diagram automorphism of order 2 of a root system of type D. Since σ differs from τ by a factor which is an inner automorphism, we conclude that σ is not inner. □

We will now focus our attention on two Clifford type EALAs of nullity 2: a “baby TKK”, and a “full lattice TKK”.

6.2 Baby TKK

Let S be a union of 3 cosets of 2ℤ^{2} in ℤ^{2}, corresponding to the coset representatives (0, 0), (0, 1), and (1, 0). Then m = 2 and r = 3. The corresponding multiloop algebra given by Theorem 6.1 is L(𝔰𝔬_{5}; σ_{0}, σ_{1}) where the matrices

σ_{0} = ( \begin{matrix}1 \\ & 1 \\ & & 1 \\ & & & 1 \\ & & & & −1 \end{matrix} ) , σ_{1} = ( \begin{matrix}1 \\ & 1 \\ & & 1 \\ & & & −1 \\ & & & & 1 \end{matrix} )

act on 𝔰𝔬_{5}(ℂ) by conjugation. Multiplying by −I does not change their action:

σ_{0} = ( \begin{matrix}−1 \\ & −1 \\ & & −1 \\ & & & −1 \\ & & & & 1 \end{matrix} ) , σ_{1} = ( \begin{matrix}−1 \\ & −1 \\ & & −1 \\ & & & 1 \\ & & & & −1 \end{matrix} ) .

Consider a Cartan subalgebra in 𝔰𝔬_{5} with a basis of coroots

h_{1} = ( \begin{matrix}& 2i \\ −2i \end{matrix} ) , h_{2} = ( \begin{matrix}& i \\ −i \\ & & & −i \\ & & i \end{matrix} ) .

The corresponding generators of the simple root spaces are:

e_{1} = (  \begin{matrix}& & −1 \\ & & i \\ 1 & −i \end{matrix} ) , f_{1} = (  \begin{matrix}& & 1 \\ & & i \\ −1 & −i \end{matrix} ) ,

e_{2} = \frac{1}{2} ( \begin{matrix}& & 1 & i \\ & & −i & 1 \\ −1 & i \\ −i & −1 \end{matrix} ) , f_{2} = \frac{1}{2} ( \begin{matrix}& & −1 & i \\ & & −i & −1 \\ 1 & i \\ −i & 1 \end{matrix} ) .

These generators satisfy the relations [h_{i}, e_{j}] = a_{ij}e_{j}, [h_{i}, f_{j}] = −a_{ij}f_{j}, [e_{i}, f_{j}] = δ_{ij} with the Cartan matrix

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

Note that

σ_{0} = exp (πi ad h_{2}) .

The Lie algebra 𝔰𝔬_{5} has an eigenspace decomposition 𝔰𝔬_{5} = 𝔰𝔬\overline{0} 5 ⊕ 𝔰𝔬\overline{1} 5 with respect to the action of σ_{0}:

𝔰𝔬\overline{j} ; 5 = \{x ∈ 𝔰𝔬_{5} | σ_{0}x = (−1)^{j}x\}

for j = 0, 1. Since σ_{0} and σ_{1} commute, the subspaces 𝔰𝔬\overline{0} 5 and 𝔰𝔬\overline{1} 5 are invariant with respect to σ_{1}. We view σ_{1} as an automorphism of the twisted loop algebra

L(𝔰𝔬_{5}; σ_{0}) = ∑_{j∈ℤ} t_{0}^{j/2}𝔰𝔬\overline{j} ; 5,

by letting it act by σ_{1}(t_{0}^{j/2}x) = t_{0}^{j/2}σ_{1}(x) for each x ∈ 𝔰𝔬\overline{j} ; 5 and j ∈ ℤ. We then extend it to an automorphism of the twisted affine algebra 𝔰̂𝔬_{5}(σ_{0}) by σ_{1}(C_{aff}) = C_{aff}.

Since σ_{0} is inner (as is every automorphism of 𝔰𝔬_{5}), the twisted loop algebra L(𝔰𝔬_{5}; σ_{0}) is isomorphic to the untwisted loop algebra ℂ[t_{0}, t_{0}^{−1}] ⊗ 𝔰𝔬_{5} [11, Prop. 8.5]. This lifts to an isomorphism of affine algebras

θ : 𝔰̂𝔬_{5} → 𝔰̂𝔬_{5}(σ_{0}),

such that

θ(t_{0}^{j} ⊗ e_{α}) = t_{0}^{j+α(h_{2})/2} ⊗ e_{α},

θ(t_{0}^{j} ⊗ h_{α}) = t_{0}^{j} ⊗ h_{α} + δ_{j,0} \frac{α(h_{2})}{2}C_{aff},

θ(C_{aff}) = C_{aff},

where e_{α} is in a root space of 𝔰𝔬_{5} with α ≠ 0, and h_{α} in the Cartan subalgebra is normalized so that [e_{α}, e_{−α}] = (e_{α}|e_{−α})h_{α}.

Using the identification θ, we transform σ_{1} into an automorphism σ̂_{1} = θ^{−1}σ_{1}θ ∈ Aut (𝔰̂𝔬_{5}). Since θ does not preserve the natural ℤ-grading of 𝔰̂𝔬_{5}, σ̂_{1} does not leave the components t_{0}^{j} ⊗ 𝔰𝔬_{5} invariant. However, it does leave invariant the Cartan subalgebra 𝔥 = (ℂh_{1} ⊕ ℂh_{2}) ⊕ ℂC_{aff} of 𝔰̂𝔬_{5}.

Let us describe the group N of automorphisms of an affine Lie algebra 𝔤̂ that leave the Cartan subalgebra 𝔥 setwise invariant [21]. First of all, N has a normal subgroup H × ℂ^{∗} of automorphisms that fix 𝔥 pointwise. Here ℂ^{∗} is the set of automorphisms {τ_{a} | a ∈ ℂ \ 0} that act by

τ_{a}(t_{0}^{j} ⊗ x) = a^{j}t_{0}^{j} ⊗ x, τ_{a}(C_{aff}) = C_{aff},

and H = {exp(ad h) | h ∈ 𝔥} consists of inner automorphisms. The quotient N/(H × ℂ^{∗}) can be presented as follows:

N/(H × ℂ^{∗}) ≅ ⟨π⟩ × (Aut(Γ) ⋉ W) ,

where π is the Chevalley involution, W is the affine Weyl group and Aut(Γ) is the group of automorphisms of the affine Dynkin diagram Γ. The elements of this factor group may be viewed as permutations of the roots of the affine Lie algebra.

Let σ be an automorphism of the affine Lie algebra 𝔤̂ leaving invariant its Cartan subalgebra. Such a Cartan subalgebra may always be found relative to any family of finite order automorphisms σ_{0}, …, σ_{N} by Appendix 6.3. Let (W, ρ) be an integrable irreducible highest weight module for 𝔤̂ with dominant integral highest weight λ relative to a fixed base of simple roots. In order to determine the thin covering of W with respect to the cyclic group generated by σ, we need to know whether the modules (W, ρ) and (W, ρ ∘ σ) are isomorphic [7]. The answer to this question does not change if we replace σ with σ ∘ μ, where μ is an automorphism of 𝔤̂ for which (W, ρ) and (W, ρ∘μ) are isomorphic.

Proposition 6.4 Let (V, ρ) be a representation of a Lie algebra L. Suppose x ∈ L, ad x is locally nilpotent on L, and ρ(x) is locally nilpotent on V. Then the representations (V, ρ) and (V, ρ ∘ exp(ad x)) are isomorphic.

Proof The isomorphism from (V, ρ) to (V, ρ ∘ exp(ad x)) is given by exp(ρ(x)). This claim is equivalent to the identity

exp(ρ(x))ρ(y) = ρ(exp((ad x)y)) exp(ρ(x)), y ∈ L,

which is well-known. □

Corollary 6.5 Let (W, ρ) be an integrable module for an affine Lie algebra 𝔤̂, and let μ be an inner automorphism of 𝔤̂. Then the modules (W, ρ) and (W, ρ∘μ) are isomorphic.

Proof This follows from the previous proposition and the fact that the Kac-Moody group (of inner automorphisms) is generated by the exponentials of the real root elements, which are locally nilpotent on integrable modules [21]. □

Lemma 6.6 Let (W, ρ) be an integrable irreducible highest weight module for 𝔤̂. Let a ∈ ℂ^{∗}. Then the modules (W, ρ) and (W, ρ ∘ τ_{a}) are isomorphic.

Proof It is easy to see that the module W admits a compatible action of the group ℂ^{∗} (see [21, Section 4]), which we will denote by

T_{a} : W → W, a ∈ ℂ^{∗}.

This can be done by requiring that ℂ^{∗} fixes the highest weight vector and satisfies the compatibility condition

T_{a}ρ(y)v = ρ(τ_{a}(y))T_{a}v.

The condition implies that T_{a} is a module isomorphism between (W, ρ) and (W, ρ ∘ τ_{a}). □

Let \overline{σ} be the image of σ in the factor-group N/W (H × ℂ^{∗}) ≅ ⟨π⟩ × Aut(Γ). We will also identify \overline{σ} with an automorphism of 𝔤̂ by viewing Aut(Γ) as a subgroup of Aut(𝔤̂).

Since W H consists of inner automorphisms of 𝔤̂, Corollary 6.5 and Lemma 6.6 imply the following lemma:

Lemma 6.7 Let (W, ρ) be an integrable irreducible highest weight module for 𝔤̂ and σ be an automorphism of 𝔤̂ leaving invariant its Cartan subalgebra. Then the 𝔤̂-modules (W, ρ ∘ σ) and (W, ρ ∘ \overline{σ}) are isomorphic. □

We now return to the setting of our particular example and calculate \overline{σ}_{1} = \overline{θ^{−1}σ_{1}θ} as an automorphism of the root system of 𝔰̂𝔬_{5}. The Dynkin diagram Γ of 𝔰̂𝔬_{5} is

\begin{matrix}α_{0} & α_{1} & α_{2} \\ • & ⇒ & • & ⇐ & • \end{matrix}

and Aut(Γ) ≅ ℤ_{2}.

We need to compute the action induced by σ̂_{1} = θ^{−1}σ_{1}θ on the simple roots α_{0}, α_{1}, α_{2} of 𝔰̂𝔬_{5}. Since σ̂_{1} leaves invariant the null root spaces, the induced automorphism of the root system fixes the null root δ. Taking into account that α_{0} = δ − 2α_{1} − α_{2}, we see that it is enough to find the action on α_{1} and α_{2}:

σ̂_{1}(e_{1}) = θ^{−1}σ_{1}θ(e_{1}) = θ^{−1}t_{0}^{α_{1}(h_{2})/2} ⊗ σ_{1}(e_{1})

= θ^{−1}t_{0}^{−1/2} ⊗ σ_{1} (  \begin{matrix}& & −1 \\ & & i \\ 1 & −i \end{matrix} ) = θ^{−1}t_{0}^{−1/2} ⊗ (  \begin{matrix}& & −1 \\ & & −i \\ 1 & i \end{matrix} )

= θ^{−1}(−t_{0}^{−1/2} ⊗ f_{1}) = −t_{0}^{−1/2+α_{1}(h_{2})/2} ⊗ f_{1} = −t_{0}^{−1} ⊗ f_{1}.

Thus we get σ̂_{1}(α_{1}) = −δ − α_{1}. Similarly,

σ̂_{1}(2e_{2}) = θ^{−1}σ_{1}θ(2e_{2}) = θ^{−1}t_{0}^{α_{2}(h_{2})/2} ⊗ σ_{1}(2e_{2})

= θ^{−1}t_{0} ⊗ σ_{1} ( \begin{matrix}& & 1 & i \\ & & −i & 1 \\ −1 & i \\ −i & −1 \end{matrix} ) = θ^{−1}t_{0} ⊗ ( \begin{matrix}& & 1 & −i \\ & & −i & −1 \\ −1 & i \\ i & 1 \end{matrix} )

= θ^{−1}t_{0} ⊗ [[e_{1}, e_{2}], e_{1}] = t_{0}^{1−(2α_{1}(h_{2})+α_{2}(h_{2}))/2} ⊗ [[e_{1}, e_{2}], e_{1}] = t_{0} ⊗ [[e_{1}, e_{2}], e_{1}].

We get σ̂_{1}(α_{2}) = δ + 2α_{1} + α_{2}, and hence σ̂_{1}(α_{0}) = 2δ − α_{2}. Let γ be the diagram automorphism that interchanges α_{0} with α_{2} and fixes α_{1}. Let r_{0}, r_{1}, r_{2} be the simple reflections generating the affine Weyl group of 𝔰̂𝔬_{5}. It is then straightforward to verify that as an automorphism of the root system,

σ̂_{1} = r_{1}r_{2}r_{0}r_{1}γ.

We conclude that \overline{σ}_{1} = γ ∈ Aut(Γ).

Proposition 6.8 Let (W, ρ) be an irreducible highest weight module for 𝔰̂𝔬_{5} of dominant integral highest weight λ. The 𝔰̂𝔬_{5}-modules (W, ρ) and (W, ρ ∘ σ̂_{1}) are isomorphic if and only if the diagram automorphism γ fixes the highest weight λ.

Proof Taking into account the fact that \overline{σ}_{1} = γ and applying Lemma 6.7, we conclude that the modules (W, ρ) and (W, ρ ∘ σ̂_{1}) are isomorphic if and only if the modules (W, ρ) and (W, ρ ∘ γ) are isomorphic. However, (W, ρ ∘ γ) is the highest weight module with the highest weight γ(λ). Since two irreducible highest weight modules are isomorphic precisely when their highest weights are equal, we obtain the claim of the proposition. □

Corollary 6.9 If the highest weight λ is not fixed by the diagram automorphism γ, then the thin covering of W with respect to the cyclic group ⟨σ̂_{1}⟩ ≅ ℤ_{2} is {W, W}.

Proof This follows from Theorem 4.4 of [7] and the proposition above. □

When the highest weight λ is fixed by the diagram automorphism γ, there is a module isomorphism ϕ_{γ} : (W, ρ) → (W, ρ ∘ γ). Concretely, we may define the action of ϕ_{γ} on the Verma module of highest weight λ by postulating that ϕ_{γ} fixes the highest weight vector and ϕ_{γ}(ρ(x)v) = ρ(γ(x))ϕ_{γ}(v), for all x ∈ 𝔰̂𝔬_{5} and v ∈ W. It is also clear that ϕ_{γ} will leave invariant the maximal submodule of this Verma module. This gives an action of γ as the operator ϕ_{γ} on W. As a result, we obtain an action of the semi-direct product ⟨γ⟩ ⋉ G on W, where G is the Kac-Moody group of 𝔰̂𝔬_{5}. Since σ̂_{1} ∈ ⟨γ⟩ ⋉ G, we realize σ̂_{1} as an order 2 operator on W. It is easy to see that the action of σ̂_{1} on W is locally finite, and hence W has decomposition

W = W\overline{0} ⊕ W\overline{1},  (6.10)

where W\overline{0}, W\overline{1} are the ±1 eigenspaces of σ̂_{1}. In this case, \{W\overline{0}, W\overline{1}\} is a thin covering of W relative to σ̂_{1}. We have now proved the following theorem:

Theorem 6.11 Let (W, ρ) be an irreducible highest weight module for 𝔰̂𝔬_{5} of integral dominant highest weight λ. View W as a module for the twisted affine algebra 𝔰̂𝔬_{5}(σ_{0}) with the action ρ ∘ θ^{−1}. Let L_{𝔰𝔩𝔙𝔦𝔯} be an irreducible highest weight module for 𝔰𝔩𝔙𝔦𝔯 as in Theorem 5.1.

(i) If γ(λ) ≠ λ, where γ is the Dynkin diagram automorphism of 𝔰̂𝔬_{5} then the space

V +_{Hyp} ⊗ L_{𝔰𝔩𝔙𝔦𝔯} ⊗ W

has the structure of an irreducible module for the “Baby TKK” EALA 𝔤_{E}(σ_{0}, σ_{1}) with the action described in Theorem 5.1.

(ii) If γ(λ) = λ, then the space

∑_{r∈ℤ} q^{r} ⊗ 𝔉 ⊗ L_{𝔰𝔩𝔙𝔦𝔯} ⊗ W\overline{r}

is an irreducible module for 𝔤_{E}(σ_{0}, σ_{1}), where \{W\overline{0}, W\overline{1}\} are as in (6.10). □

6.3 Full lattice TKK of nullity 2

Let S be the set of all 4 cosets of 2ℤ^{2} in ℤ^{2}. The corresponding multiloop algebra given by Theorem 6.1 is L(𝔰𝔬_{6}; σ_{0}, σ_{1}) where

σ_{0} = ( \begin{matrix}1 \\ & 1 \\ & & 1 \\ & & & 1 \\ & & & & −1 \\ & & & & & −1 \end{matrix} ) , σ_{1} = ( \begin{matrix}1 \\ & 1 \\ & & 1 \\ & & & −1 \\ & & & & 1 \\ & & & & & −1 \end{matrix} ) .

We consider a Cartan subalgebra in 𝔰𝔬_{6} with a basis of coroots

h_{1} = (  \begin{matrix}& i \\ −i \\ & & & i \\ & & −i \end{matrix} ) , h_{2} = ( \begin{matrix}& i \\ −i \\ & & & −i \\ & & i \end{matrix}  ) ,

h_{3} = ( \begin{matrix}& i \\ −i \\ & & & −i \\ & & i \end{matrix} ) ,

The corresponding generators of the simple root spaces are:

e_{1} = \frac{1}{2} (  \begin{matrix}& & 1 & −i \\ & & −i & −1 \\ −1 & i \\ i & 1 \end{matrix} ) , f_{1} = \frac{1}{2} (  \begin{matrix}& & −1 & −i \\ & & −i & 1 \\ 1 & i \\ i & −1 \end{matrix} ) ,

e_{2} = \frac{1}{2} ( \begin{matrix}& & 1 & i \\ & & −i & 1 \\ −1 & i \\ −i & −1 \end{matrix}  ) , f_{2} = \frac{1}{2} ( \begin{matrix}& & −1 & i \\ & & −i & −1 \\ 1 & i \\ −i & 1 \end{matrix}  ) ,

e_{3} = \frac{1}{2} (  \begin{matrix}& & 1 & i \\ & & −i & 1 \\ −1 & i \\ −i & −1 \end{matrix} ) , f_{3} = \frac{1}{2} (  \begin{matrix}& & −1 & i \\ & & −i & −1 \\ 1 & i \\ −i & 1 \end{matrix} ) .

The root system of 𝔰𝔬_{6} is of type D_{3} = A_{3} and the Lie brackets [h_{i}, e_{j}] = a_{ij}e_{j}, [h_{i}, f_{j}] = −a_{ij}f_{j} are given by the Cartan matrix

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

Note that

σ_{0} = exp (ad \frac{π}{2} i(h_{1} − h_{3})) .

Since σ_{0} is an inner automorphism of 𝔰𝔬_{5}, the twisted loop algebra L(𝔰𝔬_{6}; σ_{0}) is isomorphic to the untwisted loop algebra ℂ[t_{0}, t_{0}^{−1}]⊗𝔰𝔬_{6} [11, Prop. 8.5]. This lifts to an isomorphism θ of affine Lie algebras:

θ : 𝔰̂𝔬_{6} → 𝔰̂𝔬_{6}(σ_{0}),

where

θ(t_{0}^{j} ⊗ e_{α}) = t_{0}^{j+α(h_{1}−h_{3})/4} ⊗ e_{α},

θ(t_{0}^{j} ⊗ h_{α}) = t_{0}^{j} ⊗ h_{α} + δ_{j,0} \frac{α(h_{1} − h_{3})}{4}C_{aff},

θ(C_{aff}) = C_{aff}.

Using the identification θ, we transform σ_{1} into an automorphism σ̂_{1} = θ^{−1}σ_{1}θ of 𝔰̂𝔬_{6}. Let (W, ρ) be an integrable irreducible highest weight module for 𝔰̂𝔬_{6} with dominant integral highest weight λ. We now determine when the modules (W, ρ) and (W, ρ ∘ σ̂_{1}) are isomorphic.

Let \overline{σ}_{1} be the image of σ̂_{1} in ⟨π⟩ × Aut(Γ) under the projection N → N/W (H × ℂ^{∗}). The Dynkin diagram Γ of 𝔰̂𝔬_{6} is

α_{0} α_{3} • •

• • α_{1} α_{2} so Aut(Γ) is the dihedral group of order 8.

As in the case of the Baby TKK algebra described above, we view σ̂_{1} as an automorphism of the affine root system and calculate the action of σ̂_{1} on the simple roots α_{0}, α_{1}, α_{2}, α_{3} of 𝔰̂𝔬_{6}. After making the analogous calculations, we obtain

σ̂_{1} = r_{3}r_{0}r_{2}r_{1}η,

where η is the diagram automorphism switching α_{0} with α_{2}, and α_{1} with α_{3}.

Proposition 6.12 Let (W, ρ) be an irreducible highest weight module for 𝔰̂𝔬_{6} with a dominant integral weight λ. If η(λ) ≠ λ, then the thin covering of W with respect to the cyclic group ⟨σ̂_{1}⟩ is {W, W}. □

When η(λ) = λ, the module W and its twist by σ̂_{1} are isomorphic, as in the case of the Baby TKK algebra. The isomorphism defines a ℂ-linear action of σ̂_{1} as an order 2 operator on W, which then composes into eigenspaces W\overline{0} and W\overline{1} relative to the action. We thus obtain the analogue of Theorem 6.11:

Theorem 6.13 Let (W, ρ) be an irreducible highest weight representation of 𝔰̂𝔬_{6} of integral dominant highest weight λ.

(i) Suppose that η(λ) ≠ λ, where η is the 180^{∘} rotation automorphism of the Dynkin diagram of 𝔰̂𝔬_{6}. View W as a module for the twisted affine algebra 𝔰̂𝔬_{6}(σ_{0}) with the action ρ∘θ^{−1}. Let L_{𝔰𝔩𝔙𝔦𝔯} be an irreducible highest weight module for 𝔰𝔩𝔙𝔦𝔯 as in Theorem 5.1. Then the space

V +_{Hyp} ⊗ L_{𝔰𝔩𝔙𝔦𝔯} ⊗ W

has the structure of an irreducible module for the nullity 2 full lattice TKK EALA 𝔤_{E}(σ_{0}, σ_{1}) with the action described in Theorem 5.1.

(ii) If η(λ) = λ, then the space

∑_{r∈ℤ} q^{r} ⊗ 𝔉 ⊗ L_{𝔰𝔩𝔙𝔦𝔯} ⊗ W\overline{r}

is an irreducible module for 𝔤_{E}(σ_{0}, σ_{1}) under the action described in Theorem 5.1. □

A Appendix

Let σ_{0}, …, σ_{N} be commuting finite-order automorphisms of a finite-dimensional simple Lie algebra 𝔤. In this appendix, we show that there exists a Cartan subalgebra 𝔥 ⊆ 𝔤 which is (setwise) invariant under these automorphisms.

Lemma A.1 Let σ be an automorphism of a finite-dimensional simple Lie algebra L with Killing form (· | ·). Then (σx|σy) = (x|y) for all x, y ∈ L.

Proof Fix elements x, y ∈ L, set f = ad x ad y, and set g = ad (σx) ad (σy). Let {v_{i}} be a basis of L. For each z ∈ L, write

f(z) = ∑_{i} f_{i}(z)v_{i}

with f_{i}(z) ∈ F. Then

(x|y) = tr(f) = ∑_{i} f_{i}(v_{i}).

Since σ is an automorphism, {σv_{i}} is also a basis of L. With respect to this basis, we see that

g(σv_{i}) = [σx, [σy, σv_{i}]]

= σ[x, [y, v_{i}]] = σf(v_{i})

= ∑_{j} f_{j}(v_{i})σv_{j}.

Therefore,

(σx|σy) = tr(g) = ∑_{i} f_{i}(v_{i}),

and (x|y) = (σx|σy). □

The following lemma appears as [11, Lm 8.1] in the context of simple Lie algebras. The fact that L^{σ} is reductive had also appeared previously in [8, §1, no. 5] and [10, Chap III].

Lemma A.2 Let σ be a finite-order automorphism of a finite-dimensional reductive Lie algebra L. Let H be a Cartan subalgebra of the fixed point subalgebra L^{σ} = {x ∈ L | σx = x}. Then L^{σ} is reductive, and the centralizer ℂ_{L}(H) of H in L is a Cartan subalgebra of L.

Proof The same arguments given in [11, Lm 8.1] hold for the reductive case as well. The only exception is the justification for the σ-invariance of the Killing form, which we have already verified in Lemma A.1. □

We can now prove the existence of a σ_{1}, …, σ_{N}-invariant Cartan subalgebra:

Theorem A.3 Let σ_{1}, …, σ_{N} be commuting finite-order automorphisms of a finite-dimensional reductive Lie algebra L. Then L has a (setwise) σ_{1}, …, σ_{N}- invariant Cartan subalgebra 𝔥.

Proof We induct on the dimension of L. If the dimension of L is 0 or 1, then the theorem holds since 𝔥 = L. We can also assume that σ_{1} : L → L is not the identity map.

By Lemma A.2, the (σ_{1}, …, σ_{N}-invariant) fixed point subalgebra

L^{σ_{1}} = {x ∈ L | σ_{1}x = x}

is reductive, and by the induction hypothesis, it has a σ_{1}, …, σ_{N}-invariant Cartan subalgebra H. By Lemma A.2, the centralizer 𝔥 = ℂ_{L}(H) in L is a Cartan subalgebra of L. It is also σ_{1}, …, σ_{N}-invariant, since

[σ_{i}x, h] = σ_{i}[x, σ_{i}^{−1}(h)] ∈ σ_{i}[x, H] = 0

for all x ∈ 𝔥, h ∈ H, and i = 1, …, N. □

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