Representations of Multiloop Algebras

Michael Lau^{∗}

University of Windsor Department of Mathematics and Statistics Windsor, Ontario, Canada N9B 3P4

Abstract. We describe the finite-dimensional simple modules of all the (twisted and untwisted) multiloop algebras and classify them up to isomorphism.

MSC: 17B10, 17B65

1 Introduction

Multiloop algebras are multivariable generalizations of the loop algebras appearing in affine Kac-Moody theory. The study of these algebras and their extensions includes a substantial literature on (twisted and untwisted) multiloop, toroidal, and extended affine Lie algebras. This paper describes the finite-dimensional simple modules of multiloop algebras and classifies them up to isomorphism.

Let 𝔤 be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic zero. Suppose that σ_{1}, …, σ_{N} : 𝔤 → 𝔤 are commuting automorphisms of finite orders m_{1}, …, m_{N}, respectively. For each i, fix a primitive m_{i}th root of unity ξ_{i} ∈ F. Then 𝔤 decomposes into common eigenspaces relative to these automorphisms:

𝔤 = ⨁_{\overline{k}∈G} 𝔤\overline{k},

∗Funding from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

E-mail: mlau@uwindsor.ca

where 𝔤\overline{k} = \{x ∈ 𝔤 | σ_{i}x = ξ_{i}^{k_{i}}x\} and \overline{k} is the image of each k ∈ ℤ^{N} under the canonical map ℤ^{N} → G = ℤ/m_{1}ℤ×⋯×ℤ/m_{N}ℤ. The multiloop algebra of 𝔤, relative to these automorphisms, is the Lie algebra

L = L(𝔤; σ_{1}, …, σ_{N}) = ⨁_{k∈ℤ^{N}} 𝔤\overline{k} ⊗ F t^{k},

where F t^{k} is the span of t^{k} = t_{1}^{k_{1}} ⋯ t_{N}^{k_{N}}, and multiplication is defined pointwise. If the automorphisms σ_{1}, …, σ_{N} are all trivial, L is called an untwisted multiloop algebra. Otherwise, it is a twisted multiloop algebra.

In the one variable case (untwisted and twisted loop algebras), the finitedimensional simple modules can be deduced from ideas in the work of V. Chari and A. Pressley [8, 10, 12]. A complete list of the finite-dimensional simple modules also appears explicitly in [18]. These modules are classified up to isomorphism in [11], and a very recent paper [9] gives a detailed discussion of this problem in the twisted loop case.

A description of the finite-dimensional simple modules of the untwisted multiloop Lie algebras was first given by S.E. Rao [19]. Subsequent work by P. Batra [4] provided a complete (but redundant) list of the finitedimensional simple modules when σ_{1} is a diagram automorphism and the other automorphisms σ_{2}, …, σ_{N} are all trivial.

In the one variable case, every twisted loop algebra L(𝔤; τ ∘ γ) defined by an inner automorphism τ and a diagram automorphism γ is isomorphic to L(𝔤; γ) [15, Prop 8.5]. It was thus sufficient to consider only twists by diagram automorphisms in this setting. Unfortunately, this is far from true when the number of variables is larger than one. (See [13, Rem 5.9], for instance.) It has recently been shown that the centreless core of almost every extended affine Lie algebra is a multiloop algebra [2] (using [1] and [16]). Even for these multiloop algebras, any number of the automorphisms σ_{i} can be nontrivial, and any number of them can fail to be diagram automorphisms.

In this paper, we consider an arbitrary (twisted or untwisted) multiloop algebra L. From any ideal I of L, we construct a G-graded ideal I = I(I) of the ring of Laurent polynomials R = F [t_{1}^{±1}, …, t_{N}^{±1}]. If I is the kernel of a finite-dimensional irreducible representation, the \overline{0}-component I\overline{0} of the ideal I turns out to be a radical ideal of the \overline{0}-component of the ring R.

The resulting decomposition of I\overline{0} into an intersection of a finite number of maximal ideals produces an isomorphism

ψ_{a} : L/I → 𝔤 ⊕ ⋯ ⊕ 𝔤 (r copies)

whose composition with the quotient map π : L → L/I is evaluation at an r-tuple a = (a_{1}, …, a_{r}) of points a_{i} ∈ (F^{×})^{N}:

ψ_{a} ∘ π : x ⊗ f(t) ↦ (f(a_{1})x, …, f(a_{r})x),

for any x ⊗ f(t) ∈ L. Since the finite-dimensional simple modules of the semisimple Lie algebra 𝔤⊕⋯⊕𝔤 are the tensor products of finite-dimensional simple modules for 𝔤, we obtain a complete (but redundant) list of the finitedimensional irreducible representations of L (Corollary 4.11). Namely, any finite-dimensional simple module for L is of the form

V (λ, a) = V_{λ_{1}}(a_{1}) ⊗ ⋯ ⊗ V_{λ_{r}}(a_{r}),

where V_{λ_{i}} is the 𝔤-module of dominant integral highest weight λ_{i}, and V_{λ_{i}}(a_{i}) is the L-module obtained by evaluating elements of L at the point a_{i}, and then letting the resulting element of 𝔤 act on V_{λ_{i}}. The r-tuples a = (a_{1}, …, a_{r}) that occur in this process must satisfy the condition that the points m(a_{i}) are all distinct, where m(a_{i}) = (a_{i1}^{m_{1}}, …, a_{iN}^{m_{N}}) is determined by the orders m_{1}, …, m_{N} of the automorphisms σ_{1}, …, σ_{N}. Conversely, the L-module V (λ, a) is finite-dimensional and simple if the a_{i} satisfy this condition (Theorem 4.12).

In the second half of the paper, we establish necessary and sufficient conditions for L-modules V (λ, a) and V (μ, b) to be isomorphic. Namely, we “pull back” a triangular decomposition N_{−} ⊕ H ⊕ N_{+} of 𝔤 ⊕ ⋯ ⊕ 𝔤 to a triangular decomposition ψ_{a}^{−1}(N_{−})⊕ψ_{a}^{−1}(H)⊕ψ_{a}^{−1}(N_{+}) of L/I. The modules V (λ, a) and V (μ, b) are highest weight with respect to this decomposition of L/I, and they are isomorphic if and only if they have the same highest weights. The paper concludes with three equivalent criteria for isomorphism in terms of an explicit formula (Theorem 5.7), orbits under a group action (Corollary 5.15), and equivariant maps (Corollary 5.20). These are the first such isomorphism results for modules in any multiloop setting.

One of the most interesting features is that the triangular decomposition N_{−} ⊕H ⊕N_{+} is replaced with a new triangular decomposition ψ_{b}ψ_{a}^{−1}(N_{−})⊕ ψ_{b}ψ_{a}^{−1}(H) ⊕ ψ_{b}ψ_{a}^{−1}(N_{+}) of 𝔤^{⊕r} in the computation of the highest weight of V (μ, b). Unlike diagram automorphisms, arbitrary finite order automorphisms σ_{i} often fail to stabilize any triangular decomposition of a finitedimensional semisimple Lie algebra. This fact is reflected in the change of triangular decomposition on 𝔤^{⊕r}, and it is one of the reasons that past work considered only twists by diagram automorphisms.

Another novelty in this classification is the passage from twists by a single nontrivial automorphism σ_{1} to a family of nontrivial automorphisms σ_{1}, …, σ_{N}. Here the major obstacle to past approaches was reliance on the representation theory of the fixed point subalgebra 𝔤\overline{0} under the action of the automorphisms. While this was a great success when working with twists by a single automorphism, it cannot be used when considering twists by more than one automorphism, since the algebra 𝔤\overline{0} is then often 0. We avoid this pitfall by using a new approach that does not rely on the usual Dynkin diagram ”folding arguments”.

Finally, we expect the methods of this paper to be applicable in many related settings. For example, the arguments given here classify the finitedimensional simple modules of the Lie algebra Map(X, 𝔤) of 𝔤-valued regular functions on any complex variety X. When X is affine, they are tensor products of evaluation modules at distinct points of X. One might also consider modules of invariants under a finite group action G ↷ Map(X, 𝔤). Unfortunately, such a classification requires Lemma 2.9, which fails if not every graded component of the coordinate ring contains a unit. Incorporating twists by arbitrary actions of the group G greatly complicates matters, and we restrict our attention to the multiloop case in this paper. Another interesting direction is the classification of ℤ^{N}-graded modules of L(𝔤; σ_{1}, …, σ_{N}) with finite-dimensional graded components. (See [17, 19] for partial results.) We expect that the ideas introduced in this paper are sufficient to resolve this question, though an explicit description will likely also require the classification of thin coverings [5] or an equivalent concept.

Acknowledgements: The author thanks V. Chari, E. Neher, and P. Senesi for their careful reading and comments on an earlier draft.

Note: Throughout this paper, F will be an algebraically closed field of characteristic zero. All Lie algebras, linear spans, and tensor products will be taken over F unless otherwise explicitly indicated. We will denote the integers by ℤ, the nonnegative integers by ℤ_{+}, and the nonzero elements of F by F^{×}.

2 Multiloop algebras and their ideals

The following proposition is an immediate consequence of general facts about reductive Lie algebras [7, §6, no. 4].

Proposition 2.1 Let L be a perfect Lie algebra over F, and let ϕ : L → End V be a finite-dimensional irreducible representation. Then L/ ker ϕ is a semisimple Lie algebra.

Proof The representation ϕ descends to a faithful representation of L/ ker ϕ. By [7, Prop 6.4.5], any Lie algebra with a faithful finite-dimensional irreducible representation is reductive. Moreover, L is perfect. Therefore, L/ ker ϕ is perfect and reductive, and hence semisimple. □

We now focus our attention on multiloop algebras. Let 𝔤 be a finitedimensional simple Lie algebra over F, and let R = F [t_{1}^{±1}, …, t_{N}^{±1}] be the commutative algebra of Laurent polynomials in N variables. The untwisted multiloop algebra is the Lie algebra 𝔤 ⊗ R with (bilinear) pointwise multiplication given by

[x ⊗ f, y ⊗ g] = [x, y] ⊗ fg

for all x, y ∈ 𝔤 and f, g ∈ R. Suppose that 𝔤 is equipped with N commuting automorphisms

σ_{1}, …, σ_{N} : 𝔤 → 𝔤

of finite orders m_{1}, …, m_{N}, respectively. For each i, fix ξ_{i} ∈ F to be a primitive m_{i}th root of 1. Then 𝔤 has a common eigenspace decomposition 𝔤 = ⨁_{\overline{k}∈G} 𝔤\overline{k} where \overline{k} is the image of k = (k_{1}, …, k_{N}) ∈ ℤ^{N} under the canonical map

ℤ^{N} → G = ℤ/m_{1}ℤ × ⋯ × ℤ/m_{N}ℤ

and

𝔤\overline{k} = \{x ∈ 𝔤 | σ_{i}x = ξ_{i}^{k_{i}}x for i = 1, …, N\}.

The (twisted) multiloop algebra L = L(𝔤; σ_{1}, …, σ_{N}) is the Lie subalgebra

L = ⨁_{k∈ℤ^{N}} 𝔤\overline{k} ⊗ F t^{k} ⊆ 𝔤 ⊗ R,  (2.2)

where t^{k} = t_{1}^{k_{1}} ⋯ t_{N}^{k_{N}} is multi-index notation.

Note that R has a G-grading,

R = ⨁_{\overline{k}∈G} R\overline{k},  (2.3)

where R\overline{0} = F [t_{1}^{±m_{1}}, …, t_{N}^{±m_{N}}] and R\overline{k} = t^{k}R\overline{0} for every k ∈ ℤ^{N}. In this notation,

L = ⨁_{\overline{k}∈G} (𝔤\overline{k} ⊗ R\overline{k}) .  (2.4)

Fix an F -basis

\{x_{\overline{k}j} | j = 1, …, dim 𝔤\overline{k}\}  (2.5)

of 𝔤\overline{k} for all \overline{k} ∈ G. Then

L = ⨁_{\overline{k}∈G} ⨁_{j=1}^{dim 𝔤\overline{k}} (F x_{\overline{k}j} ⊗ R\overline{k}) .  (2.6)

Since 𝔤 is simple (hence perfect) and graded, each x_{\overline{k}j} can be expressed as a sum of brackets of homogeneous elements y, z ∈ 𝔤, with deg y + deg z = \overline{k}. For each such k ∈ ℤ^{N} and pair y, z, there exist a, b ∈ ℤ^{N} with deg y = \overline{a}, deg z = \overline{b}, and a + b = k. Then the sum of the brackets [y ⊗ t^{a}, z ⊗ t^{b}] will be x_{\overline{k}j} ⊗ t^{k}. Since these elements span L, it is clear that L is perfect.

Let π_{\overline{k}j} be the projection π_{\overline{k}j} : L → F x_{\overline{k}j} ⊗ R\overline{k}, relative to the decomposition (2.6). We will view π_{\overline{k}j} as a projection L → R\overline{k} by identifying x_{\overline{k}j} ⊗ f with f for all f ∈ R\overline{k}. Let I be an ideal of the Lie algebra L, and let I = I(I) be the ideal of R generated by

⋃_{\overline{k}∈G} ⋃_{j=1}^{dim 𝔤\overline{k}} π_{\overline{k}j}(I).  (2.7)

Note that the definition of I is independent of the choice of homogeneous basis \{x_{\overline{k}j}\} of 𝔤, and the ideal I is G-graded since its generators are homogeneous with respect to the G-grading of R. That is,

I = ⨁_{\overline{k}∈G} I\overline{k} where I\overline{k} = I ∩ R\overline{k}.

Moreover,

t^{ℓ−k}I\overline{k} ⊆ I ∩ R\overline{ℓ} = I\overline{ℓ} = t^{ℓ−k} (t^{k−ℓ}I\overline{ℓ}) ⊆ t^{ℓ−k}I\overline{k},

so

I\overline{ℓ} = t^{ℓ−k}I\overline{k},  (2.8)

for all k, ℓ ∈ ℤ^{N}. We will use the following technical lemma to show that I = L ∩ (𝔤 ⊗ I).

Lemma 2.9 Let Y = ∑_{\overline{r}∈G} ∑_{n=1}^{dim 𝔤\overline{r}} x_{\overline{r}n} ⊗ π_{\overline{r}n}(Y) ∈ I. Then

x_{\overline{k}i} ⊗ t^{k−ℓ}π_{\overline{ℓ}j}(Y) ∈ I

for all k, ℓ ∈ ℤ^{N}, 1 ≤ i ≤ dim 𝔤\overline{k}, and 1 ≤ j ≤ dim 𝔤\overline{ℓ}.

Proof The finite-dimensional simple Lie algebra 𝔤 is a finite-dimensional simple 𝔤-module (and hence a finite-dimensional simple U(𝔤)-module) under the adjoint action of 𝔤. Fix k, ℓ ∈ ℤ^{N}, i ∈ \{1, …, dim 𝔤\overline{k}\}, and j ∈ \{1, …, dim 𝔤\overline{ℓ}\}. By the Jacobson Density Theorem, there exists u ∈ U(𝔤) such that

u.x_{\overline{r}n} = \{ \begin{matrix}x_{\overline{k}i} & if \overline{r} = \overline{ℓ} and n = j \\ 0 & otherwise, \end{matrix}

for all \overline{r} ∈ G and n ∈ \{1, …, 𝔤\overline{r}\}. By the Poincaré-Birkhoff-Witt Theorem, write u = ∑_{s=1}^{a} p_{s}, where a ≥ 1 and each p_{s} is a monomial in the variables in \{x_{\overline{r}n} | \overline{r} ∈ G, n = 1, …, dim 𝔤\overline{r}\}. Considering the induced G-grading of U(𝔤), we can assume that each p_{s} is homogeneous of degree \overline{k − ℓ}. Write

p_{s} = c_{s} ∏_{\overline{r}∈G} ∏_{n=1}^{dim 𝔤\overline{r}} (x_{\overline{r}n})^{b_{\overline{r}n}^{(s)}}

where c_{s} ∈ F and b_{\overline{r}n}^{(s)} ∈ ℤ_{+}.

Since p_{s} is homogeneous of degree \overline{k − ℓ} in the G-grading of U(𝔤), we can choose α(s, \overline{r}, n, 1), α(s, \overline{r}, n, 2), …, α(s, \overline{r}, n, b_{\overline{r}n}^{(s)}) ∈ ℤ^{N} for each s ∈ \{1, …, a\}, \overline{r} ∈ G, and n ∈ \{1, …, dim 𝔤\overline{r}\}, such that

(i) \overline{r} = \overline{α(s, \overline{r}, n, 1)} = ⋯ = \overline{α(s, \overline{r}, n, b_{\overline{r}n}^{(s)})},

(ii) ∑_{\overline{r}∈G} ∑_{n=1}^{dim 𝔤\overline{r}} ∑_{b=1}^{b_{\overline{r}n}^{(s)}} α(s, \overline{r}, n, b) = k − ℓ.

Then

p̃_{s} = c_{s} ∏_{\overline{r}∈G} ∏_{n=1}^{dim 𝔤\overline{r}} ∏_{b=1}^{b_{\overline{r}n}^{(s)}} (x_{\overline{r}n} ⊗ t^{α(s,\overline{r},n,b)})

is in the universal enveloping algebra U(L) of L, which acts on I via the adjoint action of L on I, and

∑_{s=1}^{a} p̃_{s}.Y = x_{\overline{k}i} ⊗ t^{k−ℓ}π_{\overline{ℓ}j}(Y).

Thus x_{\overline{k}i} ⊗ t^{k−ℓ}π_{\overline{ℓ}j}(Y) ∈ I. □

Proposition 2.10 In the notation introduced above,

I = L ∩ (𝔤 ⊗ I)  (2.11)
= ⨁_{\overline{k}∈G} 𝔤\overline{k} ⊗ I\overline{k}.  (2.12)

Proof The second equality (2.12) and the inclusion I ⊆ L ∩ (𝔤 ⊗ I) are clear, so it remains only to verify the reverse inclusion

L ∩ (𝔤 ⊗ I) ⊆ I.

In light of (2.12), it suffices to show that x_{\overline{k}i} ⊗ f ∈ I for all \overline{k} ∈ G, i ∈ \{1, …, dim 𝔤\overline{k}\}, and f ∈ I\overline{k}.

By the definition of I, there exist Y_{\overline{ℓ}j} ∈ I and f_{\overline{ℓ}j} ∈ R\overline{k−ℓ} such that f = ∑_{\overline{ℓ}∈G} ∑_{j=1}^{dim 𝔤\overline{ℓ}} f_{\overline{ℓ}j}π_{\overline{ℓ}j} (Y_{\overline{ℓ}j}) . By Lemma 2.9,

x_{\overline{k}i} ⊗ t^{r}π_{\overline{ℓ}j} (Y_{\overline{ℓ}j}) ∈ I

for all r, ℓ ∈ ℤ^{N} satisfying \overline{r} = \overline{k − ℓ}. Since each f_{\overline{ℓ}j} ∈ R\overline{k−ℓ} is an F -linear combination of \{t^{r} | \overline{r} = \overline{k − ℓ}\}, we see that

x_{\overline{k}i} ⊗ f_{\overline{ℓ}j}π_{\overline{ℓ}j} (Y_{\overline{ℓ}j}) ∈ I

for all \overline{ℓ} ∈ G and j = 1, …, dim 𝔤\overline{ℓ}. Thus x_{\overline{k}i} ⊗ f ∈ I. □

We close this section by considering the structure of I\overline{0} ⊆ R\overline{0} in the case where I is the kernel of an irreducible finite-dimensional representation of L. Clearly I\overline{0} is an ideal of R\overline{0}. Moreover, it is a radical ideal:

Proposition 2.13 Let ϕ : L → End V be a finite-dimensional irreducible representation of the multiloop algebra L, and let I = ker ϕ. Define I = I(I) ⊆ R as above. Then the graded component I\overline{0} is a radical ideal of R\overline{0}.

Proof Suppose p is an element of \sqrt{I\overline{0}}, the radical of the ideal I\overline{0} = I ∩ R\overline{0} of R\overline{0}. Choose k ∈ ℤ^{N} so that 𝔤\overline{k} ≠ 0, and let x ∈ 𝔤\overline{k} be a nonzero element.

For y ⊗ f ∈ L, let ⟨y ⊗ f⟩ ⊆ L be the ideal (of L) generated by y ⊗ f. Let J = ⟨x ⊗ t^{k}p⟩, and note that the nth term J^{(n)} in the derived series of J satisfies

J^{(n)} ⊆ L ∩ (𝔤 ⊗ ⟨p^{n}⟩)

where ⟨p^{n}⟩ is the principal ideal of R generated by p^{n}. Since I\overline{ℓ} = t^{ℓ}I\overline{0} for all ℓ ∈ ℤ^{N} by (2.8), and since p^{n} ∈ I\overline{0} for n sufficiently large, we see that

J^{(n)} ⊆ L ∩ (𝔤 ⊗ I)

for n ≫ 0. Then by Proposition 2.10, J^{(n)} ⊆ I, so

\frac{J + I}{I} ⊆ Rad (L/I) .

Since Rad (L/I) = 0 by Proposition 2.1, we see that x ⊗ t^{k}p ∈ I. That is, p = t^{−k}(t^{k}p) ∈ t^{−k}I\overline{k} = I\overline{0}, and thus \sqrt{I\overline{0}} = I\overline{0}. □

3 Some commutative algebra

In this short section, we recall some basic commutative algebra that will be useful in the context of classifying modules for multiloop algebras. Recall that F is an algebraically closed field of characteristic zero, F^{×} = F \ 0 is its group of units, and R = F [t_{1}^{±1}, …, t_{N}^{±1}]. For any ideal I ⊆ R, let V(I) = {x ∈ (F^{×})^{N} | f(x) = 0 for all f ∈ I} be the (quasiaffine) variety corresponding to I, and let Poly(S) = {g ∈ R | g(s) = 0 for all s ∈ S} be the ideal associated with any subset S ⊆ (F^{×})^{N}.

Proposition 3.1 Let I be an ideal of R = F [t_{1}^{±1}, …, t_{N}^{±1}]. Then

Poly(V(I)) = \sqrt{I}.

Proof It is straightforward to verify that the usual proofs of the Hilbert Nullstellensatz (cf. [3, p. 85 ], for instance) also hold for this Laurent polynomial analogue. □

The following crucial lemma is an easy consequence of the Nullstellensatz (Proposition 3.1):

Lemma 3.2 Let J be a radical ideal of R, for which the quotient R/J is a finite-dimensional vector space over F. Then there exist distinct points a_{1}, …, a_{r} ∈ (F^{×})^{N} so that

J = 𝔪_{a_{1}} ∩ ⋯ ∩ 𝔪_{a_{r}},

where 𝔪_{a_{i}} = ⟨t_{1} − a_{i1}, …, t_{N} − a_{iN}⟩ is the maximal ideal corresponding to a_{i} = (a_{i1}, …, a_{iN}) for i = 1, …, r. Moreover, {a_{1}, …, a_{r}} is unique (up to permutation).

Proof Clearly, a ∈ V(J) implies that J ⊆ 𝔪_{a}, so J ⊆ ⋂_{a∈V(J)} 𝔪_{a}. Conversely, if f ∈ ⋂_{a∈V(J)} 𝔪_{a} and x ∈ V(J), then f(x) = 0 and f ∈ Poly(V(J)) = \sqrt{J} = J. Hence J = ⋂_{a∈V(J)} 𝔪_{a}.

Since J ⊆ 𝔪_{a_{1}} ∩ ⋯ ∩ 𝔪_{a_{r}} for all subsets {a_{1}, …, a_{r}} ⊆ V(J), we see that the (F -vector space) dimension of R/(𝔪_{a_{1}} ∩ ⋯ ∩ 𝔪_{a_{r}}) is bounded by dim_{F} (R/J). Take a finite collection {a_{1}, …, a_{r}} of points in V(J) for which this dimension is maximal. Then 𝔪_{a_{1}} ∩ ⋯ ∩ 𝔪_{a_{r}} ∩ 𝔪_{a_{r+1}} = 𝔪_{a_{1}} ∩ ⋯ ∩ 𝔪_{a_{r}} for all points a_{r+1} ∈ V(J), so

J = ⋂_{b∈V(J)} 𝔪_{b}

= 𝔪_{a_{1}} ∩ ⋯ ∩ 𝔪_{a_{r}} ∩ ( ⋂_{b∈V(J)} 𝔪_{b})
= 𝔪_{a_{1}} ∩ ⋯ ∩ 𝔪_{a_{r}}.

To see that {a_{1}, …, a_{r}} ⊆ (F^{×})^{N} is uniquely determined, suppose that J = 𝔪_{a_{1}} ∩ ⋯ ∩ 𝔪_{a_{r}} = 𝔪_{b_{1}} ∩ ⋯ ∩ 𝔪_{b_{s}} for some a_{1}, …, a_{r}, b_{1}, …, b_{s} ∈ (F^{×})^{N}. Then

{a_{1}, …, a_{r}} = V(𝔪_{a_{1}} ∩ ⋯ ∩ 𝔪_{a_{r}})
= V(J)
= V(𝔪_{b_{1}} ∩ ⋯ ∩ 𝔪_{b_{s}})
= {b_{1}, …, b_{s}}.
□

Note that the ideal I\overline{0} ⊆ R\overline{0} of Proposition 2.13 is radical and cofinite. Viewing R\overline{0} = F [t_{1}^{±m_{1}}, …, t_{N}^{±m_{N}}] as the ring of Laurent polynomials in the variables t_{1}^{m_{1}}, …, t_{N}^{m_{N}}, we see that

I\overline{0} = M_{a_{1}} ∩ ⋯ ∩ M_{a_{r}},  (3.3)

where \{a_{1}, …, a_{r}\} = V(I\overline{0}) is a set of distinct points in (F^{×})^{N}, and M_{a_{i}} = ⟨t_{1}^{m_{1}} − a_{i1}, …, t_{N}^{m_{N}} − a_{iN}⟩_{R\overline{0}} is the maximal ideal of R\overline{0} corresponding to the point a_{i} = (a_{i1}, …, a_{iN}). Then by the Chinese Remainder Theorem, we have the following corollary:

Corollary 3.4 Let I\overline{0} and R\overline{0} be as in Proposition 2.13. Then there exist unique (up to reordering) points a_{1}, …, a_{r} ∈ (F^{×})^{N} so that the canonical map

R\overline{0}/I\overline{0} → R\overline{0}/M_{a_{1}} × ⋯ × R\overline{0}/M_{a_{r}}  (3.5)
f + I\overline{0} ↦ (f + M_{a_{1}}, …, f + M_{a_{r}})  (3.6)

is a well-defined F -algebra isomorphism.

□

4 Classification of simple modules

We now return to classifying the finite-dimensional simple modules of multiloop algebras. As in §2, let 𝔤 be a finite-dimensional simple Lie algebra, and let ϕ : L → End V be a finite-dimensional irreducible representation of a multiloop algebra L = L(𝔤; σ_{1}, …, σ_{N}) defined by commuting automorphisms σ_{1}, …, σ_{N} : 𝔤 → 𝔤 of order m_{1}, …, m_{N}, respectively.

Letting I = ker ϕ, I = I(I), G = ℤ/m_{1}ℤ × ⋯ × ℤ/m_{N}ℤ, and R = F [t_{1}^{±1}, …, t_{N}^{±1}] be defined as in §2, we see that

L = ⨁_{\overline{k}∈G} 𝔤\overline{k} ⊗ R\overline{k} and I = ⨁_{\overline{k}∈G} 𝔤\overline{k} ⊗ I\overline{k},

by Proposition 2.10. Since I is a G-graded ideal of L, we have

L/I = ⨁_{\overline{k}∈G} ((𝔤\overline{k} ⊗ R\overline{k}) / (𝔤\overline{k} ⊗ I\overline{k}))  (4.1)
= ⨁_{\overline{k}∈G} 𝔤\overline{k} ⊗ (R\overline{k}/I\overline{k}) .  (4.2)

Each graded component R\overline{k}/I\overline{k} of R/I is an R\overline{0}-module, and it is easy to check that the map

μ_{k} : R\overline{0}/I\overline{0} → R\overline{k}/I\overline{k}  (4.3)
f + I\overline{0} ↦ t^{k}f + I\overline{k}  (4.4)

is a well-defined R\overline{0}-module homomorphism for each k ∈ ℤ^{N} and f ∈ R\overline{0}. By (2.3) and (2.8), R\overline{k} = t^{k}R\overline{0} and t^{−k}I\overline{k} = I\overline{0}, so the map μ_{k} is both surjective and injective. Hence the following lemma holds:

Lemma 4.5 Let k ∈ ℤ^{N}. Then the map μ_{k} : R\overline{0}/I\overline{0} → R\overline{k}/I\overline{k} is a welldefined isomorphism of R\overline{0}-modules. In particular, each graded component R\overline{k}/I\overline{k} has the same dimension (as a vector space):

dim (R\overline{0}/I\overline{0}) = dim (R\overline{k}/I\overline{k}) .

□

Let a_{1}, …, a_{r} ∈ (F^{×})^{N} be the (unique) points defined by Corollary 3.4, and let b_{i} = (b_{i1}, …, b_{iN}) be a point in (F^{×})^{N} such that b_{ij}^{m_{j}} = a_{ij} for all 1 ≤ i ≤ r and 1 ≤ j ≤ N. Recall that I\overline{k} = t^{k}I\overline{0} for all k ∈ ℤ^{N}, and I\overline{0} is contained in the ideal M_{a_{i}} of R\overline{0} for i = 1, …, r. Therefore, the map

ψ = ψ_{b} : L → 𝔤 ⊕ ⋯ ⊕ 𝔤 (r copies)  (4.6)
x ⊗ f ↦ (f(b_{1})x, …, f(b_{r})x)  (4.7)

descends to a well-defined Lie algebra homomorphism

\overline{ψ} : L/ ker ϕ → 𝔤 ⊕ ⋯ ⊕ 𝔤.  (4.8)

Theorem 4.9 The map \overline{ψ} : L/ ker ϕ → 𝔤 ⊕ ⋯ ⊕ 𝔤 in (4.8) is a Lie algebra isomorphism.

Proof Let k ∈ ℤ^{N}, and let

\overline{ψ}\overline{k} : 𝔤\overline{k} ⊗ (R\overline{k}/I\overline{k}) → 𝔤\overline{k} ⊕ ⋯ ⊕ 𝔤\overline{k}

be the restriction of \overline{ψ} to the graded component 𝔤\overline{k} ⊗ (R\overline{k}/I\overline{k}) of L/ ker ϕ.

Note that the map \overline{ψ} is injective if each \overline{ψ}\overline{k} is injective. In the notation of (2.5), if

u = ∑_{j=1}^{dim 𝔤\overline{k}} x_{\overline{k}j} ⊗ (t^{k}f_{j}(t) + I\overline{k})

is in the kernel of \overline{ψ}\overline{k} for some collection of f_{j} ∈ R\overline{0}, then b_{i}^{k}f_{j}(b_{i}) = 0 for all i and j. Then for all i and j, we have f_{j}(b_{i}) = 0 and f_{j} ∈ M_{a_{i}}, where M_{a_{i}} is the ideal of R\overline{0} generated by \{t_{ℓ}^{m_{ℓ}} − a_{iℓ} | ℓ = 1, …, N\}. Hence f_{j} ∈ ⋂_{i=1}^{r} M_{a_{i}} = I\overline{0}, so t^{k}f_{j}(t) ∈ t^{k}I\overline{0} = I\overline{k}, and

∑_{j=1}^{dim 𝔤\overline{k}} x_{\overline{k}j} ⊗ t^{k}f_{j}(t) ∈ 𝔤\overline{k} ⊗ I\overline{k}
⊆ ker ϕ.

Hence u = 0 in L/ ker ϕ, so \overline{ψ}\overline{k} (and thus \overline{ψ}) is injective.

By Lemma 4.5, dim (R\overline{ℓ}/I\overline{ℓ}) = dim (R\overline{0}/I\overline{0}) for all ℓ ∈ ℤ^{N}. Therefore,

dim (L/ ker ϕ) = ∑_{\overline{ℓ}∈G} (dim 𝔤\overline{ℓ}) (dim (R\overline{ℓ}/I\overline{ℓ}))
= dim (R\overline{0}/I\overline{0}) dim 𝔤.

Since F is algebraically closed, R\overline{0}/M_{a_{i}} ≅ F for every i, so the (F -vector space) dimensions satisfy

dim (R\overline{0}/I\overline{0}) = dim (R\overline{0}/M_{a_{1}} × ⋯ × R\overline{0}/M_{a_{r}})
= r,

by Corollary 3.4. Therefore, \overline{ψ} is an injective homomorphism between two Lie algebras of equal dimension, so \overline{ψ} is an isomorphism. □

The finite-dimensional simple modules over direct sums of copies of the Lie algebra 𝔤 are tensor products of finite-dimensional simple modules over 𝔤. (See [6, §7, no. 7], for instance.) We can thus conclude that the finitedimensional simple modules for multiloop algebras are pullbacks (under ψ) of tensor products of finite-dimensional simple modules over 𝔤.

Fix a Cartan subalgebra 𝔥 ⊂ 𝔤, a base Δ of simple roots, and weights λ_{i} ∈ 𝔥^{∗} for i = 1, …, r. Then we will write V_{λ_{i}}(b_{i}) for the simple 𝔤-module V_{λ_{i}} of highest weight λ_{i}, equipped with the L-action given by

(x ⊗ f(t)).v = f(b_{i})xv,

for all x ⊗ f ∈ L and v ∈ V_{λ_{i}}. The tensor product of such a family of evaluation modules will be denoted

V (λ, b) = V_{λ_{1}}(b_{1}) ⊗ ⋯ ⊗ V_{λ_{r}}(b_{r}),  (4.10)

and we will write m(b_{i}) for the point (b_{i1}^{m_{1}}, …, b_{iN}^{m_{N}}) ∈ (F^{×})^{N} for i = 1, ⋯, r. We have now proved one of our main results:

Corollary 4.11 Let V be a finite-dimensional simple module for the multiloop algebra L. Then there exist b_{1}, …, b_{r} ∈ (F^{×})^{N} and λ_{1}, …, λ_{r} dominant integral weights for 𝔤 such that V ≅ V (λ, b), where m(b_{i}) ≠ m(b_{j}) whenever i ≠ j.

□

Conversely, if the points m(b_{i}) ∈ (F^{×})^{N} are pairwise distinct, then such a tensor product of evaluation modules is simple:

Theorem 4.12 Let λ_{1}, …, λ_{r} be dominant integral weights for 𝔤, and let b_{1}, …, b_{r} ∈ (F^{×})^{N} satisfy the property that m(b_{i}) ≠ m(b_{j}) whenever i ≠ j. Then V (λ, b) is a finite-dimensional simple L-module.

Proof Let I\overline{0} be the intersection ⋂_{i=1}^{r} M_{a_{i}} of the maximal ideals M_{a_{i}} of R\overline{0}

corresponding to the points a_{i} = m(b_{i}). For any k, ℓ ∈ ℤ^{N}, we see that t^{k−ℓ}I\overline{0} = I\overline{0} if \overline{k} = \overline{ℓ} as elements of G = ℤ/m_{1}ℤ × ⋯ × ℤ/m_{N}ℤ. Thus t^{k}I\overline{0} = t^{ℓ}I\overline{0} if \overline{k} = \overline{ℓ}, so we can unambiguously define I\overline{k} = t^{k}I\overline{0} for any k ∈ ℤ^{N}.

Since a_{1}, …, a_{r} are pairwise distinct points in (F^{×})^{N}, the proof of Theorem 4.9 (in particular, the appeal to Corollary 3.4) shows that the map

ψ : L → 𝔤 ⊕ ⋯ ⊕ 𝔤 (r copies)
x ⊗ f(t) ↦ (f(b_{1})x, …, f(b_{r})x)

is surjective. Then since each V_{λ_{i}} is a simple 𝔤-module, the tensor product V_{λ_{1}} ⊗ ⋯ ⊗ V_{λ_{r}} is a simple module over 𝔤 ⊕ ⋯ ⊕ 𝔤, and the pullback V (λ, b) is a simple L-module. □

Remark 4.13 It is not difficult to verify that if m(b_{i}) = m(b_{j}) for some i ≠ j for which λ_{i} and λ_{j} are both nonzero, then V (λ, b) is not simple. However, as we do not need this fact for the classification of simple modules, we will omit its proof.

5 Isomorphism classes of simple modules

By Corollary 4.11 and Theorem 4.12, the finite-dimensional simple modules of the multiloop algebra L(𝔤; σ_{1}, …, σ_{N}) are precisely the tensor products

V (λ, a) = V_{λ_{1}}(a_{1}) ⊗ ⋯ ⊗ V_{λ_{r}}(a_{r})  (5.1)

for which all the λ_{i} ∈ 𝔥^{∗} are dominant integral, and m(a_{i}) ≠ m(a_{j}) whenever i ≠ j. If λ_{i} = 0 for some i, then V_{λ_{i}}(a_{i}) is the trivial module, and (up to isomorphism) this term can be omitted from the tensor product (5.1). With the convention that empty tensor products of L-modules are the 1-dimensional trivial module, we may assume that every λ_{i} is a nonzero dominant integral weight in (5.1).

To proceed further, we will need a lemma about how highest weights depend on triangular decompositions.

Let L be a finite-dimensional semisimple Lie algebra with Cartan subalgebra H and base of simple roots Δ ⊂ H^{∗}. Recall that the group Aut L of automorphisms of L is (canonically) a semidirect product of the group Int L of inner automorphisms and the group Out L of diagram automorphisms with respect to (H, Δ):

Aut L = Int L ⋊ Out L.  (5.2)

See [14, IX.4], for instance. Every automorphism θ can thus be decomposed as θ = τ ∘ γ with an inner part τ ∈ Int L and outer part γ ∈ Out L.

Lemma 5.3 Let H be a Cartan subalgebra of a finite-dimensional semisimple Lie algebra L, and let Δ ⊂ H^{∗} be a base of simple roots. Suppose that V is a finite-dimensional simple L-module of highest weight λ with respect to (H, Δ), and θ ∈ Aut L. Write θ = τ ∘ γ for some τ ∈ Int L and γ ∈ Out L.

Then Δ∘θ^{−1} = {α∘θ^{−1} | α ∈ Δ} is a base of simple roots for L, relative to the Cartan subalgebra θ(H) ⊂ L, and V has highest weight λ ∘ τ^{−1} with respect to (θ(H), Δ ∘ θ^{−1}).

Proof Any diagram automorphism with respect to (H, Δ) will preserve H and Δ, so V has highest weight λ with respect to (γ(H), Δ ∘ γ^{−1}) = (H, Δ). Therefore, it is enough to prove the lemma for the case where θ = τ is an inner automorphism. Since inner automorphisms are products of automorphisms of the form exp(ad x) for ad-nilpotent elements x ∈ L, we may also assume, without loss of generality, that τ = exp(ad u) for some ad-nilpotent element u.

Let ρ : L → End V be the homomorphism describing the action of L on V. Then for any v ∈ V,

τ(h).v = ( exp(ad u)(h)).v  (5.4)
= e^{ρ(u)}ρ(h)e^{−ρ(u)}v,  (5.5)

where e^{ρ(u)} denotes the matrix exponential of the endomorphism ρ(u). The map e^{ρ(u)} is invertible, so for any nonzero element

w ∈ V_{α}^{H} := {v ∈ V | h.v = α(h)v for all h ∈ H},

we see that e^{ρ(u)}w ≠ 0, and using (5.4)–(5.5),

τ(h).e^{ρ(u)}w = e^{ρ(u)}ρ(h)e^{−ρ(u)}e^{ρ(u)}w
= α(h)e^{ρ(u)}w.

That is,

e^{ρ(u)}V_{α}^{H} ⊆ V_{α∘τ^{−1}}^{τ(H)} := {v ∈ V | h.v = α ∘ τ^{−1}(h).v for all h ∈ τ(H)}.

The reverse inclusion follows similarly by considering τ^{−1} = exp(−ad u), so

e^{ρ(u)}V_{α}^{H} = V_{α∘τ^{−1}}^{τ(H)}  (5.6)

for all α ∈ H^{∗}. In the case where V is the adjoint module L, we now see that α is a root relative to H if and only if α ∘τ^{−1} is a root relative to τ(H). It follows easily that Δ ∘ τ^{−1} is a base of simple roots for L, with respect to the Cartan subalgebra τ(H).

The second part of the lemma also follows easily, since V_{λ∘τ^{−1}}^{τ(H)} = e^{ρ(u)}V_{λ}^{H} is nonzero, but

V_{λ∘τ^{−1}+α∘τ^{−1}}^{τ(H)} = e^{ρ(u)}V_{λ+α}^{H} = 0

for all α ∈ Δ. That is, the highest weight of V is λ ∘ τ^{−1}, relative to (τ(H), Δ ∘ τ^{−1}) = (θ(H), Δ ∘ θ^{−1}). □

Fix a base Δ of simple roots with respect to a Cartan subalgebra 𝔥 ⊆ 𝔤. The following theorem gives necessary and sufficient conditions for modules of the form V (λ, a) to be isomorphic.

Theorem 5.7 Let λ = (λ_{1}, …, λ_{r}) and μ = (μ_{1}, …, μ_{s}) be sequences of nonzero dominant integral weights with respect to Δ. Suppose a = (a_{1}, …, a_{r}) and b = (b_{1}, …, b_{s}) are sequences of points in (F^{×})^{N} with m(a_{i}) ≠ m(a_{j}) and m(b_{i}) ≠ m(b_{j}) whenever i ≠ j.

Then the finite-dimensional simple L-modules V (λ, a) and V (μ, b) are isomorphic if and only if r = s and there is a permutation π ∈ S_{r} satisfying the following two conditions for i = 1, …, r:

m(a_{i}) = m(b_{π(i)}) and λ_{i} = μ_{π(i)} ∘ γ_{i},

where γ_{i} is the outer part of the automorphism ω_{i} : 𝔤 → 𝔤 defined by ω_{i}(x) = (b_{π(i)}^{k}/a_{i}^{k})x for all k ∈ ℤ^{N} and x ∈ 𝔤\overline{k}.

Proof Let ϕ_{λ,a} : L → End V (λ, a) and ϕ_{μ,b} : L → End V (μ, b) be the Lie algebra homomorphisms defining the representations V (λ, a) and V (μ, b). By Theorem 4.9, the kernel of ϕ_{λ,a} is equal to the kernel of the evaluation map ψ_{a}, defined by

ψ_{a} :  L ⟶ 𝔤 ⊕ ⋯ ⊕ 𝔤
x ⊗ f ↦ (f(a_{1})x, …, f(a_{r})x)

for all x ⊗ f ∈ L. Similarly, ker ϕ_{μ,b} = ker ψ_{b}.

If the L-modules V (λ, a) and V (μ, b) are isomorphic, then ker ϕ_{λ,a} = ker ϕ_{μ,b}, so ker ψ_{a} = ker ψ_{b}. But ker ψ_{a} = ⨁_{\overline{k}∈G} 𝔤\overline{k} ⊗ I\overline{k}, where I\overline{k} = t^{k}I\overline{0} for all k ∈ ℤ^{N}, and

I\overline{0} = M_{m(a_{1})} ∩ ⋯ ∩ M_{m(a_{r})},

where M_{m(a_{i})} = ⟨t_{1}^{m_{1}} − a_{i1}^{m_{1}}, …, t_{N}^{m_{N}} − a_{iN}^{m_{N}} ⟩_{R\overline{0}} is the maximal ideal of R\overline{0} = F [t_{1}^{±m_{1}}, …, t_{N}^{±m_{N}}] corresponding to the point m(a_{i}) = (a_{i1}^{m_{1}}, …, a_{iN}^{m_{N}}). Since ker ψ_{a} = ker ψ_{b}, we see that (in the notation of §3):

{m(a_{1}), …, m(a_{r})} = V(M_{m(a_{1})} ∩ ⋯ ∩ M_{m(a_{r})})
= V(I\overline{0})
= V(M_{m(b_{1})} ∩ ⋯ ∩ M_{m(b_{s})})
= {m(b_{1}), …, m(b_{s})}.

Hence r = s, and there is a permutation π ∈ S_{r} such that m(a_{i}) = m(b_{π(i)}) for i = 1, …, r. We will write π(b) = (b_{π(1)}, …, b_{π(r)}).

Let 𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+} be the triangular decomposition of 𝔤 relative to Δ. Assuming that r = s and m(a_{i}) = m(b_{π(i)}) for all i, view V_{λ} = V_{λ_{1}} ⊗⋯⊗V_{λ_{r}} and V_{π(μ)} = V_{μ_{π(1)}} ⊗ ⋯ ⊗ V_{μ_{π(r)}} as highest weight modules of the semisimple Lie algebra 𝔤^{⊕r}, relative to the triangular decomposition

𝔤^{⊕r} = (𝔫_{−}^{⊕r}) ⊕ (𝔥^{⊕r}) ⊕ (𝔫_{+}^{⊕r}).  (5.8)

The highest weights of V_{λ} and V_{π(μ)} are λ and π(μ) = (μ_{π(1)}, …, μ_{π(r)}), respectively, where λ(h_{1}, …, h_{r}) = ∑_{i} λ_{i}(h_{i}) for all (h_{1}, …, h_{r}) ∈ 𝔥^{⊕r}, and π(μ) ∈ (𝔥^{⊕r})^{∗} is defined analogously.

We can pull back the triangular decomposition (5.8) via the isomorphism \overline{ψ}_{a} : L/ ker ψ_{a} → 𝔤^{⊕r} defined in (4.8). Then V (λ, a) and V (μ, b) are irreducible highest weight modules of the semisimple Lie algebra L/ ker ψ_{a}, relative to the triangular decomposition

L/ ker ψ_{a} = \overline{ψ}_{a}^{−1}(𝔫_{−}^{⊕r}) ⊕ \overline{ψ}_{a}^{−1}(𝔥^{⊕r}) ⊕ \overline{ψ}_{a}^{−1}(𝔫_{+}^{⊕r}).  (5.9)

The L-modules V (λ, a) and V (μ, b) are isomorphic if and only if they have the same highest weights relative to the decomposition (5.9). Since \overline{ψ}_{a} maps the decomposition (5.9) to the decomposition (5.8), the highest weight of V (λ, a) is clearly λ ∘ \overline{ψ}_{a} : \overline{ψ}_{a}^{−1}(𝔥^{⊕r}) → F.

The highest weight of V (μ, b) is ν ∘ \overline{ψ}_{π(b)}, where ν ∈ (\overline{ψ}_{π(b)}\overline{ψ}_{a}^{−1}(𝔥^{⊕r}))^{∗} is the highest weight of V_{π(μ)} relative to the new triangular decomposition

𝔤^{⊕r} = \overline{ψ}_{π(b)}\overline{ψ}_{a}^{−1}(𝔫_{−}^{⊕r}) ⊕ \overline{ψ}_{π(b)}\overline{ψ}_{a}^{−1}(𝔥^{⊕r}) ⊕ \overline{ψ}_{π(b)}\overline{ψ}_{a}^{−1}(𝔫_{+}^{⊕r}).  (5.10)

Let \overline{ψ}_{π(b)}\overline{ψ}_{a}^{−1} = τ ∘ γ be a decomposition into an inner automorphism τ and a diagram automorphism γ with respect to (𝔥^{⊕r}, Δ). By Lemma 5.3, ν = π(μ)∘τ^{−1}, so the two modules V (λ, a) and V (μ, b) are isomorphic if and only if λ ∘ \overline{ψ}_{a} = π(μ) ∘ τ^{−1} ∘ \overline{ψ}_{π(b)} on \overline{ψ}_{a}^{−1}(𝔥^{⊕r}). That is, V (λ, a) ≅ V (μ, b) if and only if

λ = π(μ) ∘ γ  (5.11)

on 𝔥^{⊕r}. To finish the proof, it is enough to write down an explicit formula for the automorphism \overline{ψ}_{π(b)}\overline{ψ}_{a}^{−1} = τ ∘ γ of 𝔤^{⊕r}.

For each x ∈ 𝔤, let x^{i} = (0, …, x, …, 0) ∈ 𝔤^{⊕r} (x in the ith position). If k ∈ ℤ^{N} and x ∈ 𝔤\overline{k}, then we see that

\overline{ψ}_{a}^{−1}(x^{i}) = a_{i}^{−k}x ⊗ t^{k}f_{i}(t) + ker ψ_{a}

in L/ ker ψ_{a}, for any f_{i}(t) ∈ R\overline{0} with f_{i}(a_{j}) = δ_{ij} for all j = 1, …, r. Since f_{i} ∈ R\overline{0} = F [t_{1}^{±m_{1}}, …, t_{N}^{±m_{N}}] and m(a_{j}) = m(b_{π(j)}) for all j, we see that f_{i}(b_{π(j)}) = δ_{ij}, and

\overline{ψ}_{π(b)}\overline{ψ}_{a}^{−1}(x^{i}) = (\frac{b_{π(i)}^{k}}{a_{i}^{k}} ) x^{i}.
□

Theorem 5.7 may also be interpreted in terms of a group action on the space of parameters (λ, a) defining the finite-dimensional simple modules of L. Let G^{r} = G × ⋯ × G (r factors), where G is the finite abelian group G = ⟨σ_{1}⟩ × ⋯ × ⟨σ_{N}⟩, as before. Note that G acts on (F^{×})^{N} via the primitive m_{i}th roots of unity ξ_{i} used in the definition of L:

(σ_{1}^{c_{1}}, …, σ_{N}^{c_{N}}).(d_{1}, …, d_{N}) = (ξ_{1}^{c_{1}}d_{1}, …, ξ_{N}^{c_{N}} d_{N}),

for any (c_{1}, …, c_{N}) ∈ ℤ^{N} and (d_{1}, …, d_{N}) ∈ (F^{×})^{N}. Form the semidirect product G^{r} ⋊ S_{r} by letting the symmetric group S_{r} act on G^{r} (on the left) by permuting the factors of G^{r}:

π(ρ_{1}, …, ρ_{r}) = (ρ_{π(1)}, …, ρ_{π(r)}),

for all π ∈ S_{r} and ρ_{i} ∈ G. This semidirect product acts on the space of ordered r-tuples of points in the torus (F^{×})^{N} by letting G^{r} act diagonally and letting S_{r} permute the points:

ρπa = (ρ_{1}.a_{π(1)}, … ρ_{r}.a_{π(r)}),  (5.12)

for all ρ = (ρ_{1}, …, ρ_{r}) ∈ G^{r}, π ∈ S_{r}, and r-tuples a = (a_{1}, …, a_{r}) of points a_{i} ∈ (F^{×})^{N}.

The group G^{r} ⋊ S_{r} also acts on the space of r-tuples λ of nonzero dominant integral weights. For each ρ = (ρ_{1}, …, ρ_{r}) ∈ G^{r} write ρ_{i} = (σ_{1}^{ρ_{i1}}, …, σ_{N}^{ρ_{iN}}) for some nonnegative integers ρ_{ij}. Let the ρ_{i} act on 𝔤 by

ρ_{i}(x) = σ_{1}^{ρ_{i1}} ⋯ σ_{N}^{ρ_{iN}} x,

for all x ∈ 𝔤, and on the weights λ_{i} by

ρ_{i}(λ_{i}) = λ_{i} ∘ γ(ρ_{i}^{−1}),

where γ(ρ_{i}^{−1}) is the outer part of the automorphism ρ_{i}^{−1} : 𝔤 → 𝔤. Then G^{r} ⋊ S_{r} acts on each λ = (λ_{1}, …, λ_{r}) by

ρπλ = (λ_{π(1)} ∘ γ(ρ_{1}^{−1}), …, λ_{π(r)} ∘ γ(ρ_{r}^{−1})).  (5.13)

Combining (5.12) and (5.13) gives an action of G^{r} ⋊S_{r} on the set of pairs (λ, a), where λ is an r-tuple of nonzero dominant integral weights λ_{i} and a is an r-tuple of points a_{i} ∈ (F^{×})^{N} with m(a_{i}) ≠ m(a_{j}) whenever i ≠ j. Namely, let

ρπ(λ, a) = (^{ρπ}λ,^{ρπ} a) .  (5.14)

In terms of this action, the isomorphism classes of the finite-dimensional simple L-modules are labelled by the orbits of the groups G^{r} ⋊ S_{r}.

Corollary 5.15 Let λ = (λ_{1}, …, λ_{r}) and μ = (μ_{1}, …, μ_{s}) be sequences of nonzero dominant integral weights with respect to Δ. Suppose a = (a_{1}, …, a_{r}) and b = (b_{1}, …, b_{s}) are sequences of points in (F^{×})^{N} with m(a_{i}) ≠ m(a_{j}) and m(b_{i}) ≠ m(b_{j}) whenever i ≠ j.

Then V (λ, a) and V (μ, b) are isomorphic if and only if r = s and (λ, a) =^{ρπ} (μ, b) for some (ρ, π) ∈ G^{r} ⋊ S_{r}.

Proof Note that m(a_{i}) = m(b_{π(i)}) if and only if the coordinates a_{ij} of a_{i} = (a_{i1}, …, a_{iN}) differ from the coordinates b_{π(i)j} of b_{π(i)} by an m_{j}th root of unity. Since ξ_{j} is a primitive m_{j}th root of unity, this happens if and only if there are integers ρ_{ij} so that a_{ij} = ξ_{j}^{ρ_{ij}}b_{π(i)j}. In terms of group actions, this is precisely the existence of ρ_{i} = (σ_{1}^{ρ_{i1}}, …, σ_{N}^{ρ_{iN}}) ∈ G with a_{i} = ρ_{i}.b_{π(i)}. In other words, m(a_{i}) = m(b_{π(i)}) for all i if and only if a =^{ρπ} b for some ρ ∈ G^{r} and π ∈ S_{r}.

Since ξ_{j}^{ρ_{ij}} = \frac{a_{ij}}{b_{π(i)j}}, we see that

ρ_{i}^{−1}(x) = σ_{1}^{−ρ_{i1}} ⋯ σ_{N}^{−ρ_{iN}} x

= ξ_{1}^{−ρ_{i1}k_{1}} ⋯ ξ_{N}^{−ρ_{iN} k_{N}} x

= \frac{b_{π(i)}^{k}}{a_{i}^{k}} x,

for all k ∈ ℤ^{N} and x ∈ 𝔤\overline{k}. Therefore, the automorphism ω_{i} of Theorem 5.7 is equal to ρ_{i}^{−1}, and λ =^{ρπ} μ is equivalent to the condition that λ_{i} = μ_{π(i)}∘γ_{i} for every i. □

For any diagram automorphism σ_{1}, the finite-dimensional simple modules for the twisted (single) loop algebra L(𝔤; σ_{1}) were classified in [9]. Recently, P. Senesi [20] has reinterpreted this work in terms of finitely supported σ_{1}-equivariant maps F^{×} → P_{+}, where P_{+} is the set of nonzero dominant integral weights of 𝔤 with respect to a fixed Cartan subalgebra and base of simple roots. Theorem 5.7 and Corollary 5.15 can be used to extend this perspective to the multiloop setting.

Let λ = (λ_{1}, …, λ_{r}) and a = (a_{1}, …, a_{r}) be as in Theorem 5.7. Each evaluation module V_{λ_{i}}(a_{i}) corresponds to a map

χ_{λ_{i},a_{i}} : (F^{×})^{N} → P_{+}  (5.16)
x ↦ δ_{x,a_{i}}λ_{i}.  (5.17)

The isomorphism class [λ, a] of the tensor product V (λ, a) can then be identified with the sum of all the characters χ_{η_{0},c_{0}} for which the pair (η_{0}, c_{0}) = (μ_{1}, b_{1}) for some μ = (μ_{1}, …, μ_{r}) and b = (b_{1}, …, b_{r}) with (μ, b) in the G^{r} ⋊ S_{r}-orbit of (λ, a). That is, we let

χ_{[λ,a]} = ∑_{g∈G} ∑_{i=1}^{r} χ_{λ_{i}∘γ(g^{−1}),g.a_{i}}.  (5.18)

This process associates a finitely supported G-equivariant map

χ_{[λ,a]} : (F^{×})^{N} → P_{+}  (5.19)

with each isomorphism class of finite-dimensional simple L(𝔤; σ_{1}, …, σ_{N})- modules. From Corollary 5.15 and the construction of χ_{[λ,a]}, it is easy to see that distinct isomorphism classes get sent to distinct functions.

Conversely, any finitely supported G-equivariant map f : (F^{×})^{N} → P_{+} corresponds to an isomorphism class [λ, a] of finite-dimensional simple Lmodules as follows. By G-equivariance, the support supp f of f decomposes into a disjoint union of G-orbits. Choose representatives a_{1}, …, a_{r} ∈ (F^{×})^{N} to label each G-orbit in supp f. Since the G-orbits are disjoint, m(a_{i}) ≠ m(a_{j}) whenever i ≠ j, and by definition of f, λ := (f(a_{1}), …, f(a_{r})) is an r-tuple of nonzero dominant integral weights. Then by Theorem 4.12, V (λ, a) is a finite-dimensional simple L-module, and by Corollary 5.15, the isomorphism class [f] := [λ, a] of this module is independent of the choice of orbit representatives a_{1}, …, a_{r}. It is now straightforward to verify that χ_{[f]} = f for all finitely supported G-equivariant maps f : (F^{×})^{N} → P_{+}.

Corollary 5.20 The isomorphism classes of the finite-dimensional simple L-modules are in bijection with the finitely supported G-equivariant maps (F^{×})^{N} → P_{+}.

□

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