Quantum Affine Algebras at Roots of Unity

Let Uq(ĝ) be the quantized universal enveloping algebra of the affine Lie algebra ĝ associated to a finite-dimensional complex simple Lie algebra g, and let U res q (ĝ) be the C[q, q−1]-subalgebra of Uq(ĝ) generated by the q-divided powers of the Chevalley generators. Let U res (ĝ) be the Hopf algebra obtained from U res q (ĝ) by specialising q to a non-zero complex number of odd order. We classify the finite-dimensional irreducible representations of U res (ĝ) in terms of highest weights. We also give a “factorisation” theorem for such representations: namely, any finite-dimensional irreducible representation of U res (ĝ) is isomorphic to a tensor product of two representations, one factor being the pull-back of a representation of ĝ by Lusztig’s Frobenius homomorphism F̂r : U res (ĝ) → U(ĝ), the other factor being an irreducible representation of the Frobenius kernel. Finally, we give a concrete construction of all of the finite-dimensional irreducible representations of U res (ŝl2). The proofs make use of several interesting new identities in Uq(ĝ). 0. Introduction Let Uq(g) be the Drinfel’d–Jimbo quantum group associated to a symmetrizable Kac–Moody algebra g. Thus, Uq(g) is a Hopf algebra over the field C(q) of rational functions of an indeterminate q, and is defined by certain generators and relations (which are written down in Proposition 1.1 below for the cases of interest in this paper). Roughly speaking, one thinks of Uq(g) as a family of Hopf algebras depending on a “parameter” q. To make this precise, one constructs a “specialisation” U (g) of Uq(g), for a non-zero complex number , as follows. If ∈ C is transcendental, define U (g) = Uq(g)⊗C(q) C, via the algebra homomorphism C(q) → C that takes q to . If, on the other hand, is algebraic, the latter homomorphism does not exist, and one proceeds by first constructing a C[q, q−1]-form of Uq(g), i.e. a C[q, q−1](Hopf) subalgebra Ũq(g) of Uq(g) such that Uq(g) = Ũq(g)⊗C[q,q−1] C(q). Then one defines U (g) = Ũq(g) ⊗C[q,q−1] C, via the algebra homomorphism C[q, q−1] → C that takes q to . Two such C[q, q−1]-forms have been studied in the literature. They lead to the same algebra U (g) when is not a root of unity, but different algebras, with very different representation theories, when is a root of unity. In the “non-restricted” form, one takes Ũq(g) to be the C[q, q−1]-subalgebra of Uq(g) generated by the Chevalley generators ei, fi of Uq(g). The finite-dimensional representations of the non-restricted specialisation U (g) have been studied by Received by the editors April, 30, 1997. 1991 Mathematics Subject Classification. Primary 17B67. The first author was partially supported by NATO and EPSRC (GR/K65812). The second author was partially supported by NATO and EPSRC (GR/L26216) . c ©1997 American Mathematical Society 280 QUANTUM AFFINE ALGEBRAS AT ROOTS OF UNITY 281 De Concini, Kac and Procesi [10], [11], when g is finite-dimensional, and by Beck and Kac [2], when g is (untwisted) affine. In the “restricted” form, one takes Ũq(g) to be the C[q, q−1]-subalgebra of Uq(g) generated by the divided powers ei /[r]q! and f r i /[r]q!, for all r > 0, where [r]q ! denotes a q-factorial. When g is finite-dimensional and is a root of unity, the structure and representation theory of the restricted specialisation U res (g) was worked out by Lusztig (see [5], [15] and the references there). It is the purpose of the present work to study the finite-dimensional representations of U res (g) when is a root of unity and g is (untwisted) affine. Part of our work may be regarded as the quantum analogue of Garland’s paper [13]. A crucial role is played in [13] by a certain identity (Lemma 7.5) that is needed to prove a suitable triangular decomposition of the restricted form of U(ĝ) (analogous to U res q (ĝ)). The proof of the analogue of Garland’s lemma in the quantum case (Lemma 5.1) is, however, more difficult than in the classical situation because the generators of the “positive part” of U res (ĝ) do not commute, whereas their classical analogues do commute. Moreover, Garland makes use of a natural derivation of U(ĝ) which turns out to have no straightforward analogue in the quantum situation. Lemma 5.1 is one of several interesting new identities in U res q (ĝ) that we obtain below. One of these (see (19)) shows an unexpected (and as yet unexplained) connection between U res q (ŝl2) and Young diagrams. (This relation is invisible in the classical situation considered by Garland.) Once the triangular decomposition of U res (ĝ) is available, we are able to give, in Theorem 8.2, an abstract highest weight description of its finite-dimensional irreducible representations. It turns out that there is a natural one-to-one correspondence between the finite-dimensional irreducible representations of U res (ĝ) when is a root of unity, and those of U (ĝ) when is not a root of unity, although corresponding representations have different dimensions, in general (the representations of U (ĝ) when is not a root of unity were classified in [4], [5] and [6]). This is exactly parallel to the situation for U res (g) when dim(g) < ∞, where the finite-dimensional irreducible representations are parametrised by dominant integral weights whether or not is a root of unity (see [5], Chapter 11, and [15]). One of the most important results about the finite-dimensional irreducible representations V of U res (g) when dim(g) <∞ and is a root of unity of order ` asserts that V factorises into the tensor product of a representation whose highest weight is divisible by ` and one whose highest weight is “less than `”, in the sense that its value on every simple coroot of g is less than ` (see [5] and [15]). In Section 9, we prove an analogue of this result for U res (ĝ) (Theorem 9.1). It is well known that there is a close relationship between finite-dimensional representations of quantum affine algebras and affine Toda theories (see [7] and the references there). The value of is determined by the “coupling constant” of the associated theory. Since the representation theory of U res (ĝ) depends crucially on whether is a root of unity or not, one would expect that the structure of affine Toda theories will be different at certain special values of the coupling constant. This does indeed appear to be the case (T. J. Hollowood, private communication), but we shall leave further discussion of this point to another place. 282 VYJAYANTHI CHARI AND ANDREW PRESSLEY 1. Preliminaries In this section and the next, we recall certain facts about g and ĝ and their associated quantum groups that will be needed later. See [1], [5] and [15] for further details. Let (aij)i,j∈I be the Cartan matrix of the finite-dimensional complex simple Lie algebra g, let Î = I ∐ {0}, and let (aij)i,j∈Î be the generalised Cartan matrix of the untwisted affine Lie algebra ĝ of g. Let (di)i∈Î be the coprime positive integers such that the matrix (diaij)i,j∈Î is symmetric. Let P̌ be the lattice over Z with basis (λ̌i)i∈I , let α̌j = ∑ i∈I ajiλ̌i (j ∈ I), and let Q̌ = ∑ i∈I Zα̌i ⊆ P̌ . The root lattice Q = HomZ(P̌ ,Z) has basis the simple roots (αi)i∈I of g, where αi(λ̌j) = δij . Similarly, the weight lattice P = HomZ(Q̌,Z) has basis the fundamental weights (λi)i∈I of g, where λi(α̌j) = δij . Let P = {λ ∈ P | λ(α̌i) ≥ 0 for all i ∈ I}, and let Q = ∑ i∈I N.αi. For i ∈ I, define si : P̌ → P̌ by si(x) = x− αi(x)α̌i, and let W be the group of automorphisms of P̌ generated by the si. Then, W acts on Q by si(ξ) = ξ−ξ(α̌i)αi, for ξ ∈ Q. The root and coroot systems of g are given, respectively, by R = ⋃ i∈I Wαi, Ř = ⋃ i∈I Wα̌i. There is a partial order on R such that α ≤ β if and only if β − α is a linear combination of the αi (i ∈ I) with non-negative integer coefficients. The correspondence αi ↔ α̌i extends to a W -invariant correspondence R ↔ Ř, written α ↔ α̌, such that α(α̌) = 2 for all α ∈ R. Define sα : P̌ → P̌ by sα(x) = x− α(x)α̌, for all α ∈ R. Let Ŵ = W ×̃P̌ be the semi-direct product group defined using the W action on P̌ . For s ∈ W write s for (s, 0) ∈ Ŵ , and for x ∈ P̌ write x for (1, x) ∈ Ŵ , where 1 is the identity element of W . Let θ be the highest root of g with respect to ≤, and write s0 for (sθ, θ̌) ∈ Ŵ . Let W̃ be the (normal) subgroup of Ŵ generated by the si for i ∈ Î, and let T = Ŵ/W̃ . Then, T is a finite group isomorphic to a subgroup of the group of diagram automorphisms of ĝ, i.e. the bijections τ : Î → Î such that aτ(i)τ(j) = aij for all i, j ∈ Î. Moreover, there is an isomorphism of groups Ŵ ∼= T ×̃W̃ , where the semi-direct product is defined using the action of T in W̃ given by τ.si = sτ(i) (see [1]). If w ∈ Ŵ , a reduced expression for w is an expression w = τsi1si2 . . . sin with τ ∈ T , i1, i2, . . . , in ∈ Î and n minimal. The universal enveloping algebra U(g) (resp. U(ĝ)) is the associative algebra over C with generators ei , hi for i ∈ I (resp. i ∈ Î) and defining relations hihj = hjhi, hie ± j − ej hi = ±aijej , ei e − j − ej ei = δijhi, 1−aij ∑ r=0 (−1) ( 1− aij r ) (ei ) ej (e ± i ) 1−aij−r = 0 if i 6= j, where i, j ∈ I (resp. i, j ∈ Î). Let q be an indeterminate, let C(q) be the field of rational functions of q with complex coefficients, and let C[q, q−1] be the ring of Laurent polynomials. For QUANTUM AFFINE ALGEBRAS AT ROOTS OF UNITY 283 r, n ∈ N, n ≥ r, define [n]q = q − q−n q − q−1 , [n]q! = [n]q[n− 1]q . . . [2]q[1]q, [ n r ] q = [n]q! [r]q ![n− r]q ! .