Coset Construction for Winding Subalgebras and Applications

In this paper we review the coset construction for winding subalgebras of affine Lie algebras. We classify all cosets of central charge ĉ < 1 and calculate their branching rules. The corresponding character identities give certain ‘doubling formulae’ for the affine characters. We discuss some applications of our construction, in particular we find a simple proof of a crucial identity needed for the computation of the level-2 root multiplicities of the hyperbolic Kac-Moody algebra E10. ADP-96-39/M48 q-alg/9610013 ‡ Supported by an Australian Research Council (QEII) Fellowship Let g be a simple, finite-dimensional Lie algebra. Consider the (untwisted) affine Lie algebra ĝk defined by ĝk = (g ⊗ C[t, t ])⊕ Ck ⊕ Cd with commutators [x(m), y(n)] = [x, y](m+ n) + km (x|y) δm,−n , [d, x(n)] = −nx(n) , [k, x(n)] = [k, d] = 0 , (1) where we have written x(n) = x⊗ t. We follow the conventions of [1]. In particular ( | ) denotes the Killing form on both g and h, normalized such that (θ|θ) = 2 for a long root θ of g. The integrable highest weight modules L(Λ) of ĝ at level k are parametrized by dominant integral weights Λ such that ∑ ai (Λ, α ∨ i ) = k. The proof of the following theorem is standard Theorem 1. i. For every j ∈ N we have an embedding ĝjk →֒ ĝk defined by x(n) 7→ x(jn). ii. Let L(Λ) be an integrable highest weight module of ĝk with highest weight vector vΛ. Considered as a ĝjk module, L(Λ) is integrable (and hence fully reducible). Moreover, U(ĝjk) · vΛ ∩ L(Λ) is an irreducible integrable ĝjk module. (Note: ĝjk is called a winding subalgebra of ĝk [2].) As an example, consider ĝ = ŝl2. We have, under ĝk →֒ ĝ1 (k ≥ 2), e.g., U(ĝk) · vΛ0 ∩ L(Λ0) ∼= L(kΛ0) , (2) and U(ĝk) · vΛ1 ∩ L(Λ1) ∼= L((k − 1)Λ0 +Λ1) . (3) Other irreducible modules can be projected out by considering other maximal vectors (vectors in L(Λ) on the Weyl orbit of the highest weight vector), e.g. U(ĝk) · vr0Λ0 ∩ L(Λ0) ∼= L((k − 2)Λ0 + 2Λ1) . (4) We can actually do more than projecting out the irreducible modules. The ĝk modules decompose with respect to a direct sum of ĝjk and a coset Virasoro algebra. The construction is a slight modification of the standard coset construction [3,4]. Recall that the Virasoro algebra, Vc, is generated by {L(m) |m ∈ Z} and a central charge c with relations [L(m), L(n)] = (m− n)L(m+ n) + c 24 m(m − 1) δm,−n , (5) Any (positive energy) module of ĝk can be extended to a module of the semi-direct sum Vc ⊕ ĝk by means of the affine Sugawara construction [3] L(m) = 1 2(k + h∨) ∑ n∈Z : x(m− n)x(n) : , (6)