Branching problem on winding subalgebras of affine Kac-Moody algebras A1.1 and A2.2
Consider an affine Kac-Moody algebra g with Cartan subalgebra h. Given Λ in the set P + of dominant integral weights of g, we denote by L(Λ) the integrable highest weight g-module with highest weight Λ. For µ ∈ h * , we denote by L(Λ) µ the corresponding weight space. Consider the support Γ(g, h) of the decompositions of the L(Λ) as a h-module: Γ(g, h) = {(Λ, µ) : L(Λ) µ = {0}}. Consider now the winding subalgebra g[u] (for some positive integer u). The winding subalgebra g[u] is isomorphic to g but with a nontrivial embedding in g depending on the parameter u. Given λ in the set Ṗ + of dominant integral weights of g[u], we denote by L(λ) the integrable highest weight g[u]-module with highest weight λ. Then the g-module L(Λ) decomposes as a direct sums of simple g[u]-modules L(λ) with finite multiplicities. In this paper, we are interested in the supports of this decomposition, i.e., the set of pairs (Λ, λ) in P + ×Ṗ + such that the integrable highest weight g[u]-modules L(λ) is a submodule of L(Λ). We show that both Γ(g, h) and Γ(g, g[u]) are semigroups. Moreover, for the cases A (1) 1 and A (2) 2 , we determine explicitly Γ(g, h). Finaly, we describe explicit subsets of P + ×Ṗ + where the two semigroups coincide.