On the Structure of Some Modules for g(A_I~(1))

Let A be the affine matrix of type A_1~(1), D_l~(1) or E_l~(1), g(A) the Kac-MoodyLie algebra associated with A. For any integerable g(A)-module V, suppose thatP(V) is the set of weights of V. Let λ∈P(V) and let a_i be a simple root of g(A),M =- p_i(λ), q_i(λ)] the closed interval of intergers of all t ∈ z such that λ ta_i ∈P(V). We define the category D as follows.Its objects are integerable g(A)-modules V such that for any λ∈P(V), 0≤i≤l, p_i(λ) q_i(λ)<∞.The morphismsin D are homomorphisms of g(A)-modules. Let