The Restricted Kirillov–reshetikhin Modules for the Current and Twisted Current Algebras

In this paper we define and study a family of Z+–graded modules for the polynomial valued current algebra g[t] and the twisted current algebra g[t] associated to a finite–dimensional classical simple Lie algebra g and a non–trivial diagram automorphism of g. The modules which we denote as KR(mωi) and KR (mωi) respectively are indexed by pairs (i,m), where i is a node of the Dynkin diagram and m is a non–negative integer, and are given by generators and relations. These modules are indecomposable, but usually reducible, and we describe their Jordan–Holder series by giving the corresponding graded decomposition as a direct sum of irreducible modules for the underlying finite–dimensional simple Lie algebra. Moreover, we prove that the modules are finite–dimensional and hence restricted, i.e., there exists an integer n ∈ Z+ depending only on g and σ such that (g ⊗ t )KR(mωi) = 0. It turns out that this graded decomposition is exactly the one predicted in [9, Appendix A], [10, Section 6] coming from the study of the Bethe Ansatz in solvable lattice models.