Classification of irreducible integrable highest weight modules for current Kac–Moody algebras

Abstract

This paper classifies irreducible, integrable highest weight modules for "current Kac-Moody Algebras" with finite dimensional weight spaces. We prove that these modules turn out to be modules of appropriate direct sums of finitely many copies of Kac-Moody Lie algebras.Mathematics Subject Classification. 17B65, 17B66

Modules for loop Affine-Virasoro algebras

“…Suppose V (ψ) has finite dimensional weight spaces. As in the proof of Lemma 2.3 of [6] there exists a co-finite ideal of I 1 of A such that h(I 1 ))v = 0.…”

“…See [1,15] and references there in. In the theory of Toroidal Lie algebras, the study of irreducible modules with finite dimensional weight spaces has been reduced to the study of modules for the loop affine Lie algebras [4] and [6]. On the other hand the semidirect product of Virasoro algebra and affine Lie algebra is very important and occurs in physics literature [12,13].…”

“…Now it is easy to see that V (ψ) has finite dimensional weight spaces as it is a module for τ (A/I). See [6] for details.…”

Spectral Characters of a Class of Integrable Representations of Toroidal Lie Algebras

“…A T (g)-module is said to be integrable if it is the direct sum of (possibly infinite) finite-dimensional sl 2 (β)-modules for all real roots β of T (g). Integrable representations of toroidal Lie algebras and their quotients have been studied in several papers [CL,L,B,BR,PB,FL,R1,R3,R4,R5]. In [R3] the irreducible integrable T (g)-modules having finite-dimensional weight spaces have been classified.…”