A class of irreducible integrable modules for the extended baby TKK algebra

Abstract

The baby TKK algebra is a core of the extended affine Lie algebra of type A 1 over a semilattice in ‫ޒ‬ 2 . In this paper, we classify the irreducible integrable weight modules for the extended baby TKK algebra under the assumption that its center acts nontrivially.

Modules for double affine Lie algebras

“…The structure of IVM's is characterized by the generalized IVM's, and we show that they are irreducible if the second center is nonzero. When the second center is zero, they are similar to the evaluation modules of the affine Lie algebras, for which we also generalize several results from the Tits-Kantor-Koecher algebras [1].…”

“…Write λ = λ − rδ 2 , where λ is the component of ∆ĝ 1 . We can assume that r is minimum so that v λ = 0, otherwise we can replace λ by…”

“…Using the method of [1], we generalize some results of TKK modules to the highest weight Tmodules under the condition that λ(c 1 ) = λ(c 2 ) = 0.…”

“…Thus there exists a nonzero polynomial g = Rao [15] and Chang-Tan [1] have shown respectively that irreducible integrable modules for toroidal and TKK modules with finite dimensional weight spaces and c 1 > 0, c 2 = 0 are highest weight modules. In general our modules do not seem to be of highest weight type when c 1 = c 2 = 0.…”

Integrable modules for Lie tori

“…They turn out to be highest weight modules for direct sum of finitely many affine Kac-Moody Lie algebras. Some very special cases are done in [12,15,17] and [9].…”