Modules for the core of extended affine Lie algebras of type A₁ with coordinates in rank 2 quantum tori

Lin Weiqiang and Su Yucai

Vol. 242 (2009), No. 1, 143–166

DOI: 10.2140/pjm.2009.242.143

Abstract

We construct a Lie algebra L from rank 3 quantum tori and show that it is isomorphic to the core of extended affine Lie algebras of type A₁. Then we construct two classes—which turn out to be exhaustive—of irreducible ℤ-graded highest weight L-modules and give necessary and sufficient conditions for these modules to have finite-dimensional homogeneous subspaces. As a consequence, we also determine all the irreducible ℤ-graded L-modules with nonzero center and finite-dimensional homogeneous subspaces.

Keywords

core of extended affine Lie algebras, graded-module, the generalized highest weight module, the highest weight module, quantum torus

Mathematical Subject Classification 2000

Primary: 17B68

Secondary: 17B10, 17B65

Milestones

Received: 8 October 2008

Accepted: 8 June 2009

Published: 1 September 2009