Imaginary Verma Modules for the Extended Affine Lie Algebra sl₂(ℂ_q)

Abstract

We consider one of the most natural extended affine Lie algebras, the algebra sl ₂(ℂ _q ), over the quantum torus ℂ _q and begin a theory of its representations. In particular, we study a class of imaginary Verma modules, obtain for them a criterion of irreducibility and describe their submodule structure when the modules are in “general position”.

Key Words:

• Extended affine Lie algebras
• Verma module
• Quantum Torus

Mathematical Subject Classification:

• 17 B67
• 17 B65

Acknowledgments

This work was partially done during the stay of the third author at the University of São Paulo as a visiting professor and was completed during his visit to the Fields Institute. The financial support of Fapesp of Brazil, Proc. 97/05415-0, and the Fields Institute is gratefully acknowledged. The third author is grateful to B. Allison for many useful discussions on the subject. The first author was partially supported by CNPq of Brazil, Proc. 301115/95-8.