Finite-Dimensional Representations of Twisted Hyper-Loop Algebras

Abstract

We investigate the category of finite-dimensional representations of twisted hyper-loop algebras, i.e., the hyperalgebras associated to twisted loop algebras over finite-dimensional simple Lie algebras. The main results are the classification of the irreducible modules, the definition of the universal highest-weight modules, called the Weyl modules, and, under a certain mild restriction on the characteristic of the ground field, a proof that the simple modules and the Weyl modules for the twisted hyper-loop algebras are isomorphic to appropriate simple and Weyl modules for the nontwisted hyper-loop algebras, respectively, via restriction of the action.

Key Words:

• Finite-dimensional representations
• Hyperalgebras
• Loop algebras

2000 Mathematics Subject Classification:

• Primary 17B10, 17B65
• Secondary 20G42

ACKNOWLEDGMENTS

The results of this paper were obtained during the Ph.D. studies of the first author under the supervision of the second author. We thank A. Engler and P. Brumatti for useful discussions and helpful references about Henselian discrete valuation rings.

Partially supported by the FAPESP grant 2007/07456-9 (A. B.), the CNPq grant 306678/2008-0, and the MATH-AmSud project 12MATH-01 (A. M.).

Notes

Communicated by K. Misra.