Frontiers of Mathematics in China

, Volume 2, Issue 1, pp 127–142

On Whittaker modules over a class of algebras similar to U(sl2)

  • Tang Xin 
Research Article

DOI: 10.1007/s11464-007-0009-2

Cite this article as:
Tang, X. Front. Math. China (2007) 2: 127. doi:10.1007/s11464-007-0009-2

Abstract

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A1 and finding interesting families of new noetherian rings, a class of algebras similar to U(sl2) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.

Keywords

algebras similar to U(sl2)Whittaker modelWhittaker modules

MSC

17B3520G0517B37

Copyright information

© Higher Education Press and Springer-Verlag 2007

Authors and Affiliations

  • Tang Xin 
    • 1
  1. 1.Department of Mathematics & Computer ScienceFayetteville State UniversityFayettevilleUSA