On Whittaker modules over a class of algebras similar to U(sl _{2})
 Xin Tang
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Motivated by the study of invariant rings of finite groups on the first Weyl algebras A _{1} and finding interesting families of new noetherian rings, a class of algebras similar to U(sl _{2}) 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.
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 Title
 On Whittaker modules over a class of algebras similar to U(sl _{2})
 Journal

Frontiers of Mathematics in China
Volume 2, Issue 1 , pp 127142
 Cover Date
 20070301
 DOI
 10.1007/s1146400700092
 Print ISSN
 16733452
 Online ISSN
 16733576
 Publisher
 Higher Education Press
 Additional Links
 Topics
 Keywords

 algebras similar to U(sl 2)
 Whittaker model
 Whittaker modules
 17B35
 20G05
 17B37
 Authors

 Xin Tang ^{(1)}
 Author Affiliations

 1. Department of Mathematics & Computer Science, Fayetteville State University, Fayetteville, NC, 28301, USA