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Weight modules over generalized Witt algebras with 1-dimensional weight spaces

Weight modules over generalized Witt algebras with 1-dimensional weight spaces Abstract. In this paper, indecomposable and irreducible weight representations with 1dimensional weight spaces for simple generalized Witt algebras over any field of characteristic 0 are classified. There are five classes of such nontrivial indecomposable modules. 2000 Mathematics Subject Classification: 17B10, 17B20, 17B65, 17B66, 17B68. §1 Introduction Throughout this paper we use the definitions and notations in [DZ1], and assume that F is an arbitrary field of characteristic 0. Let me start with the definition of generalized Witt algebras. Let A be an abelian group, T a vector space over F . We denote by F ½A the group algebra of A over F . The elements t x , x A A, form a basis of F ½A with the multiplication t x t y ¼ t xþy . We shall write 1 instead of t 0 . The tensor product W ¼ F ½A nF T is a free left F ½A-module. We shall usually denote an arbitrary element of T by q, and for simplicity we write t x q instead of t x n q. We now fix a pairing j : T Â A ! F , which is F -linear in the first http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Weight modules over generalized Witt algebras with 1-dimensional weight spaces

ZHAO, K.
Forum Mathematicum , Volume 16 (5) – Sep 16, 2004
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Publisher
de Gruyter
Copyright
Copyright © 2004 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2004.034
Publisher site
See Article on Publisher Site

Abstract

Abstract. In this paper, indecomposable and irreducible weight representations with 1dimensional weight spaces for simple generalized Witt algebras over any field of characteristic 0 are classified. There are five classes of such nontrivial indecomposable modules. 2000 Mathematics Subject Classification: 17B10, 17B20, 17B65, 17B66, 17B68. §1 Introduction Throughout this paper we use the definitions and notations in [DZ1], and assume that F is an arbitrary field of characteristic 0. Let me start with the definition of generalized Witt algebras. Let A be an abelian group, T a vector space over F . We denote by F ½A the group algebra of A over F . The elements t x , x A A, form a basis of F ½A with the multiplication t x t y ¼ t xþy . We shall write 1 instead of t 0 . The tensor product W ¼ F ½A nF T is a free left F ½A-module. We shall usually denote an arbitrary element of T by q, and for simplicity we write t x q instead of t x n q. We now fix a pairing j : T Â A ! F , which is F -linear in the first

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Forum Mathematicum – de Gruyter

Published: Sep 16, 2004

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