Block型李代数B(q)的表示.docVIP

Block 型李代数 B(q) 的表示 苏育才1，夏春光2，许莹2 1 同济大学数学系，上海 200092 2 中国科技大学吴文俊数学重点实验室，合肥 230026 摘要：受 Mathieu 关于 Virasoro 代数 Harish-Chandra 模的一个著名定理的启发，我们证明 了一类 Block 型李代数 B(q) 的拟有限模是最高/最低权模或者是一致有界模，其中参数 q 是 任意非零复数。我们还给出了拟有限最高权模及不可约中间序列模的分类。特别地，我们得 到，不可约中间序列模是中间序列 Vir-模的非平凡扩张如果 q 是负整数的一半，其中 Vir 是 与 Virasoro 代数同构的 B(q) 的子代数。 关键词：拟有限表示；Block 型李代数；Virasoro 代数；拟有限表示；最高权表示 中图分类号： O152.5 Representations of Lie algebras B(q) of Block type 1 1  Department of Mathematics, Tongji University, Shanghai 200092 2 Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Hefei 230026 Abstract: Intrigued by a well-known theorem of Mathieu’s on Harish-Chandra modules over the Virasoro algebra, we show that any quasi?nite irreducible module over a class of Block type Lie algebras B(q) is either a highest or lowest weight module, or else a uniformly bounded module, where the parameter q is a nonzero complex number. We also classify quasi?nite irreducible highest weight B(q)-modules and irreducible B(q)-modules of the intermediate series. In particular, we obtain that an irreducible B(q)-module of the intermediate series may be a nontrivial extension of a Vir-module of the intermediate series if q is half of a negative integer, where Vir is a subalgebra of B(q) isomorphic to the Virasoro algebra. Key words: Quasi?nite representations; Block type Lie algebras; Virasoro algebra; Quasi?nite representations; highest weight representations 基金项目： NSF grant noof China, Research Fund for the Doctoral Program of Higher Education of China (no. 20093402110044) 作者简介： Su Yucai (1963-), male, professor, major research direction: Lie theory. XU Ying (1985-), female, Ph.D.student, major research direction: Lie theory. Correspondence author: XIA Chunguang (1984-), male, Ph.D.student, major research direction: Lie theory. -1- 0 Introduction Since a class of in?nite dimensional simple Lie algebras were introduced by Block [1], gen- eralizations of Lie algebras of this type (usually referred to as Block ty