华东师范大学学报(自然科学版) ›› 2017, Vol. ›› Issue (3): 1-19.doi: 10.3969/j.issn.1000-5641.2017.03.001

• 数学 •    下一篇

有限维李超代数的模表示

杨恒云, 姚裕丰   

  1. 上海海事大学 数学系, 上海 201306
  • 收稿日期:2017-03-01 出版日期:2017-05-25 发布日期:2017-05-18
  • 通讯作者: 姚裕丰,男,副教授,研究方向为李理论及表示理论.E-mail:yfyao@shmtu.edu.cn E-mail:yfyao@shmtu.edu.cn
  • 作者简介:杨恒云,女,副教授,研究方向为李理论及表示理论.E-mail:hyyang@shmtu.edu.cn
  • 基金资助:
    国家自然科学基金(11571008,11671138);上海市自然科学基金(16ZR1415000)

On modular representations of finite-dimensional Lie superalgebras

YANG Heng-yun, YAO Yu-feng   

  1. Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China
  • Received:2017-03-01 Online:2017-05-25 Published:2017-05-18

摘要: 研究了特征大于2的代数闭域上有限维李超代数的表示.证明了有限维李超代数的单模都是有限维的,并且所有单模的维数有上界.进一步,一个有限维李超代数可以嵌入到一个有限维限制李超代数.给出了有限维限制李超代数g上单模的判定准则,定义了g的一个限制李超子代数,得到了该子代数的单模同构类和g的单模同构类之间的一个双射.这些结果是素特征域上李代数相关理论的推广.

关键词: 李超代数, 表示, p-包络, p-特征

Abstract: In this paper, we studied representations of finite-dimensional Lie superalgebras over an algebraically closed field F of characteristic p > 2. It was shown that simple modules of a finite-dimensional Lie superalgebra over F are finite-dimensional, and there exists an upper bound on the dimensions of simple modules. Moreover, a finite-dimensional Lie superalgebra can be embedded into a finite-dimensional restricted Lie superalgebra. We gave a criterion on simplicity of modules over a finite-dimensional restricted Lie superalgebra g, and defined a restricted Lie super subalgebra, then obtained a bijection between the isomorphism classes of simple modules of g and those of this restricted subalgebra. These results are generalization of the corresponding ones in Lie algebras of prime characteristic.

Key words: Lie superalgebra, representation, p-envelope, p-character

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