有限维李超代数的模表示

doi:10.3969/j.issn.1000-5641.2017.03.001

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

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

Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China

Received:2017-03-01 Online:2017-05-25 Published:2017-05-18

摘要/Abstract

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

中图分类号:

O152.5

杨恒云, 姚裕丰. 有限维李超代数的模表示[J]. 华东师范大学学报(自然科学版), 2017, (3): 1-19.

YANG Heng-yun, YAO Yu-feng. On modular representations of finite-dimensional Lie superalgebras[J]. Journal of East China Normal University(Natural Sc, 2017, (3): 1-19.

参考文献

[1] KAC V G. Lie superalgebras [J]. Advances in Mathematics, 1977, 26(1): 8-96.
[2] SHU B, WANG W Q. Modular representations of the ortho-symplectic supergroups [J]. Proceedings of the London Mathematical Society, 2008, 96(1): 251-271.
[3] WANG W Q, ZHAO L. Representations of Lie superalgebras in prime characteristic I [J]. Proceedings of the London Mathematical Society, 2009, 99(1): 145-167.
[4] WANG W Q, ZHAO L. Representations of Lie superalgebras in prime characteristic II: The queer series [J]. Journal of Pure and Applied Algebra, 2011, 215: 2515-2532.
[5] ZHANG C W. On the simple modules for the restricted Lie superalgebra sl(n|1) [J]. Journal of Pure and Applied Algebra, 2009, 213: 756-765.
[6] ZHENG L S. Classical Lie superalgebras in prime characteristic and their representations [D]. Shanghai: East China Normal University, 2009.
[7] SHU B, ZHANG C W. Restricted representations of the Witt superalgebras [J]. Journal of Algebra, 2010, 324: 652-672.
[8] SHU B, ZHANG C W. Representations of the restricted Cartan type Lie superalgebra W(m, n, 1) [J]. Algebras and Representation Theory, 2011, 14: 463-481.
[9] SHU B, YAO Y F. Character formulas for restricted simple modules of the special superalgebras [J]. Mathema-tische Nachrichten, 2012, 285: 1107-1116.
[10] YAO Y F. On restricted representations of the extended special type Lie superalgebra [J]. Monatshefte für Mathematik, 2013, 170: 239-255.
[11] YAO Y F. Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian type [J]. Science China Mathematics, 2013, 56: 239-252.
[12] YAO Y F, SHU B. Restricted representations of Lie superalgebras of Hamiltonian type [J]. Algebras and Repre-sentation Theory, 2013, 16: 615-632.
[13] YAO Y F, SHU B. A note on restricted representations of the Witt superalgebras [J]. Chinese Annals of Math-ematics, Series B, 2013, 34: 921-926.
[14] YUAN J X, LIU W D. Restricted Kac modules of Hamiltonian Lie superalgebras of odd type [J]. Monatshefte für Mathematik, 2015, 178: 473-488.
[15] WANG S J, LIU W D. On restricted representations of restricted contact Lie superalgebras of odd type [J]. Journal of Algebra and Its Applications, 2016, 15(4): 1650075, 14pages.
[16] JANTZEN J C. Representations of Lie algebras in prime characteristic [C]//Proceedings of the NATO ASI Representation Theories and Algebraic Geometry. Montreal, 1997, 514: 185-235.
[17] STRADE H, FARNSTEINER R. Modular Lie Algebras and Their Representations [M]. New York: Marcel Dekker, 1988.
[18] HUMPHREYS J E. Introduction to Lie Algebras and Representation Theory [M]. New York: Springer, 1972.
[19] JACOBSON N. Lie Algebras [M]. New York: Interscience, 1962.
[20] PETROGRADSKI V. Identities in the enveloping algebras for modular Lie superalgebras [J]. Journal of Algebra, 1992, 145: 1-21.

有限维李超代数的模表示

On modular representations of finite-dimensional Lie superalgebras

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