有限维李超代数的模表示

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${\mathfrak{g}}$上单模的判定准则，定义了${\mathfrak{g}}$的一个限制李超子代数，得到了该子代数的单模同构类和${\mathfrak{g}}$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.
- 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 $\mathbb{F}$ of characteristic p > 2. It was shown that simple modules of a finite-dimensional Lie superalgebra over $\mathbb{F}$ 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 ${\mathfrak{g}}$, and defined a restricted Lie super subalgebra, then obtained a bijection between the isomorphism classes of simple modules of ${\mathfrak{g}}$ and those of this restricted subalgebra. These results are generalization of the corresponding ones in Lie algebras of prime characteristic.
- Lie superalgebra /
- representation /
- p-envelope /
- p-character

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[1]	KAC V G. Lie superalgebras [J]. Advances in Mathematics, 1977, 26(1): 8-96. doi: 10.1016/0001-8708(77)90017-2
[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. doi: 10.1112/plms/pdm040
[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. doi: 10.1112/plms.2009.99.issue-1
[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. doi: 10.1016/j.jpaa.2011.02.011
[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. doi: 10.1016/j.jpaa.2008.09.005
[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. doi: 10.1016/j.jalgebra.2010.04.032
[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. doi: 10.1007/s10468-009-9198-6
[9]	SHU B, YAO Y F. Character formulas for restricted simple modules of the special superalgebras [J]. Mathema-tische Nachrichten, 2012, 285: 1107-1116. doi: 10.1002/mana.v285.8/9
[10]	YAO Y F. On restricted representations of the extended special type Lie superalgebra [J]. Monatshefte für Mathematik, 2013, 170: 239-255. doi: 10.1007/s00605-012-0414-9
[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. doi: 10.1007/s11425-012-4486-8
[12]	YAO Y F, SHU B. Restricted representations of Lie superalgebras of Hamiltonian type [J]. Algebras and Repre-sentation Theory, 2013, 16: 615-632. doi: 10.1007/s10468-011-9322-2
[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. doi: 10.1007/s11401-013-0800-1
[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. doi: 10.1007/s00605-014-0700-9
[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. doi: 10.1142/S0219498816500754
[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. doi: 10.1016/0021-8693(92)90173-J