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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.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.
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Key words:
- Lie superalgebra /
- representation /
- p-envelope /
- p-character
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