On modular representations of finite-dimensional Lie superalgebras

Abstract

In this paper, we studied representations of finite-dimensional Lie superalgebras over an algebraically closed field \begin{document} $\mathbb{F}$ \end{document} of characteristic p > 2. It was shown that simple modules of a finite-dimensional Lie superalgebra over \begin{document} $\mathbb{F}$ \end{document} 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 \begin{document} ${\mathfrak{g}}$ \end{document} , and defined a restricted Lie super subalgebra, then obtained a bijection between the isomorphism classes of simple modules of \begin{document} ${\mathfrak{g}}$ \end{document} and those of this restricted subalgebra. These results are generalization of the corresponding ones in Lie algebras of prime characteristic.