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Irreducible modules of modular Lie superalgebras and super version of the first Kac–Weisfeiler conjecture

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  11 December 2023

Bin Shu
Bin Shu*
Affiliation:
School of Mathematical Sciences, Ministry of Education Key Laboratory of Mathematics and Engineering Applications & Shanghai Key Laboratory of PMMP, East China Normal University, No. 500 Dongchuan Road, Shanghai 200241, China
*
e-mail: bshu@math.ecnu.edu.cn
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Abstract

Suppose $\mathfrak {g}=\mathfrak {g}_{\bar 0}+\mathfrak {g}_{\bar 1}$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $\mathbf {k}$ of characteristic $p>2$. In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(\mathfrak {g})$ of $\mathfrak {g}$, as a super generalization of the celebrated first Kac–Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras.

Keywords

Modular Lie superalgebrasmaximal dimensions of irreducible modulesthe first Kac–Weisfeiler conjecture

MSC classification

Primary: 17B50: Modular Lie (super)algebras
Secondary: 17B20: Simple, semisimple, reductive (super)algebras 17B30: Solvable, nilpotent (super)algebras
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 20
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 12071136, 11771279, and 12271345), supported in part by Science and Technology Commission of Shanghai Municipality (Grant No. 22DZ2229014).

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