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Representation Theory

ISSN 1088-4165



Varieties of elementary abelian Lie algebras and degrees of modules

Authors: Hao Chang and Rolf Farnsteiner
Journal: Represent. Theory 25 (2021), 90-141
MSC (2020): Primary 17B50, 16G10
Published electronically: February 11, 2021
MathSciNet review: 4214336
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Abstract: Let $(\mathfrak {g},[p])$ be a restricted Lie algebra over an algebraically closed field $k$ of characteristic $p\!\ge \!3$. Motivated by the behavior of geometric invariants of the so-called $(\mathfrak {g},[p])$-modules of constant $j$-rank ($j \in \{1,\ldots ,p\!-\!1\}$), we study the projective variety $\mathbb {E}(2,\mathfrak {g})$ of two-dimensional elementary abelian subalgebras. If $p\!\ge \!5$, then the topological space $\mathbb {E}(2,\mathfrak {g}/C(\mathfrak {g}))$, associated to the factor algebra of $\mathfrak {g}$ by its center $C(\mathfrak {g})$, is shown to be connected. We give applications concerning categories of $(\mathfrak {g},[p])$-modules of constant $j$-rank and certain invariants, called $j$-degrees.

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Additional Information

Hao Chang
Affiliation: School of Mathematics and Statistics, Central China Normal University, 430079 Wuhan, People’s Republic of China
ORCID: 0000-0002-0137-0586

Rolf Farnsteiner
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany
MR Author ID: 194225

Received by editor(s): April 22, 2020
Received by editor(s) in revised form: December 5, 2020
Published electronically: February 11, 2021
Additional Notes: The first author was partially supported by the National Natural Science Foundation of China (No. 11801204).
Dedicated: Dedicated to Jens Carsten Jantzen on the occasion of his 70th birthday
Article copyright: © Copyright 2021 American Mathematical Society