Varieties of elementary abelian Lie algebras and degrees of modules

Chang, Hao; Farnsteiner, Rolf

doi:10.1090/ert/559

Varieties of elementary abelian Lie algebras and degrees of modules
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by Hao Chang and Rolf Farnsteiner

Represent. Theory 25 (2021), 90-141

DOI: https://doi.org/10.1090/ert/559

Published electronically: February 11, 2021

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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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