Elementary Subalgebrs of Lie Algebras
Jon F. Carlson, Eric M. Friedlander, and Julia Pevtsova

TL;DR
This paper explores the geometric structure of varieties of elementary subalgebras in Lie algebras, introducing invariants that relate to support varieties and analyzing their properties and examples.
Contribution
It initiates the study of these varieties for various ranks and introduces new invariants, expanding understanding of their geometric and representation-theoretic significance.
Findings
Identification of special cases with interesting geometric structures
Introduction of local radical and socle rank invariants
Examples of modules with constant rank functions
Abstract
We initiate the investigation of the projective varieties of elementary subalgebras of dimension of a (-restricted) Lie algebra for various . These varieties are the natural ambient varieties for generalized support varieties for restricted representations of . We identify these varieties in special cases, revealing their interesting and varied geometric structures. We also introduce invariants for a finite dimensional -module , the local -radical rank and local -socle rank, functions which are lower/upper semicontinuous on . Examples are given of -modules for which some of these rank functions are constant.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic structures and combinatorial models · Algebraic Geometry and Number Theory
