Varieties of elementary subalgebras of submaximal rank in type A

TL;DR

This paper investigates the geometric structure of varieties of elementary subalgebras in Lie algebras of type A, identifying their irreducible components for submaximal rank cases.

Contribution

It characterizes the irreducible components of the variety of elementary subalgebras of submaximal rank in type A Lie algebras, advancing understanding of their geometric properties.

Findings

Identifies irreducible components of E(rkp(g)-1,g) in type A

Provides a detailed description of elementary subalgebras of submaximal rank

Advances geometric understanding of elementary subalgebra varieties

Abstract

Let G be a connected simple algebraic group over an algebraically closed field k of characteristic p > 0, and g := Lie(G). We additionally assume that G is standard and is of type An. Motivated by the investigation of the geometric properties of the varieties E(r, g) of r-dimensional elementary subalgebras of a restricted Lie algebra g, we will show in this article the irreducible components of E(rkp(g)-1,g) when rkp(g) is the maximal dimension of an elementary subalgebra of g.