Abelian ideals with given dimension in Borel subalgebras

TL;DR

This paper refines Peterson's theorem by determining the distribution of dimensions of abelian ideals within Borel subalgebras of finite-dimensional simple Lie algebras, providing deeper algebraic insights.

Contribution

It introduces a detailed dimensional distribution of abelian ideals, extending Peterson's theorem to include more Lie algebra invariants.

Findings

Distribution of abelian ideal dimensions in Borel subalgebras determined

Refinement of Peterson's theorem with additional invariants

Enhanced understanding of Lie algebra structure

Abstract

A well-known Peterson's theorem says that the number of abelian ideals in a Borel subalgebra of a rank-$r$ finite dimensional simple Lie algebra is exactly $2^r$. In this paper, we determine the dimensional distribution of abelian ideals in a Borel subalgebra of finite dimensional simple Lie algebras, which is a refinement of the Peterson's theorem capturing more Lie algebra invariants.