On the dimensions of commutative subalgebras and subgroups

TL;DR

This paper investigates how the dimensions of finite-dimensional associative or Lie algebras relate to their commutative subalgebras, establishing quadratic growth bounds and extending these results to Lie groups.

Contribution

It proves that the functions bounding algebra dimensions based on commutative subalgebras grow quadratically, providing new estimates for Lie groups.

Findings

Functions bounding algebra dimensions have quadratic growth.

Derived estimates for Lie group dimensions based on abelian subgroups.

Extended results to associative and Lie algebras.

Abstract

We consider the functions that bound the dimensions of finite-dimensional associative or Lie algebras in terms of the dimensions of their commutative subalgebras. It is proved that these functions have quadratic growth. As a result, we also get similar estimates for the dimension of a Lie group with bounded dimensions of its abelian Lie subgroups.