On the dimensions of commutative subalgebras and subgroups of nilpotent algebras and Lie groups of class 2

TL;DR

This paper establishes bounds on the dimensions of finite-dimensional nilpotent associative and Lie algebras of class 2 based on their commutative subalgebras, and extends these results to complex nilpotent Lie groups.

Contribution

It derives explicit functions bounding the dimensions of nilpotent algebras and groups of class 2 in terms of their commutative subalgebras, providing new quantitative insights.

Findings

Derived functions bounding algebra dimensions by commutative subalgebra sizes

Computed similar bounds for complex nilpotent Lie groups of class 2

Enhanced understanding of the structure of nilpotent algebras and groups

Abstract

We obtain the functions that bound the dimensions of finite dimensional nilpotent associative or Lie algebras of class 2 over an algebraically closed field in terms of the dimensions of their commutative subalgebras. As a result, we also compute a similar function for complex nilpotent Lie groups of class 2.