On the values of commutativity degree of Lie algebras

TL;DR

This paper investigates the possible values of the commutativity degree in Lie algebras, introduces an asymptotic version, and explores specific families with high asymptotic commutativity.

Contribution

It determines the range of commutativity degrees, defines and computes the asymptotic commutativity degree for certain Lie algebras, and proves the existence of families with maximal asymptotic degree.

Findings

Identified possible values of commutativity degree.

Computed asymptotic commutativity degrees for specific Lie algebras.

Established existence of Lie algebra families with asymptotic degree equal to 1/q.

Abstract

In this paper, the possible values of commutativity degree of Lie algebras are determined. Also, we define the asymptotic commutativity degree of Lie algebras and obtain the asymptotic commutativity degree for some of them. Moreover, we prove the existence of a family of Lie algebras such that the asymptotic commutativity degree is equal to 1\qk for all q greater than 2 and a positive integer k.