A characterisation of Lie algebras via algebraic exponentiation

TL;DR

This paper characterizes Lie algebras among non-associative algebras using algebraic exponentiation, showing that Lie algebras uniquely form a non-abelian locally algebraically cartesian closed category over an infinite field of characteristic not 2.

Contribution

It proves that the variety of Lie algebras is the only non-associative algebra variety with a non-abelian LACC structure over such fields, and extends the characterization to n-algebras in general.

Findings

Lie algebras are uniquely characterized as non-abelian LACC categories over infinite fields of characteristic not 2.

In characteristic 2, specific identities lead to similar LACC categories for certain non-associative algebras.

The paper provides a complete classification of algebraic varieties with LACC structure in this context.

Abstract

In this article we describe varieties of Lie algebras via algebraic exponentiation, a concept introduced by Gray in his Ph.D. thesis. For $\mathbb{K}$ an infinite field of characteristic different from $2$, we prove that the variety of Lie algebras over $\mathbb{K}$ is the only variety of non-associative $\mathbb{K}$-algebras which is a non-abelian locally algebraically cartesian closed (LACC) category. More generally, a variety of $n$-algebras $\mathcal{V}$ is a non-abelian (LACC) category if and only if $n=2$ and $\mathcal{V}=\mathsf{Lie}_\mathbb{K}$. In characteristic $2$ the situation is similar, but here we have to treat the identities $xx=0$ and $xy=-yx$ separately, since each of them gives rise to a variety of non-associative $\mathbb{K}$-algebras which is a non-abelian (LACC) category.