A characterisation of Lie algebras via algebraic exponentiation

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.

The ancillary files contain the code used in the proofs. Final version to appear in Advances in Mathematics

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Cited by in corpus (11)

• A characterisation of Lie algebras amongst anti-commutative algebras
• Algebras with representable representations
• Weak representability of actions of non-associative algebras
• An introduction to regular categories
• Non-associative algebras
• Algebraic exponentiation for Lie algebras
• Associativity and the cosmash product in operadic varieties of algebras
• A characterisation of Lie algebras using ideals and subalgebras
• A universal Kaluzhnin--Krasner embedding theorem
• On some properties of $\mathsf{Lie}$-centroids of Leibniz algebras
• On the capablility of Hom-Lie algebras