A characterisation of Lie algebras amongst anti-commutative algebras

Abstract

Let $\mathbb{K}$ be an infinite field. We prove that if a variety of anti-commutative $\mathbb{K}$-algebras - not necessarily associative, where $xx=0$ is an identity - is locally algebraically cartesian closed, then it must be a variety of Lie algebras over $\mathbb{K}$. In particular, ${\mathsf{Lie}}_{\mathbb{K}}$ is the largest such. Thus, for a given variety of anti-commutative $\mathbb{K}$-algebras, the Jacobi identity becomes equivalent to a categorical condition: it is an identity in~$\mathcal{V}$ if and only if $\mathcal{V}$ is a subvariety of a locally algebraically cartesian closed variety of anti-commutative $\mathbb{K}$-algebras. This is based on a result saying that an algebraically coherent variety of anti-commutative $\mathbb{K}$-algebras is either a variety of Lie algebras or a variety of anti-associative algebras over $\mathbb{K}$.

Final version to appear in Journal of Pure and Applied Algebra

References in corpus (2)

• A torsion theory in the category of cocommutative Hopf algebras
• A characterisation of Lie algebras via algebraic exponentiation

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• Algebraic exponentiation for Lie algebras
• On the "three subobjects lemma" and its higher-order generalisations
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• Algebraic logoi
• A universal Kaluzhnin--Krasner embedding theorem
• Categorical-algebraic aspects of Heyting semilattices
• On some properties of $\mathsf{Lie}$-centroids of Leibniz algebras