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arXiv · 1701.05493

A characterisation of Lie algebras amongst anti-commutative algebras

Xabier García-Martínez·Tim Van der Linden
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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 $\mathbb{K}$.

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Xabier García-Martínez, Tim Van der Linden. 2017-01-19. A characterisation of Lie algebras amongst anti-commutative algebras. https://doi.org/10.1016/j.jpaa.2019.02.018

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