A characterisation of Lie algebras amongst anticommutative algebras
Abstract
Let $\mathbb{K}$ be an infinite field. We prove that if a variety of anticommutative $\mathbb{K}$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 anticommutative $\mathbb{K}$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 anticommutative $\mathbb{K}$algebras. This is based on a result saying that an algebraically coherent variety of anticommutative $\mathbb{K}$algebras is either a variety of Lie algebras or a variety of antiassociative algebras over $\mathbb{K}$.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.05493
 Bibcode:
 2017arXiv170105493G
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Category Theory;
 08C05;
 17A99;
 18B99;
 18A22;
 18D15
 EPrint:
 Final version to appear in Journal of Pure and Applied Algebra