A characterisation of Lie algebras amongst anti-commutative algebras

Abstract

Let K be an infinite field. We prove that if a variety of anti-commutative 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 K. In particular, LieK is the largest such. Thus, for a given variety of anti-commutative K-algebras, the Jacobi identity becomes equivalent to a categorical condition: it is an identity in V if and only if V is a subvariety of a locally algebraically cartesian closed variety of anti-commutative K-algebras. This is based on a result saying that an algebraically coherent variety of anti-commutative K-algebras is either a variety of Lie algebras or a variety of anti-associative algebras over K.