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, Lie K 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.

Original languageEnglish
Pages (from-to)4857-4870
Number of pages14
JournalJournal of Pure and Applied Algebra
Volume223
Issue number11
DOIs
Publication statusPublished - Nov 2019

    Research areas

  • Algebraic exponentiation, Algebraically coherent, locally algebraically cartesian closed, semi-abelian category, Anti-associative, anti-commutative algebra, Lie algebra

ID: 46315911