A characterisation of Lie algebras amongst anti-commutative algebras

Garcia Martinez, Xabier;Van der Linden, Tim
(2019) Journal of Pure and Applied Algebra — Vol. 223, p. 4857-4870 (2019)

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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, 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.
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Garcia Martinez, X., & Van der Linden, T. (2019). A characterisation of Lie algebras amongst anti-commutative algebras. Journal of Pure and Applied Algebra, 223, 4857-4870. https://doi.org/10.1016/j.jpaa.2019.02.018 (Original work published 2019)