2017/01/31 by Xabier García‐Martínez, Xabier García-Martínez, Tim Van der Linden · 22 citations
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Algebraically closed field #Associative property #Commutative property #Homotopy and Cohomology in Algebraic Topology #Identity (music) #Mathematics #Non-associative algebra #Pure mathematics #Subvariety #Variety (cybernetics) #math.CT #math.RA #msc:08C05 #msc:17A99 #msc:18A22 #msc:18B99 #msc:18D15
paper · pdf · doi:10.1016/j.jpaa.2019.02.018
published in Journal of Pure and Applied Algebra 223(11), 4857-4870 (Elsevier BV) · Final version to appear in Journal of Pure and Applied Algebra
arxiv created 2019/01/23 · openalex publication_date 2019/02/28 · arxiv updated 2019/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let \mathbbK be an infinite field. We prove that if a variety of anti-commutative \mathbbK-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 \mathbbK. In particular, Lie_\mathbbK is the largest such. Thus, for a given variety of anti-commutative \mathbbK-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 \mathbbK-algebras. This is based on a result saying that an algebraically coherent variety of anti-commutative \mathbbK-algebras is either a variety of Lie algebras or a variety of anti-associative algebras over \mathbbK.