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A characterisation of Lie algebras via algebraic exponentiation

Journal Contribution - Journal Article

In this article we describe varieties of Lie algebras via algebraic exponentiation, a concept introduced by Gray in his Ph.D. thesis. For K an infinite field of characteristic different from 2, we prove that the variety of Lie algebras over K is the only variety of non-associative K-algebras which is a non-abelian locally algebraically cartesian closed (LACC) category. More generally, a variety of n-algebras V is a non-abelian (LACC) category if and only if n=2 and V=Lie K. In characteristic 2 the situation is similar, but here we have to treat the identities xx=0 and xy=−yx separately, since each of them gives rise to a variety of non-associative K-algebras which is a non-abelian (LACC) category.

Journal: Adv. Math.
ISSN: 0001-8708
Volume: 341
Pages: 92-117
Publication year:2019
Keywords:Algebraic exponentiation, Lie algebra, Locally algebraically cartesian closed category
CSS-citation score:2
Accessibility:Open