A characterisation of Lie algebras amongst anti-commutative algebras

Abstract

Let K be an infinite field. We prove that if a variety of anticommutative 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.

A characterisation of Lie algebras via algebraic exponentiation

“…(Actually, he proved it for Lie algebras over a commutative ring with unit.) In our article [19] we managed to find a partial converse to this theorem: we showed that the condition (LACC) characterises Lie algebras amongst K-algebras with an alternating multiplication (xx " 0). The result is a simple categorical description of the Jacobi identity.…”

“…In Section 2 we start by recalling some definitions and basic properties of varieties of non-associative algebras, with in particular an analysis of the objects B5X in this context. This leads to Theorem 2.8 which provides a summary of the main result of [19]. Section 3 is devoted to the commutative case.…”

“…Abusing terminology, we make no distinction between the polynomials and the equivalence class to which they belong. See [19] for further details on this issue.…”

“…Here the cartesian product functor Bˆp´q is replaced by a functor B5p´q which happens not to be induced by a monoidal product. However, both the examples and the general theory seem to indicate that in the setting of semi-abelian categories [23], this is the right thing to consider [3,5,11,19,2,12,16].…”

“…The aim of our present article is not to give an overview of this theory-for that we refer the reader to [19] and the references there-but rather to prove one specific result: amongst all varieties of non-associative algebras over an infinite field, there exists precisely one variety that forms a non-abelian category whose objects are exponentiable with respect to the functors B5p´q, namely the category of Lie algebras. (As we shall see, there is actually one exception to this rule: when the characteristic of the field is 2, there are two solutions, since we then have to distinguish between the identities xx " 0 and xy "´yx.)…”