A Seifert–van Kampen theorem in non-abelian algebra

Duckerts-Antoine, Mathieu; Linden, Tim Van der

doi:10.4310/hha.2018.v20.n2.a5

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2019

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“…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].…”

Section: Introductionmentioning

confidence: 99%

A characterisation of Lie algebras via algebraic exponentiation

García-Martínez

Linden

2019

Advances in Mathematics

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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.

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