Leibniz algebra (Rev #3, changes) in nLab

Showing changes from revision #2 to #3: Added | Removed | Changed

Given a commutative unital ring kk (usually a field), a Lebniz kk-algebra AA is a particular kind of nonassociative algebra over kk which is somewhat more general than a Lie algebra over kk.

A left Leibniz kk-algebra is kk-module LL equipped with a bracket, which is a kk-linear map [,]:A⊗A→A[,]:A\otimes A \to A satisfying the left Leibniz identity

[a,[b,c]]=[[a,b],c]+[b,[a,c]] [a, [b,c]] = [[a,b],c]+[b,[a,c]]

In other words, the left adad-map, a↦(ad la=[a,−]:L→L)a \mapsto (ad_l a = [a,-]:L\to L) is a derivation of LL as a nonassociative algebra. Similarly, there are right Leibniz algebras, for which the right adad-map ad r:a↦[−,a]:L→Lad_r :a\mapsto [-,a]:L\to L is a derivation. In the presence of antisymmetry, the left Leibniz identity is equivalent to the Jacobi identity, though this is not true in general; thus a Lie algebra is precisely an antisymmetric (or alternating) Leibniz algebra. There is a remarkable obesrvation of Loday and Pirashvili that in the Loday–Pirashvili tensor category of linear maps with (exotic) “infinitesimal tensor product”, the category of internal Lie algebras has the category of, say left, Leibniz kk-algebras as a full subcategory.

Loday introduced Leibniz algebras, because of considerations in algebraic K-theory which lead to a refinement of the Chevalley-Eilenberg Chevalley–Eilenberg complex for Lie algebra homology: in a refinement, a sort of noncommutative phenomena appeared with a new complex leading to so-called Leibniz homology. Some people dislike the term (left/right) Leibniz algebra (which is allegedly due to Loday), and prefer other names, including ‘Loday algebras’ and many longer descriptive names.