Abstract
AbstractA Hopf algebra object in Loday and Pirashvili's category of linear maps entails an ordinary Hopf algebra and a Yetter–Drinfel'd module. We equip the latter with a structure of a braided Leibniz algebra. This provides a unified framework for examples of racks in the category of coalgebras discussed recently by Carter, Crans, Elhamdadi and Saito.
Highlights
The subject of the present paper is the relation between racks, Leibniz algebras and Yetter–Drinfel’d modules
The problem of the integration of Leibniz algebras arose, i.e., the problem of nding an object that is to a Leibniz algebra what a Lie group is to its Lie algebra
In this case, is recovered as the Yetter–Drinfel’d module of left invariant elements, with trivial right coaction and right action being induced by the right g-module structure on
Summary
The subject of the present paper is the relation between racks, Leibniz algebras and Yetter–Drinfel’d modules. An augmented rack (or a crossed -module) can be de ned as a Yetter–Drinfel’d module over a group , viewed as a Hopf algebra object in the symmetric monoidal category (Set, ×). It is a right -set together with a -equivariant map : → , where carries the right adjoint action of. Our aim here is to directly relate Leibniz algebras to Yetter–Drinfel’d modules, starting with the fact that the universal enveloping algebra of a Leibniz algebra gives rise to a Hopf algebra object in the category LM of linear maps [17], see Section 2.3.
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