Racks, Leibniz algebras and Yetter-Drinfel'd modules
Ulrich Kraehmer, Friedrich Wagemann
TL;DR
This paper explores the relationship between Hopf algebra objects, Leibniz algebras, and Yetter-Drinfel'd modules, providing a unified framework that connects these algebraic structures and includes examples of racks in coalgebra categories.
Contribution
It introduces a braided Leibniz algebra structure on Yetter-Drinfel'd modules derived from Hopf algebra objects, unifying various algebraic concepts and examples.
Findings
Yetter-Drinfel'd modules can be equipped with braided Leibniz algebra structures
Provides a unified framework for racks in coalgebra categories
Connects Hopf algebra objects with Leibniz and rack structures
Abstract
A 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.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology