On Lie Algebras in the Category of Yetter-Drinfeld Modules

TL;DR

This paper develops a generalized notion of Lie algebras within the braided monoidal category of Yetter-Drinfeld modules over a Hopf algebra, extending classical Lie algebra concepts to this categorical setting.

Contribution

It introduces a new definition of Lie algebras in the Yetter-Drinfeld module category, including n-ary operations and universal enveloping algebras, expanding the algebraic framework.

Findings

Primitive elements form a Lie algebra in the new sense.

The Yetter-Drinfeld module of derivations is a Lie algebra.

Universal enveloping algebras are braided Hopf algebras.

Abstract

The category of Yetter-Drinfeld modules over a Hopf algebra (with bijektive antipode over a field) is a braided monoidal category. Given a Hopf algebra in this category then the primitive elements of this Hopf algebra do not form an ordinary Lie algebra anymore. We introduce the notion of a (generalized) Lie algebra in the category of Yetter-Drinfeld modules such that the set of primitive elements of a Hopf algebra is a Lie algebra in this sense. It has n-ary partially defined Lie multiplications on certain symmetric submodules of n- fold tensor products. They satisfy antisymmetry and Jacobi identities. Also the Yetter-Drinfeld module of derivations of an associative algebra in the category of Yetter- Drinfeld modules is a Lie algebra. Furthermore for each Lie algebra in the category of Yetter-Drinfeld modules there is a universal enveloping algebra which turns out to be a (braided) Hopf algebra in this category.