On Lie Algebras in the Category of Yetter-Drinfeld Modules

Bodo Pareigis

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On Lie Algebras in the Category of Yetter-Drinfeld Modules

Bodo Pareigis

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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.

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Bodo Pareigis. 1996-12-17. On Lie Algebras in the Category of Yetter-Drinfeld Modules. https://arxiv.org/abs/q-alg/9612023

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