dorsal/arxiv
View SchemaOn Lie Algebras in the Category of Yetter-Drinfeld Modules
| Authors | Bodo Pareigis |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9612023 |
| URL | https://arxiv.org/abs/q-alg/9612023 |
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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"abstract": "The category of Yetter-Drinfeld modules over a Hopf algebra (with bijektive\nantipode over a field) is a braided monoidal category. Given a Hopf algebra in\nthis category then the primitive elements of this Hopf algebra do not form an\nordinary Lie algebra anymore.\n We introduce the notion of a (generalized) Lie algebra in the category of\nYetter-Drinfeld modules such that the set of primitive elements of a Hopf\nalgebra is a Lie algebra in this sense. It has n-ary partially defined Lie\nmultiplications on certain symmetric submodules of n- fold tensor products.\nThey satisfy antisymmetry and Jacobi identities. Also the Yetter-Drinfeld\nmodule of derivations of an associative algebra in the category of Yetter-\nDrinfeld modules is a Lie algebra. Furthermore for each Lie algebra in the\ncategory of Yetter-Drinfeld modules there is a universal enveloping algebra\nwhich turns out to be a (braided) Hopf algebra in this category.",
"arxiv_id": "q-alg/9612023",
"authors": [
"Bodo Pareigis"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "On Lie Algebras in the Category of Yetter-Drinfeld Modules",
"url": "https://arxiv.org/abs/q-alg/9612023"
},
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