dorsal/arxiv
View SchemaOn Lie Algebras in Braided Categories
| Authors | Bodo Pareigis |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9612002 |
| URL | https://arxiv.org/abs/q-alg/9612002 |
Abstract
The set of primitive elements of a Hopf algebra in the braided category of group graded vector spaces (with a commutative group) carry the structure of a generalized Lie algebra. In particular the graded derivations of an associative algebra carry this Lie algebra structure. The Lie multiplications consist of certain n-ary partially defined multiplications satisfying generalized antisymmetry and Jacobi identities. This generalizes the concept of Lie super algebras and Lie color algebras. We show that universal enveloping algebras in the braided category exist. They are (braided) Hopf algebras. This explains many constructions of noncommutative noncocommutative Hopf algebras in the literature.
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"abstract": "The set of primitive elements of a Hopf algebra in the braided category of\ngroup graded vector spaces (with a commutative group) carry the structure of a\ngeneralized Lie algebra. In particular the graded derivations of an associative\nalgebra carry this Lie algebra structure. The Lie multiplications consist of\ncertain n-ary partially defined multiplications satisfying generalized\nantisymmetry and Jacobi identities. This generalizes the concept of Lie super\nalgebras and Lie color algebras.\n We show that universal enveloping algebras in the braided category exist.\nThey are (braided) Hopf algebras. This explains many constructions of\nnoncommutative noncocommutative Hopf algebras in the literature.",
"arxiv_id": "q-alg/9612002",
"authors": [
"Bodo Pareigis"
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"title": "On Lie Algebras in Braided Categories",
"url": "https://arxiv.org/abs/q-alg/9612002"
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