1996/12/17 by Bodo Pareigis, Pareigis, Bodo
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.q-alg/9612023
openalex publication_date 1996/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.