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On Lie algebras in braided categories. (English) Zbl 0880.16022
Budzyński, Robert (ed.) et al., Quantum groups and quantum spaces. Lectures delivered during the minisemester, Warsaw, Poland, December 1, 1995. Warszawa: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 40, 139-158 (1997).
Let $$G$$ be an abelian group. For any bicharacter of $$G$$, the category of $$G$$-graded vector spaces becomes a braided monoidal category. The notion of a Lie algebra in this category is defined, generalizing the concepts of Lie super and color algebras. The universal enveloping algebra of such a Lie algebra is a Hopf algebra in the category. The biproducts of this Hopf algebra with the group algebra are noncommutative noncocommutative Hopf algebras, recovering some known examples.
For the entire collection see [Zbl 0865.00041].

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17A70 Superalgebras 17B70 Graded Lie (super)algebras 16S30 Universal enveloping algebras of Lie algebras 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 16S40 Smash products of general Hopf actions
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