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Double Quantum Groups

✍ Scribed by Daniela Hobst; Bodo Pareigis

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
325 KB
Volume
242
Category
Article
ISSN
0021-8693

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✦ Synopsis

The construction of the Drinfeld double D H of a finite dimensional Hopf algebra H was one of the first examples of a quasitriangular Hopf algebra whose category of modules M M is braided. The braided category of Yetter᎐Drinfeld DŽ H . modules Y Y D D H is the analogue for infinite dimensional Hopf algebras. It uses a H strong dependence between the H-module and the H-comodule structures. We C Ž . generalize this construction to the category M M of entwined modules, that is, A A-modules and C-comodules over Hopf algebras A and C where the structures are C Ž . only related by an entwining map : C m A ª A m C. We show that M M is A braided iff there is an r-map r: C m C ª A m A satisfying suitable axioms that generalize the axioms of an R-matrix. For finite dimensional C there is a Ž . quasitriangular Hopf algebra structure on Hom C, A , the quantum group double, generalizing the construction of the Drinfeld double.

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