dorsal/arxiv
View SchemaDoubles of Quasi-Quantum Groups
| Authors | Frank Hausser, Florian Nill |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9708023 |
| URL | https://arxiv.org/abs/q-alg/9708023 |
Abstract
Drinfeld showed that any finite dimensional Hopf algebra \G extends to a quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the construction of a so--called diagonal crossed product developed by the authors, we generalize this result to the case of quasi--Hopf algebras \G. As for ordinary Hopf algebras, as a vector space the ``quasi--quantum double'' \D(\G) is isomorphic to the tensor product of \G and its dual \dG. We give explicit formulas for the product, the coproduct, the R--matrix and the antipode on \D(\G) and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi--Hopf algebra. In particular \D(\G) becomes an associative algebra containing \G as a quasi--Hopf subalgebra. On the other hand, \dG \otimes 1 is not a subalgebra of \D(\G) unless the coproduct on \G is strictly coassociative. It is shown that the category of finite dimensional representations of \D(\G) coincides with what has been called the double category of \G--modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi--quantum doubles in terms of a Tannaka--Krein--like reconstruction procedure. The whole construction is shown to generalize to weak quasi--Hopf algebras with \D(\G) now being linearly isomorphic to a subspace of \dG \otimes \G.
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"abstract": "Drinfeld showed that any finite dimensional Hopf algebra \\G extends to a\nquasitriangular Hopf algebra \\D(\\G), the quantum double of \\G. Based on the\nconstruction of a so--called diagonal crossed product developed by the authors,\nwe generalize this result to the case of quasi--Hopf algebras \\G. As for\nordinary Hopf algebras, as a vector space the ``quasi--quantum double\u0027\u0027 \\D(\\G)\nis isomorphic to the tensor product of \\G and its dual \\dG. We give explicit\nformulas for the product, the coproduct, the R--matrix and the antipode on\n\\D(\\G) and prove that they fulfill Drinfeld\u0027s axioms of a quasitriangular\nquasi--Hopf algebra. In particular \\D(\\G) becomes an associative algebra\ncontaining \\G as a quasi--Hopf subalgebra. On the other hand, \\dG \\otimes 1 is\nnot a subalgebra of \\D(\\G) unless the coproduct on \\G is strictly\ncoassociative. It is shown that the category of finite dimensional\nrepresentations of \\D(\\G) coincides with what has been called the double\ncategory of \\G--modules by S. Majid [M2]. Thus our construction gives a\nconcrete realization of Majid\u0027s abstract definition of quasi--quantum doubles\nin terms of a Tannaka--Krein--like reconstruction procedure. The whole\nconstruction is shown to generalize to weak quasi--Hopf algebras with \\D(\\G)\nnow being linearly isomorphic to a subspace of \\dG \\otimes \\G.",
"arxiv_id": "q-alg/9708023",
"authors": [
"Frank Hausser",
"Florian Nill"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Doubles of Quasi-Quantum Groups",
"url": "https://arxiv.org/abs/q-alg/9708023"
},
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