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• W2963128662 abstract "We construct a new class of finite-dimensional C⁎-quantum groupoids at roots of unity q=eiπ/ℓ, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C⁎-category of the category of tilting modules of the non-semisimple quantum group Uq(slN) of Drinfeld, Jimbo and Lusztig. As an algebra, the C⁎-groupoid is a quotient of Uq(slN). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociative, but satisfies axioms of the weak quasi-Hopf C⁎-algebras: quasi-coassociativity and non-unitality of the coproduct. There are also a multiplicative counit, an antipode, and an R-matrix. For this, we give a general construction of quantum groupoids for complex simple Lie algebras g≠E8 and certain roots of unity. Our main tools here are Drinfeld's coboundary associated to the R-matrix, related to the algebra involution, and certain canonical projections introduced by Wenzl, which yield the coproduct and Drinfeld's associator in an explicit way. Tensorial properties of the negligible modules reflect in a rather special nature of the associator. We next reduce the proof of the categorical equivalence to the problems of establishing semisimplicity and computing dimension of the groupoid. In the case g=slN we construct a (non-positive) Haar-type functional on an associative version of the dual groupoid satisfying key non-degeneracy properties. This enables us to complete the proof." @default.
• W2963128662 created "2019-07-30" @default.
• W2963128662 creator A5074630519 @default.
• W2963128662 creator A5084651587 @default.
• W2963128662 date "2017-12-01" @default.
• W2963128662 modified "2023-10-16" @default.
• W2963128662 title "Quasi-coassociative C ⁎ -quantum groupoids of type A and modular C ⁎ -categories" @default.
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• W2963128662 doi "https://doi.org/10.1016/j.aim.2017.10.025" @default.
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