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Integrity, integral closedness and finiteness over their centers of the coordinate algebras of quantum groups at \(p^ \nu\)-th roots of unity. (English) Zbl 0838.17012

The author proves that coordinate algebras dual to Lusztig’s quantum enveloping algebras at \(p^\nu\)-th roots of 1 are integral domains where \(p\) is a prime number not dividing the coefficients of the Cartan matrix. From this he deduces the same result if the parameter is transcendental. He also proves after a suitable \(p\)-adic completion that these coordinate algebras are finite over their centers and integrally closed in the sense of De Concini, Kac, and Procesi.
In an appendix, written jointly with N. Andruskiewitsch, a Frobenius morphism is constructed in the non-simply connected case. The paper may be regarded as a contribution to the geometric theory of quantum group duals.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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