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Noncommutative Tannaka theory. (Théorie tannakienne non commutative.) (French) Zbl 0808.18005

The author extends the Tannaka-Krein results over a field to the noncommutative situation. More precisely, a 1-1-correspondence is established between (not necessarily symmetric) Tannakian categories and transitive quantum groupoids. When the base field \(k\) is perfect, a quantum groupoid over \(\text{Spec} (B)\) is transitive precisely when it is projective and faithfully flat over \(B \bigotimes_ kB\). Moreover, the fibre functor of the corresponding Tannakian category is unique up to quantum isomorphisms.

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
20G15 Linear algebraic groups over arbitrary fields
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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