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.


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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