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

Bruguières, Alain

Commun. Algebra 22, No. 14, 5817-5860 (1994).

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.

Reviewer: F.Borceux (Louvain-La-Neuve)

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

Keywords:

Tannakian categories; quantum groupoids