A covariant Stinespring theorem

Abstract: We will introduce a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T:= Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras (a.k.a C*-dynamical systems) can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X: T -> M1, Y: T -> M2, we show that covariant channels between the corresponding G-C*-algebras can be 'dilated' to isometries t: X -> Y \otimes E, where E: M2 -> M1 is some 'environment' 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimising the quantum dimension of the environment is unique up to a unitary. When G is a compact group this implies and generalises previous covariant Stinespring-type theorems.

We will also discuss some results relating to rigid C*-2-categories, including that any connected semisimple rigid C*-2-category is equivalent to Mod(T) for some rigid C*-tensor category T. (Here semisimple means not just semisimplicity of Hom-categories but also idempotent splitting for 1-morphisms, direct sums for objects, etc.)

This talk is based on the paper arXiv:2108.09872.

category theoryoperator algebrasquantum algebra

Audience: researchers in the topic

( paper )

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Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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