Monoidal category of C*-algebras

TL;DR

This paper characterizes when the category of C*-algebras with quantum group actions admits a monoidal structure, linking it to the existence of a quasitriangular quantum group and unitary R-matrices.

Contribution

It establishes a precise correspondence between monoidal structures on C*-algebras with quantum group actions and quasitriangular quantum groups, using natural properties without assuming a specific form.

Findings

Monoidal structure exists iff the quantum group is quasitriangular.

Monoidal structures correspond bijectively to unitary R-matrices.

The characterization relies on natural properties of the monoidal product.

Abstract

We consider the category of C*-algebras equipped with actions of a locally compact quantum group. We show that this category admits a monoidal structure satisfying certain natural conditions if and only if the group is quasitriangular. The monoidal structures are in bijective correspondence with unitary R-matrices. To prove this result we use only very natural properties imposed on considered monoidal structures. We assume that monoidal product is a crossed product, monoidal product of injective morphisms is injective and that monoidal product reduces to the minimal tensor product when one of the involved C*-algebras is equipped with a trivial action of the group. No a priori form of monoidal product is used.