Fusion rules for representations of compact quantum groups

TL;DR

This paper surveys recent developments in the fusion semirings of compact quantum groups, exploring their structure, deformations, and implications for classification, with conjectures and comments on related invariants.

Contribution

It provides a comprehensive overview of fusion rules for compact quantum groups, introducing the concept of R^+-deformations and discussing their applications and related conjectures.

Findings

Fusion semirings are key to understanding quantum group representations.

R^+-deformations preserve fusion semiring structures.

Several conjectures and classification remarks are proposed.

Abstract

We give a survey of some recent results on the fusion semirings of compact quantum groups (computations of and applications to discrete quantum groups) by using the following simplifying terminology: we say that a compact quantum group G is an R^+-deformation of a compact quantum group H if their fusion semirings are isomorphic. The paper contains also some easy related results (with proofs), two conjectures and many remarks and comments, some of them concerning classification by invariants related to R^+.