Grothendieck rings of universal quantum groups

TL;DR

This paper computes the Grothendieck rings of finite-dimensional comodules for various universal quantum groups, revealing their structure as non-commutative polynomial rings and providing detailed multiplication tables.

Contribution

It determines the Grothendieck rings for several universal quantum groups, extending known results and offering detailed algebraic structures.

Findings

Grothendieck rings are isomorphic to non-commutative polynomial rings

Explicit multiplication tables are provided for these rings

Results generalize and parallel previous work on quantum groups

Abstract

We determine the Grothendieck ring of finite-dimensional comodules for the free Hopf algebra on a matrix coalgebra, and similarly for the free Hopf algebra with bijective antipode and other related universal quantum groups. The results turn out to be parallel to those for Wang and Van Daele's deformed universal compact quantum groups and Bichon's generalization of those results to universal cosovereign Hopf algebras: in all cases the rings are isomorphic to those of non-commutative polynomials over certain sets, these sets varying from case to case. In most cases we are able to give more precise information about the multiplication table of the Grothendieck ring.