Half-commutative orthogonal Hopf algebras

TL;DR

This paper introduces a general construction method for half-commutative orthogonal Hopf algebras using crossed products, linking them to self-transpose compact subgroups of unitary groups and describing their fusion rules.

Contribution

It provides a universal construction for all half-commutative orthogonal Hopf algebras based on crossed products with compact subgroups.

Findings

All such Hopf algebras can be obtained via the proposed construction.

Fusion rules of these algebras are expressed in terms of the subgroup's fusion rules.

The method generalizes previous examples by Banica and Speicher.

Abstract

A half-commutative orthogonal Hopf algebra is a Hopf *-algebra generated by the self-adjoint coefficients of an orthogonal matrix corepresentation $v=(v_{ij})$ that half commute in the sense that $abc=cba$ for any $a,b,c \in \{v_{ij}\}$. The first non-trivial such Hopf algebras were discovered by Banica and Speicher. We propose a general procedure, based on a crossed product construction, that associates to a self-transpose compact subgroup $G \subset U_n$ a half-commutative orthogonal Hopf algebra $\mathcal A_*(G)$. It is shown that any half-commutative orthogonal Hopf algebra arises in this way. The fusion rules of $\mathcal A_*(G)$ are expressed in term of those of $G$.