Half-commutative orthogonal Hopf algebras

Julien Bichon and Michel Dubois-Violette

Vol. 263 (2013), No. 1, 13–28

DOI: 10.2140/pjm.2013.263.13

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 ∈{v_ij}. The first nontrivial 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 ⊂ U_n a half-commutative orthogonal Hopf algebra 𝒜_∗(G). It is shown that any half-commutative orthogonal Hopf algebra arises in this way. The fusion rules of 𝒜_∗(G) are expressed in term of those of G.

Keywords

Hopf algebras, quantum groups, compact groups

Mathematical Subject Classification 2010

Primary: 20G42, 22C05, 16T05

Milestones

Received: 28 February 2012

Accepted: 15 May 2012

Published: 14 May 2013