Half-commutative orthogonal Hopf algebras

Abstract

A half-commutative orthogonal Hopf algebra is a Hopf �-algebra generated by the self-adjoint coefficients of an orthogonal matrix corepresentation v = (vij) that half commute in the sense that abc = cba for any a,b,c 2 {vij}. 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 GUn 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.