Half-commutative orthogonal Hopf algebras

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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 ∈ {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 G ⊂ Uₙ 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.