Complex Quantum Groups and Their Real Representations

Abstract

We define *-Hopf algebras Fun(SLq(N, ℂ; ε1,…, ε_N_)), Fun(Oq(N, ℂ; ε1,…, ε_N_)) and Fun(Spq(n, ℂ; ε1,…, ε2_n_)) as the real complexifications of *-Hopf algebras Fun(SUq(N, ℂ; ε1,…, ε_N_)), Fun(Oq(N, ℂ; ε1,…, ε_N_)) and Fun(Spq(N, ℂ; ε1,…, ε2_n_)) of [RTF] (for q > 0). Such construction can be done for each coquasitriangular (CQT) *-Hopf algebra A. The obtained object Aℂℝ is also a CQT *-Hopf algebra. We describe the theory of corepresentations of Aℂℝ in terms of such a theory for A.