On the classification of compact quantum groups U θ (2)

In this paper, we show that the corepresentation of compact group Uθ(2) on the vector space is determined by its infinitesimal generators B0, B2, A0, A1 and A2, where θ is an irrational number. We also show that B0 and B2 commute with Aj, j = 0, 1, 2, so B0, A0, A1 and A2 are sl(2, ℂ) loop algebra. Then we exhibit all irreducible representations of Uθ(2), which are different from those of group U(2), and use above results to give the classification of quantum groups Uθ(2), which is analogous to that of irrational algebra Aθ. At the same time, we also give all the forms of automorphisms on quantum group Uθ(2).