Phùng Hô Hai: On the representation theory of matrix quantum groups of type A

We study representations of quantum groups defined in terms of solutions of the Yang-Baxter equation which also satisfies the Hecke equation (x+1)(x−q) = 0. Typical examples of such solutions are the Drinfeld-Jimbo solutions of series A, which define the well-known quantum group GL_{q}(n), and their super analog found by Manin, which determine the quantum super groups GL_{q}(m|n). The representation theory of these particular quantum groups have been studied by many authors. The main result in the case q is not a root of unity essentially says that the representation categories them do not differ so much from those of the corresponding groups, i.e., of GL(n) or GL(m|n). Given an arbitrary solution of the Yang-Baxter equation and the Hecke equation with the parameter q being not a root of unity, we show that the representation category of the associated quantum group is equivalent as an abelian braided monoidal category to the representation category of a quantum super group GL_{q}(m|n) mentioned above, where the pair (m, n) is uniquely defined and can by constructively computed from the given solution. In particular we can classisfy all simple representations.