ON THE THEORY OF QUANTUM GROUPS

By using the results of S. L. Woronowicz, we show that for the twisted version of the classical compact matrix groups, the Hopf algebra Ah of representative elements is isomorphic as a co-algebra to the Hopf algebra AO of representative functions on the classical group. As a consequence, Ah can be identified with AO as a co-algebra but with an associative product, called the star-product, which is a deformation of the original commutative product of AO. Furthermore, the construction of this star product from the original product is connected to the Fourier transformation in a manner which is similar to the construction of quantum mechanics from classical mechanics on phase space. In fact, we shall describe the analog of the Weyl correspondence. © 1990 Kluwer Academic Publishers.