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 algebraA _h of representative elements is isomorphic as a co-algebra to the Hopf algebraA _O of representative functions on the classical group. As a consequence,A _h can be identified withA _O as a co-algebra but with an associative product, called the star-product, which is a deformation of the original commutative product ofA _O . 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.