by requiring $q$ to be a root of unity. In this case the representation theory of the quantum group is significantly different from the classical one. The first surprise is that when $q^N=1$ there are only $N-1$ distinct finite dimensional irreps with spins $j=0,1/2,1,\ldots ,(N-1)/2$, this last one irrep being singular. This truncation of the representation theory of quantum groups was put in correspondence [@ags] with the previous truncation of the operator algebra found by BPZ in the minimal models.

With this historical perspective in mind, what we have suggested by introducing what we call the Renormalization Quantum Group Method (qRG for short) [@q-germanyo] is to establish the connection of the truncation of states characteristic of the Real-Space RG methods (the other tool introduced by Wilson) with the special features of the (1+1)-dimensional systems exemplified by the RCFT. The connection we have found can be casted into the following squematical form,

$$\mbox{$q$RG-truncation} \leftrightarrow \mbox{RCFT}$$

whose precise content will be explained in the following sections with several examples.

Physicists working in field theory and condensed matter generalized the Real Space Renormalization Group methods introduced by Wilson [@wilson] to other problems by using the Kadanoff’s concept of block [@drell], [@jullien]. The Block 