"Fusion" of the representations of quantum groups in the root of unity Let us take some quantum group .It is known that it is possible to find vertex operator algebra such that the category or representations is equivalent to the category of representations of the quantum group.Actually the tensor structure in the category of representations of quantum group is also has the VOA counterpart.But the VOA corresponding to the quantum group is not unique -there are a lot of them.Such VOA defines some additional structures on the category of representations of quantum group . The "fusion" product is one of such structures.I will try to explain what is happening in the simplest sl(2) case.