On the one hand there is the notion of quantum groups due to Drinfeld and Jimbo. In there notion a quantum group is defined as a deformation of the universal enveloping algebra of a semisimple Lie algebra or, more generally, a Kac–Moody algebra. On the other hand there is the notion of quantum groups due to Woronowicz. He introduced compact matrix quantum groups and his notion can be assigned to the theory of C*-algebras.
My question is, do these notions sometimes coincide? And another question is when do they coincide?
I am very grateful for any advice
