COHERENT STATES FOR QUANTUM COMPACT-GROUPS

Coherent states are introduced and their properties are discussed for simple quantum compact groups A(l),B-l, C-l and D-l. The multiplicativ e form of the canonical element for the quantum double is used to intr oduce the holomorphic coordinates on a general quantum dressing orbit. The coherent state is interpreted as a holomorphic function on this o rbit with values in the carrier Hilbert space of an irreducible repres entation of the corresponding quantized enveloping algebra. Using Gaus s decomposition, the commutation relations for the holomorphic coordin ates on the dressing orbit are derived explicitly and given in a compa ct R-matrix formulation (generalizing this way the q-deformed Grassman n and Bag manifolds). The antiholomorphic realization of the irreducib le representations of a compact quantum group (the analogue of the Bor el-Weil construction) is described using the concept of coherent state . The relation between representation theory and non-commutative diffe rential geometry is suggested.