MORE ON QUANTUM GROUPS FROM THE QUANTIZATION POINT-OF-VIEW

Star products on the classical double group of a simple Lie group and on corresponding symplectic groupoids are given so that the quantum do uble and the ''quantized tangent bundle'' are obtained in the deformat ion description. ''Complex'' quantum groups and bicovariant quantum Li e algegras are discussed from this point of view. Further we discuss t he quantization of the Poisson structure on the symmetric algebra S(g) leading to the quantized enveloping algebra U(h)(g) as an example of biquantization in the sense of Turaev. Description of U(h)(g) in terms of the generators of the biocovariant differential calculus on F(G(q) ) is very convenient for this purpose. Finaly we interpret in the defo rmation framework some well known properties of compact quantum groups as simple consequences of corresponding properties of classical compa ct Lie groups. An analogue of the classical Kirillov's universal chara cter formula is given for the unitary irreducible representation in th e compact case.