In this paper, we consider
the problem of quantizing the canonical multiplicative Poisson structure on SU(2) by
C∗-algebraic deformation, a notion introduced by Rieffel, and show that there is such
a deformation which is also a coalgebra homomorphism. Parallel to the algebraic
development of quantum group theory, Woronowicz successfully quantized the group
structure of SU(2) (and other groups) through deformation in the context of Hopf
C∗-algebras. It is known that there exists a C∗-algebraic deformation quantization of
the multiplicative Poisson structure on SU(2) which is ‘compatible’ with
Woronowicz’s deformation (of the group structure) on the C∗-algebra level. Although
that deformation preserves the important symplectic leaf structure on SU(2) in a
natural way, it does not preserve the group struture in the sense that it is not a
coalgebra homomorphism. We show that the Weyl transformation introduced by
Dubois-Violette gives a different C∗-algebraic deformation quantization which is
compatible with Woronowicz’s deformation and does preserve the group
structure.
