Non-compact quantum groups associated with Abelian subgroups

LetG be a Lie group. For any Abelian subalgebra $$\mathfrak{h}$$ of the Lie algebra g ofG, and any $$r \in \mathfrak{h}\Lambda \mathfrak{h}$$ , the difference of the left and right translates ofr gives a compatible Poisson bracket onG. We show how to construct the corresponding quantum group, in theC *-algebra setting. The main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onC *-algebras.