Commutative Poisson algebras from deformations of noncommutative algebras (Q6622491)

It is well known that ``taking the classical limit'', the counterpart of deformation quantization, may fail for a noncommutative phase space \(\mathcal{A}\). The difficulty in the noncommutative case is that the limit of commutators of operators provides a Hamiltonian structure which may be incompatible with the multiplication in \(\mathcal{A}\), making it impossible to obtain a Poisson algebra. \N\NThe authors introduce a generalization of the Hamiltonian structure, which deals with the above difficutly. Specifically, they view \(\mathcal{A}\) as a Poisson module over a commutative Poisson algebra structure on \(\Pi(\mathcal{A}) = Z(\mathcal{A}) \times (\mathcal{A} / Z\mathcal{A})\), which arises naturally from the deformation.