Quantization of Poisson Manifolds from the Integrability of the Modular Function

Abstract

We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C * -algebras. This setting allows very singular polarizations. In particular we consider the case when the modular function is multiplicatively integrable, i.e. when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the Bohr-Sommerfeld conditions.We apply this procedure to the case of a family of Poisson structures on CP n , seen as Poisson homogeneous spaces of the standard Poisson-Lie group SU (n + 1). We show that a bihamiltoniam system on CP n defines a multiplicative integrable model on the symplectic groupoid; we compute the Bohr-Sommerfeld groupoid and show that it satisfies the needed properties for applying Renault theory. We recover and extend Sheu's description of quantum homogeneous spaces as groupoid C * -algebras.

Multiplicative integrable models from Poisson–Nijenhuis structures

“…In Section 3 we prove the main result of this paper in Proposition 3.1: we show how the algebra of hamiltonian forms can be lifted to the symplectic groupoid integrating P 2 to a Poisson subalgebra; if the PN structure is of maximal rank then this subalgebra is abelian and defines a multiplicative integrable model. This result explains the construction of [2].…”

“…The task of this note is to make clear the role of maximal rank Poisson Nijenhuis structures in the definition of multiplicative integrable models. We show that the construction of the multiplicative integrable model of [2] is actually valid in general for all maximal rank PN structures.…”

“…In this case, the integrable model comes from the canonical hierarchy of a PN structure long ago introduced by [7] for all compact hermitian symmetric spaces. In [2] we proved that the integrable model can be lifted from the complex projective space to a multiplicative integrable model on the symplectic groupoid and computed the Bohr-Sommerfeld groupoid.…”

“…Thanks to this compatibility, the application of geometric quantization methods should define a non commutative algebra on the space of polarized sections. The basic remark underlying [2] is that this perspective, rather than putting additional conditions on the difficult task of finding suitable polarizations, can allow more singular choices, namely singular real polarizations. Indeed, one can look for a singular real polarization such that the space of lagrangian leaves inherits the structure of topological groupoid.…”

“…By applying our result, we conclude that there exists a multiplicative integrable model on the symplectic groupoid integrating π + tω −1 kks for each t. This model has been studied so far only for the case of complex projective spaces. Indeed it has been shown in [2] that in this case the multiplicative integrable model defines a lagrangian fibration and the Bohr-Sommerfeld groupoid has been computed. The properties of the multiplicative integrable model on the symplectic groupoid in the general case will be the object of further investigation.…”