Lie-Rinehart algebras, descent, and quantization
by
Johannes Huebschmann
USTL, UFR de Math, CNRS-UMR 8524, F-59655 Villeneuve d'Ascq Cedex, France

The quantization problem for a Poisson algebra is related to that of constructing suitable representations of a Lie-Rinehart algebra associated with the Poisson algebra in a natural fashion, as explained in our paper entitled ``Poisson cohomology and quantization'', J. für die reine und angewandte Mathematik 408 (1990), 57-113. In the presence of singularities, such a Lie-Rinehart algebra does not arise from a Lie algebroid. The old question whether reduction after quantization coincides with quantization after reduction thus appears as a descent problem for (i) suitable Lie-Rinehart algebras defined in terms of Poisson structures and (ii) for the requisite additional data including the notions of prequantum module and polarization. In the Kähler case this question admits a positive answer: For a positive Kähler manifold with a hamiltonian action of a compact Lie group, when suitable additional conditions are imposed, reduction after quantization coincides with quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond but the (invariant) unreduced and reduced quantum observables as well. On the reduced level, the resulting classical phase space involves in general singularities and is a stratified Kähler space (see math.dg/0104213 for details), and the appropriate quantum phase space is a costratified Hilbert space (a manuscript in preparation, entitled Kähler quantization and reduction, will provide the necessary details) in such a way that the costratified structure reflects the stratification. Examples of stratified Kähler spaces arise from the closures of holomorphic nilpotent orbits in a semisimple Lie algebra of hermitian type, including angular momentum zero reduced spaces; in this case, the Lie-Rinehart structure on the reduced level admits a direct description in terms of the Lie structure of the Lie algebra of hermitian type. Other examples of stratified Kähler spaces arise from representations of compact Lie groups as well as from spaces of possibly twisted representations of the fundamental group of a surface in a compact Lie group. Particular examples of reduced quantum spaces are singular Fock spaces.

Date received: May 28, 2002