Lie-Rinehart algebras, descent, and quantization

TL;DR

This paper explores Lie-Rinehart algebras as a framework for understanding the interaction between functions and vector fields, addressing quantization and reduction through a categorical and descent perspective.

Contribution

It introduces Lie-Rinehart algebras as a unifying structure to analyze quantization and reduction problems in geometric and algebraic contexts.

Findings

Lie-Rinehart algebras generalize the interaction between functions and vector fields.

They provide a categorical framework for quantization and reduction.

The approach addresses whether Kähler quantization commutes with reduction.

Abstract

A Lie-Rinehart algebra consists of a commutative algebra and a Lie algebra with additional structure which generalizes the mutual structure of interaction between the algebra of functions and the Lie algebra of smooth vector fields on a smooth manifold. Lie-Rinehart algebras provide the correct categorical language to solve the problem whether Kaehler quantization commutes with reduction which, in turn, may be seen as a descent problem.