K\"ahler-Poisson algebras

TL;DR

This paper introduces Kähler-Poisson algebras as algebraic analogues of smooth functions on Kähler manifolds, establishing their properties and connections to classical differential geometry.

Contribution

It defines Kähler-Poisson algebras, proves their key properties, and shows how to construct them from Poisson algebras via localization, with illustrative examples.

Findings

Module of inner derivations is finitely generated and projective

Existence of a unique metric and torsion-free connection with classical curvature symmetries

Every Poisson algebra can be localized to a Kähler-Poisson algebra

Abstract

We introduce K\"ahler-Poisson algebras as analogues of algebras of smooth functions on K\"ahler manifolds, and prove that they share several properties with their classical counterparts on an algebraic level. For instance, the module of inner derivations of a K\"ahler-Poisson algebra is a finitely generated projective module, and allows for a unique metric and torsion-free connection whose curvature enjoys all the classical symmetries. Moreover, starting from a large class of Poisson algebras, we show that every algebra has an associated K\"ahler-Poisson algebra constructed as a localization. At the end, detailed examples are provided in order to illustrate the novel concepts.