Morita Theory in Deformation Quantization

TL;DR

This paper explores Morita theory in deformation quantization, analyzing various equivalences of star product algebras on Poisson manifolds and classifying their Morita equivalence classes.

Contribution

It provides a comprehensive classification of Morita equivalent star products on Poisson and symplectic manifolds, connecting different types of Morita equivalences.

Findings

Complete classification of Morita equivalent star products on Poisson manifolds

Classification of covariantly Morita equivalent star products on symplectic manifolds

Relations between ring-theoretic, *-Morita, and strong Morita equivalences

Abstract

Various aspects of Morita theory of deformed algebras and in particular of star product algebras on general Poisson manifolds are discussed. We relate the three flavours ring-theoretic Morita equivalence, $^*$-Morita equivalence, and strong Morita equivalence and exemplify their properties for star product algebras. The complete classification of Morita equivalent star products on general Poisson manifolds is discussed as well as the complete classification of covariantly Morita equivalent star products on a symplectic manifold with respect to some Lie algebra action preserving a connection.