Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds

TL;DR

This paper classifies invariant star products on symplectic manifolds up to equivariant Morita equivalence, providing a comprehensive understanding of their algebraic and geometric structures under symmetry actions.

Contribution

It computes the equivariant Picard groupoid for star product algebras and classifies these algebras up to equivariant Morita equivalence on symplectic manifolds.

Findings

Complete classification of invariant star products up to equivariant Morita equivalence

Explicit computation of the equivariant Picard groupoid

Analysis of three types of Morita theory in this context

Abstract

In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, *-equivalence and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita equivalence.