Covariant Strong Morita Theory of Star Product Algebras

TL;DR

This paper reviews recent advances in the Morita theory of *-algebras over ordered rings, focusing on representation theory, positivity, and symmetry via Hopf algebra actions, with tools for computing Picard groupoids.

Contribution

It provides a comprehensive review of Morita theory for *-algebras over ordered rings, including the role of positivity and symmetry in representations.

Findings

Established Morita theory for *-algebras over rings C = R(i).

Developed tools to compute Picard groupoids and their orbits.

Clarified the role of positivity and symmetry in representation theory.

Abstract

In this note we recall some recent progress in understanding the representation theory of *-algebras over rings C = R(i) where R is ordered and i^2 = -1. The representation spaces are modules over auxiliary *-algebras with inner products taking values in this auxiliary *-algebra. The ring ordering allows to implement positivity requirements for the inner products. Then the representations are required to be compatible with the inner product. Moreover, one can add the notion of symmetry in form of Hopf algebra actions. For all these notions of representations there is a well-established Morita theory which we review. The core of each version of Morita theory is the corresponding Picard groupoid for which we give tools to compute and determine both the orbits and the isotropy groups.