Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces

TL;DR

This paper explores non-formal deformation quantizations of Ricci-type symplectic symmetric spaces, establishing a correspondence with Schwartz operator multipliers and demonstrating the relation between formal star products and tempered non-formal quantizations.

Contribution

It introduces the concept of non-formal tempered deformation quantization for Ricci-type symmetric spaces and links formal star products to tempered quantizations.

Findings

Tempered deformation quantizations correspond to Schwartz operator multipliers.

Invariant formal star products are asymptotic expansions of tempered non-formal quantizations.

The paper provides an example illustrating non-formal quantization challenges with large symmetry groups.

Abstract

Ricci-type symplectic manifolds have been introduced and extensively studied by M. Cahen et al.. In this note, we describe their deformation quantizations in the split solvable symmetric case. In particular, we introduce the notion of non-formal tempered deformation quantization on such a space. We show that the set of tempered deformation quantizations is in one-to-one correspondence with the space of Schwartz operator multipliers on the real line. Moreover we prove that every invariant formal star product on a split Ricci-type solvable symmetric space is an asymptotic expansion of a tempered non-formal quantization. This note illustrates and partially reviews through an example a problematic studied by the author regarding non-formal quantization in presence of large groups of symmetries.