# Algebraic symplectic reduction and quantization of singular spaces

## TL;DR

This paper explores algebraic methods for singular symplectic spaces resulting from non-regular group actions, focusing on deformation quantization and providing explicit constructions for certain singular Poisson spaces.

## Contribution

It introduces an algebraic approach to singular reduction and explicitly constructs deformation quantization for specific singular Poisson spaces.

## Key findings

- Deformation quantization converges for flat phase space with classical moment map.
- Explicit deformation quantization constructed for some singular Poisson spaces.
- Singular symplectic spaces can be effectively quantized using algebraic methods.

## Abstract

The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular surfaces is the focus. For some examples of singular Poisson spaces the deformation quantization is explicitly constructed. In is shown that for the flat phase space with the classical moment map and the orthogonal group action the deformation quantization converges for the entire arguments of exponential type.