A simple construction of associative deformations

TL;DR

This paper introduces a straightforward method for constructing associative algebra deformations using multiplicative coresolutions, providing explicit formulas for extending first-order deformations to all orders, with applications in deformation quantization.

Contribution

It presents a new simple approach to formal deformations of associative algebras leveraging coresolutions, enabling explicit recursive formulas for deformation extensions.

Findings

Certain first-order deformations extend to all orders

Explicit recurrent formulas for deformation extension

Application to deformation quantization of noncommutative Poisson structures

Abstract

We propose a simple approach to formal deformations of associative algebras. It exploits the machinery of multiplicative coresolutions of an associative algebra A in the category of A-bimodules. Specifically, we show that certain first-order deformations of A extend to all orders and we derive explicit recurrent formulas determining this extension. In physical terms, this may be regarded as the deformation quantization of noncommutative Poisson structures on A.