Nearly associative deformation quantization

TL;DR

This paper investigates non-associative algebras for deformation quantization of Poisson brackets, revealing that alternative algebras must satisfy Jacobi identities and are essentially associative, while flexible algebras can be more general.

Contribution

It demonstrates that alternative deformation quantization algebras enforce Jacobi identities, whereas flexible algebras can accommodate non-Jacobi Poisson brackets.

Findings

Alternative algebras require Jacobi identities.

Flexible algebras exist for any Poisson bracket.

Most deformation quantization algebras are associative.

Abstract

We study several classes of non-associative algebras as possible candidates for deformation quantization in the direction of a Poisson bracket that does not satisfy Jacobi identities. We show that in fact alternative deformation quantization algebras require the Jacobi identities on the Poisson bracket and, under very general assumptions, are associative. At the same time, flexible deformation quantization algebras exist for any Poisson bracket.