$L_\infty$-derivations and the argument shift method for deformation quantization algebras

TL;DR

This paper introduces a novel method for constructing commutative subalgebras in deformation quantization algebras using $L_$-derivations and the argument shift technique, linking Poisson structures with Hochschild cochains.

Contribution

It develops an analogous construction to the argument shift method for deformation quantization algebras utilizing $L_$-differentiations of Hochschild cochains.

Findings

Provides a new way to generate commutative subalgebras from the center of deformed algebras.

Connects the argument shift method with $L_$-derivations in Hochschild cohomology.

Extends classical Poisson algebra techniques to deformation quantization contexts.

Abstract

The argument shift method is a well-known method for generating commutative families of functions in Poisson algebras from central elements and a vector field, verifying a special condition with respect to the Poisson bracket. In this notice we give an analogous construction, which gives one a way to create commutative subalgebras of a deformed algebra from its center (which is as it is well known describable in the terms of the center of the Poisson algebra) and an $L_\infty$-differentiation of the algebra of Hochschild cochains, verifying some additional conditions with respect to the Poisson structure.