New Universal Deformation Formulas for deformation quantization

TL;DR

This paper introduces new universal deformation formulas for associative algebras, expanding the toolkit for deformation quantization by utilizing specific 2-cocycles and their exponentials, with applications to smooth functions on manifolds.

Contribution

It presents novel universal deformation formulas based on certain 2-cocycles and their exponentials, applicable to associative algebras over the rationals.

Findings

Defined a class of basic representable 2-cocycles using commuting derivations.

Constructed formal deformations via exponential of 2-cocycles.

Applied formulas to rational quantization of smooth functions on manifolds.

Abstract

Universal Deformation Formulas (UDFs) for the deformation of associative algebras play a key role in deformation quantization. Here we present examples for certain classes of infinitesimals. A basic representable 2-cocycle $F$ of an associative algebra $\mathcal A$ is one for which there exist commuting derivations $D,\dots, D_n$ of $\mathcal A$ such that $F = \sum_{ij}a_{ij}D_i \smile D_j$, where the $a_{ij}$ are central elements of $\mathcal A$. When $\mathcal A$ is defined over the rationals, there is a natural definition of the exponential of such a cocycle. With this $\exp \hbar F$ defines a formal one-parameter family of deformations of $\mathcal A$, where $\hbar$ is a deformation parameter. The rational quantization of smooth functions on a smooth manifold using a bivector field as an infinitesimal deformation is a special case.