New Universal Deformation Formulas for deformation quantization

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 $F={\sum }_{ij}{a}_{ij}{D}_i⌣D_j$, 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 $\exp \hslash F$ defines a formal one-parameter family of deformations of $\mathcal{A}$, where $\hslash$ 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.

The given exponential formula for quantization fails in general

References in corpus (1)

• Deformation quantization of Poisson manifolds, I

Cited by in corpus (1)

• Cohomology and deformations of dendriform algebras, and ${\mathrm{Dend}}_{\infty }$-algebras