A note on the Koszul complex in deformation quantization

TL;DR

This paper provides a proof of the existence of an $A_$-quasi-isomorphism linking the bimodule $K$ from previous work to the Koszul complex of $ ext{S}(V^*)$, within the context of deformation quantization.

Contribution

It establishes an $A_$-quasi-isomorphism between two bimodules related to the Koszul complex in deformation quantization, clarifying their homotopy equivalence.

Findings

Proves the existence of an $A_$-quasi-isomorphism

Connects bimodule $K$ to the Koszul complex $ ext{K}(V)$

Enhances understanding of algebraic structures in deformation quantization

Abstract

The aim of this short note is to present a proof of the existence of an $A_\infty$-quasi-isomorphism between the $A_\infty$-$\mathrm S(V^*)$-$\wedge(V)$-bimodule $K$, introduced in \cite{CFFR}, and the Koszul complex $\mathrm K(V)$ of $\mathrm S(V^*)$, viewed as an $A_\infty$-$\mathrm S(V^*)$-$\wedge(V)$-bimodule, for $V$ a finite-dimensional (complex or real) vector space.