A note on the Koszul complex 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 $A_{\infty }$-$\mathrm{S}(V^{\ast })$-$\wedge (V)$-bimodule $K$, introduced in \cite{CFFR}, and the Koszul complex $\mathrm{K}(V)$ of $\mathrm{S}(V^{\ast })$, viewed as an $A_{\infty }$-$\mathrm{S}(V^{\ast })$-$\wedge (V)$-bimodule, for $V$ a finite-dimensional (complex or real) vector space.

8 pages, 1 figure; a more conceptual proof of the existence of an $A_{\infty }$-bimodule structure on the tensor product of $A_{\infty }$-bimodules has been added; the proof of Theorem 3.5 has been corrected; some typos have been corrected; acknowledgments modified; added a final remark on the generalization of the result to the framework of \cite{CFFR}