The Chevalley--Eilenberg complex and deformation quantization in presence of two branes

Abstract

In this note, we prove that, for a finite-dimensional Lie algebra $\mathfrak{g}$ over a field $\mathbb{K}$ of characteristic 0 which contains $\mathbb{C}$, the Chevalley--Eilenberg complex $\mathrm{U}(\mathfrak{g})\otimes \wedge (\mathfrak{g})$, which is in a natural way a deformation quantization of the Koszul complex of $\mathrm{S}(\mathfrak{g})$, is $A_{\infty }$-quasi-isomorphic to the deformation quantization of the $A_{\infty }$-bimodule $K=\mathbb{K}$ provided by the Formality Theorem in presence of two branes (CFFR).

1 figure; according to remarks of M. Duflo and the anonymous referee, we have added Subsections 4.1, 4.2 and 4.3; we have also changed the title and updated the bibliography; some misprints have also been corrected

References in corpus (5)

• A Counterexample to the Quantizability of Modules
• Bimodules and branes in deformation quantization
• Biquantization of symmetric pairs and the quantum shift
• Deformation quantization with generators and relations
• A note on the Koszul complex in deformation quantization