On the compatibility between cup products, the Alekseev--Torossian connection and the Kashiwara--Vergne conjecture

Abstract

For a finite-dimensional Lie algebra $\mathfrak{g}$ over a field $\mathbb{K}\supset \mathbb{C}$, we deduce from the compatibility between cup products Kontsevich (2003, Section 8) and from the main result of Shoikhet (2001) an alternative way of re-writing Kontsevich product $\star$ on $\mathrm{S}(\mathfrak{g})$ by means of the Alekseev--Torossian flat connection (Alekseev and Torossian, 2010). We deduce a similar formula directly from the Kashiwara--Vergne conjecture (Kashiwara and Vergne, 1978).

8 pages, 1 figure; notation changed; corrected many other misprints recently noticed; comments are very welcome!

References in corpus (2)

• Deformation quantization with generators and relations
• Two-colored noncommutative Gerstenhaber formality and infinity Duflo isomorphism