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

Abstract

For a finite-dimensional Lie algebra g over a field K⊃C, we deduce from the compatibility between cup products (Kontsevich, 2003, [6, Section 8]), on the one hand, and from the Kashiwara–Vergne conjecture (Kashiwara and Vergne, 1978, [4]), on the other hand, alternative ways of re-writing Kontsevich product ⋆ on S(g).In this first part, we fix notation and conventions, revise the main features of Kontsevichʼs star product and examine the Kashiwara–Vergne conjecture and its relationship with Kontsevichʼs star product.