On the Kontsevich and the Campbell-Baker-Hausdorff deformation quantizations of a linear Poisson structure

Abstract

For the Kirillov-Poisson structure on the vector space $\g^*$, where $\g$ is a finite-dimensional Lie algebra, it is known at least two canonical deformations quantization of this structure: they are the M. Kontsevich universal formula [K], and the formula, arising from the classical Campbell-Baker-Hausdorff formula [Ka]. It was proved in [Ka] that the last formula is exactly the part of Kontsevich's formula consisting of all the admissible graphs without (oriented) cycles between the vertices of the first type. It follows from the CBH-theorem that this part of Kontsevich's formula defines an associative product (in the case of a linear Poisson structure). The aim of these notes is to prove the last result directly, using the methods analogous to [K] instead of the CBH-formula. We construct an $L_\infty$-morphism $\U_\lin\colon[T^\ndot_\poly]_\lin\to D^\ndot_\poly$ from the \dg Lie algebra of polyvector fields with linear coefficients to the \dg Lie algebra of polydifferential operators, which is not equal to the restriction of the Formality $L_\infty$-morphism $\U\colon T^\ndot_\poly\to D^\ndot_\poly$ [K] to the subalgebra $[T^\ndot_\poly]_\lin$. For a bivector field $\alpha$ with linear coefficients such that $[\alpha,\alpha]=0$ the corresponding solution $\U_\lin(\alpha)$ of the Maurer-Cartan equation in $D^\ndot_\poly$ defines exactly the CBH-quantization,in the case of the harmonic angle map [K], Sect.2.We prove the associativity of the restricted Kontsevich formula (in the linear case) also for any angle map [K], Sect.6.2.