On the Kontsevich and the Campbell–baker–hausdorff Deformation Quantizations of a Linear Poisson Structure

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∞-morphism Ulin : [T • poly]lin → D • 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∞-morphism U : T • poly → D • poly [K] to the subalgebra [T • poly]lin. For a bivector field α with linear coefficients such that [α,α] = 0 the corresponding solution Ulin(α) of the Maurer–Cartan equation in D • 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. 1. L∞-morphisms, the Maurer–Cartan equation, and ∗-products Let F : T • poly → D • poly be an L∞-morphism from the dg Lie algebra of polyvector fields on R to the dg Lie algebra of polydifferential operators on R, and let F1 : T • poly → D • poly F2 : ∧ 2 T • poly → D • poly[−1] F3 : ∧ 3 T • poly → D • poly[−2] . . . . . . . . . . . . . . . . . . . . . . . be its Taylor components. Than any solution α ∈ T 1 poly of the Maurer–Cartan equation (i.e. α is a bivector field such that [α,α] = 0) defines a solution F(α) ∈ D1 poly of the Maurer–Cartan equation in D• poly (F(α) ∈ HomC(C ∞(Rd)⊗2 → C∞(Rd))) as follows: F(α) = F1(α) + 1 2 F2(α,α) + 1 6 F3(α,α, α) + . . .+ 1 n! Fn(α, . . . , α) + . . . (1)