An explicit two step quantization of Poisson structures and Lie bialgebras (Q1989851)

The article under review provides new explicit formulas for the deformation quantization of Poisson manifolds, as well as Lie bialgebras. These formulas hold for infinite dimensions. Moreover, it is shown that Kontsevich's celebrated formality theorem [\textit{M. Kontsevich}, Lett. Math. Phys. 66, No. 3, 157--216 (2003; Zbl 1058.53065)] follows. The authors use the \(\text{Lie}_{\infty}\) algebra structure on polyvector fields of \(\mathbb{R}^{n}\) (including \(n=\infty\)) introduced by \textit{B. Shoikhet} [Sel. Math., New Ser. 24, No. 2, 1691--1728 (2018; Zbl 1393.53092)]. Moreover, Shoikhet [loc. cit.] constructed an explicit map from Maurer-Cartan elements of this \(\text{Lie}_{\infty}\) algebra to star-products in \(C^{\infty}(\mathbb{R}^n)[[\hbar]]\). (The second author in [Commun. Math. Phys. 334, No. 3, 1649--1666 (2015; Zbl 1307.05088)] proved uniqueness.) More precisely, recall that a Poisson bracket on \(\mathbb{R}^n\) is a Maurer-Cartan element on the algebra of polyvector fields, with respect to the Shouten bracket. It is known that deformation quantization of such brackets is classified up to homotopy by Drinfeld' associators (hence there is an action of the Grothendieck-Teichmüller group). Given a Drinfeld' associator, the authors provide an explicit 1-1 correpondence between Poisson brackets as such, and Maurer-Cartan elements in terms of Shoikhet's \(\text{Lie}_{\infty}\) structure. The explicit transcendental formula provided for this correspondence is distinct from the formula provided by Shoicket [loc. cit.] using Kontsevich's formality theorem. Whence, the formula given by the authors consists in a new deformation quantization formula. In a similar fashion, the authors also treat the deformation quantization of finite-dimensional Lie bialgebras, providing an explicit transcendental formula, which improves results by \textit{P. Etingof} and \textit{D. Kazhdan} [Sel. Math., New Ser. 2, No. 1, 1--41 (1996; Zbl 0863.17008)], \textit{D. Tamarkin} [Geom. Funct. Anal. 17, No. 2, 537--604 (2007; Zbl 1163.17024)], as well as \textit{P. Ševera} [Sel. Math., New Ser. 22, No. 3, 1563--1581 (2016; Zbl 1366.17015)]