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Register a new userKoszul duality in deformation quantization and Tamarkins approach to Kontsevich formality
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Authors
Boris Shoikhet
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Let $alpha$ be a quadratic Poisson bivector on a vector space $V$. Then one can also consider $alpha$ as a quadratic Poisson bivector on the vector space $V^*[1]$. Fixed a universal deformation quantization (prediction some weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras $S(V^*)$ and $Lambda(V)$. These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on $alpha$, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the framework of the Tamarkins theory [T1].
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Let $alpha$ be a polynomial Poisson bivector on a finite-dimensional vector space $V$ over $mathbb{C}$. Then Kontsevich [K97] gives a formula for a quantization $fstar g$ of the algebra $S(V)^*$. We give a construction of an algebra with the PBW property defined from $alpha$ by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra $T(V^*)$ by relations $x_iotimes x_j-x_jotimes x_i=R_{ij}(hbar)$ where $R_{ij}(hbar)in T(V^*)otimeshbar mathbb{C}[[hbar]]$, $R_{ij}=hbar Sym(alpha_{ij})+mathcal{O}(hbar^2)$, with one relation for each pair of $i,j=1...dim V$. We prove that the constructed algebra obeys the PBW property, and this is a generalization of the Poincar{e}-Birkhoff-Witt theorem. In the case of a linear Poisson structure we get the PBW theorem itself, and for a quadratic Poisson structure we get an object closely related to a quantum $R$-matrix on $V$. At the same time we get a free resolution of the deformed algebra (for an arbitrary $alpha$). The construction of this PBW algebra is rather simple, as well as the proof of the PBW property. The major efforts should be undertaken to prove the conjecture that in this way we get an algebra isomorphic to the Kontsevich star-algebra.
In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(mathfrak g,r)$ on a smooth manifold $M$. Using the formality of polydifferential operators on Lie algebroids we obtain a deformation quantization of $M$ together with a quantum group $mathscr{U}_hbar(mathfrak{g})$ and a map of associated DGLAs. This motivates a definition of quantum action in terms of $L_infty$-morphisms which generalizes the one given by Drinfeld.
We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs, thus characterizing to which extent these spaces are different from oriented orbifolds. We also provide a relation between the cohomology of the space of metric ribbon graphs, known to be homotopy equivalent to the moduli space of Riemann surfaces, and the cohomology of a certain sheaf on the space of usual metric graphs.
This paper is based on the authors paper Koszul duality in deformation quantization, I, with some improvements. In particular, an Introduction is added, and the convergence of the spectral sequence in Lemma 2.1 is rigorously proven. Some informal discussion in Section 1.5 is added.
In this review an overview on some recent developments in deformation quantization is given. After a general historical overview we motivate the basic definitions of star products and their equivalences both from a mathematical and a physical point of view. Then we focus on two topics: the Morita classification of star product algebras and convergence issues which lead to the nuclear Weyl algebra.
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