相关论文: Koszul duality in deformation quantization, I

A systematic computational approach for the explicit construction of any quantum Hopf algebra (U_z(g),\Delta_z) starting from the Lie bialgebra (g,\delta) that gives the first-order deformation of the coproduct map \Delta_z is presented.…

In this paper we define an algebra structure on the vector space $L(\Sigma)$ generated by links in the manifold $\Sigma \times [0,1]$ where $\Sigma $ is an oriented surface. This algebra has a filtration and the associated graded algebra…

This dissertation is an exposition of Kontsevich's proof of the formality theorem and the classification of deformation quantisation on a Poisson manifold. We begin with an account of the physical background and introduce the Weyl-Moyal…

Whenever a given Poisson manifold is equipped with discrete symmetries the corresponding algebra of invariant functions or the algebra of functions twisted by the symmetry group can have new deformations, which are not captured by…

We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation.…

We quantize the Poisson-Lie group SL(2,R)^* as a bialgebra using the product of Kontsevich. The coproduct is a deformation of the coproduct that comes from the group structure. The resulting bialgebra structure is isomorphic to the quantum…

We prove that when Kontsevich's deformation quantization is applied on weight homogeneous Poisson structures, the operators in the $\ast-$ product formula are weight homogeneous. We then consider the linear Poisson case…

In his celebrated paper Kontsevich has proved a theorem which manifestly gives a quantum product (deformation quantization formula) and states that changing coordinates leads to gauge equivalent star products. To illuminate his procedure,…

Based on work done by Bonechi, Cattaneo, Felder and Zabzine on Poisson sigma models, we formally show that Kontsevich's star product can be obtained from the twisted convolution algebra of the geometric quantization of a Lie 2-groupoid, one…

Braverman and Gaitsgory gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the Poincare-Birkhoff-Witt property when its homogeneous version is Koszul. We widen their viewpoint and consider a quotient…

Let $\mathbb R^{m|n}$ be the usual super space. It is known that the algebraic functions on $\mathbb R^{m|n}$ is a Koszul algebra, whose Koszul dual algebra, however, is not the set of functions on $\mathbb R^{n|m}$, due to the…

We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string…

Kontsevich's formula for a deformation quantization of Poisson structures involves a Feynman series of graphs, with the weights given by some complicated integrals (using certain pullbacks of the standard angle form on a circe). We explain…

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C*-algebra may be regarded as a result of a quantization procedure.…

In this article we establish an explicit link between the classical theory of deformations \`a la Gerstenhaber -- and a fortiori with the Hochschild cohomology-- and (weak) PBW-deformations of homogeneous algebras. Our point of view is of…

The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson manifold $W$ by the ideal $I$ of…

Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the second quantization of a quantum cluster algebra, which means the…

This paper is based on the author's 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…

Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex"…

Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We…