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Register a new userExotic deformation quantization
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Authors
V. Ovsienko
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We consider formal deformations of the Poisson algebra of functions (with singularities) on $T^*M$ which are Laurent polynomials of fibers. Tn the case: $dim M=1$ ($M=S^1, {bf R}$), there exists a non-trivial $star$-product on this algebra non-equivalent to the standard Moyal product.
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We give an explicit local formula for any formal deformation quantization, with separation of variables, on a Kahler manifold. The formula is given in terms of differential operators, parametrized by acyclic combinatorial graphs.
Deformation quantization conventionally is described in terms of multidifferential operators. Jet manifold technique is well-known provide the adequate formulation of theory of differential operators. We extended this formulation to the multidifferential ones, and consider their infinite order jet prolongation. The infinite order jet manifold is endowed with the canonical flat connection that provides the covariant formula of a deformation star-product.
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.
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.
We study deformation quantization of nonassociative algebras whose associator satisfies some symmetric relations. This study is expanded to a larger class of nonassociative algebras includind Leibniz algebras. We apply also to this class the rule of polarization-depolarization.
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