An invariant formula for a star product with separation of variables
type: Item
instance of: scholarly article
title: An invariant formula for a star product with separation of variables
zbMATH Open document ID: 1250.53080
published in: Journal of Geometry and Physics
publication date: 2012-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.5832
review text: Let \(M\) be a pseudo-Kähler manifold with metric tensor \(g_{k\overline{l}}\). Its inverse is a Kähler-Poisson tensor, denoted by \(g^{\overline{l}k}\). Let \(\{\cdot,\cdot\}\) be the Poisson bracket corresponding to the tensor \(g^{\overline{l}k}\). A \(\ast\)-product on \(M\) is an associative product on \(C^\infty(M)\nu\) such that \(f\ast g=fg+\sum_{k\geq1}\nu^kB_k(f,g)\) where \(B_1(f,g)-B_1(g,f)=i\{f,g\}\). A \(\ast\)-product on \(M\) is called to be with separation of variables if \(B_1(f,g)=g^{\overline{l}k}\frac{\partial f}{\partial\overline{z}^l}\frac{\partial g}{\partial z^k}\). Introducing a locally defined total symbol \(\sigma(A)\) of a differential operator \(A\) on \(M\), the author provides a closed formula for the total symbol \(\sigma(L_f)\), where \(L_f(g)=f\ast g\) is the left \(\ast\)-multiplication operator with respect to deformation quantization with separation of variables on \(M\), and then proves an invariant formula with respect to such a product.
reviewed by: Panagiotis Batakidis
Mathematics Subject Classification ID: 53D55
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 70G45
zbMATH DE Number: 6086943
zbMATH Keywords: deformation quantization with separation of variables
zbMATH Keywords: Kähler connection
zbMATH Keywords: covariant tensors
author: A. V. Karabegov
MaRDI profile type: Publication
OpenAlex ID: W2080380973
arXiv ID: 1107.5832
cites work: Deformation theory and quantization. I: Deformations of symplectic structures
cites work: Deformation quantization of Poisson manifolds
cites work: Deformation quantizations with separation of variables on a Kähler manifold
cites work: A Fedosov star product of the Wick type for Kähler manifolds
cites work: A universal formula for deformation quantization on Kähler manifolds
cites work: An explicit formula for the Berezin star product
DOI: 10.1016/J.GEOMPHYS.2012.06.009
Recommended article: On Gammelgaard's formula for a star product with separation of variables
Recommended article: An explicit formula for the Berezin star product
Recommended article: Star products with separation of variables admitting a smooth extension
Recommended article: A new invariant for \(G\)-invariant star products
Recommended article: An algebra of distributions related to a star product with separation of variables
Recommended article: Parametrizing equivalence classes of invariant star products
Recommended article: Projectively Invariant Star Products
Recommended article: Invariant theory for the orthogonal group via star products
Recommended article: Homogeneous star products and closed integral formulas
Recommended article: Invariant star products on \(S^2\) and the canonical trace
description: scientific article; zbMATH DE number 6086943