Jacobi structure revisited
type: Item
instance of: scholarly article
zbMATH Open document ID: 0998.53054
DOI: 10.1088/0305-4470/34/49/316
author: J. Grabowski
author: Giuseppe Marmo
published in: Journal of Physics A: Mathematical and General
publication date: 2002-06-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0111148
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 17B63
zbMATH DE Number: 1709278
zbMATH Keywords: Jacobi structure
zbMATH Keywords: Lie algebroid
zbMATH Keywords: graded Lie brackets
zbMATH Keywords: Gerstenhaber-Jacobi algebras
MaRDI profile type: Publication
OpenAlex ID: W3104144441
Wikidata QID: Q56689274
arXiv ID: math/0111148
title: Jacobi structure revisited
review text: The authors introduce notions of Jacobi algebroids and Jacobi bialgebroids, which turn out to be equivalent to objects already studied by \textit{D. Iglesias} and \textit{J. C. Marrero} in [J. Geom. Phys. 40, No. 2, 176-199 (2001; Zbl 1001.17025)]. NEWLINENEWLINENEWLINEThe paper results from an attempt to understand the Lie algebroid structure on \(T^{\ast}M\oplus_{M} \mathbb R\) usually associated with a Jacobi structure on a manifold \(M\). Since a Jacobi bracket is just a Lie bracket on \(V=C^{\infty}(M)\) given by a bilinear first order differential operator, the authors begin with studying the structure of the algebra \({\mathcal A}\text{Diff}_{1}(V)\) of skew-linear first order differential operators on \(V\), endowed with the Richardson-Nijenhuis bracket. The resulting algebra is a Gerstenhaber algebra, but with a graded bracket satisfying NEWLINE\[NEWLINE[X,Y\wedge Z]=[X,Y]\wedge Z+(-1)^{|X|(|Y|+1)}Y\wedge [X,Z]- \widetilde{D}(X)\wedge Y\wedge ZNEWLINE\]NEWLINE where \(\widetilde{D}\) is a graded linear map of degree -1, instead of the Leibniz rule. Such algebras, are called Gerstenhaber-Jacobi algebras. NEWLINENEWLINENEWLINEIt is already known that Schouten-Nijenhuis algebras i.e. Gerstenhaber structures on the Grassmann algebra \({\mathcal A}(L)\) associated with a vector bundle \(L\) are in one to one correspondence with Lie algebroids (see the paper of \textit{Y. Kosmann-Schwarzbach} [Acta Appl. Math. 41, 153-165 (1995; Zbl 0837.17014)]). In view of this identification, the authors define the Jacobi-algebroids as Schouten-Jacobi algebras, i.e Gerstenhaber-Jacobi algebras on \({\mathcal A}(L)\). They then prove that this notion is equivalent to the notion of a Lie algebroid with the presence of a 1-cocycle as defined by Iglesias and Marrero.NEWLINENEWLINENEWLINEIn the last sections, the authors develop a lifting procedure which transports the Schouten-Jacobi bracket on \({\mathcal A}(L)\) into the Schouten bracket of multivector fields on the total space …
reviewed by: Angela Gammella-Mathieu
description: scientific article; zbMATH DE number 1709278