On the computation of the Lichnerowicz--Jacobi cohomology.
type: Item
instance of: scholarly article
title: On the computation of the Lichnerowicz--Jacobi cohomology.
zbMATH Open document ID: 1092.53060
DOI: 10.1016/S0393-0440(02)00056-6
author: Manuel de León
author: Juan Carlos Marrero
author: Belén López
author: Edith Padrón
published in: Journal of Geometry and Physics
publication date: 2003-03-23T00:00:00Z
review text: Let \((\Lambda,E)\) be a Jacobi structure on a manifold \(M\). This defines a canonical Lie algebroid structure on the first jet bundle \(J^1(M,\mathbb{R})\). The corresponding Lie algebroid cohomology (with trivial coefficients) is called by the authors the Lichnerowicz-Jacobi cohomology of \((M,\Lambda,E)\) (LJ-cohomology, in short). First, the authors give an easy proof that the LJ-cohomology is invariant under conformal tranformations of the form: \[ \Lambda_f=f\Lambda,\quad E_f=\Lambda(df)+fE. \] where \(f\in C^\infty(M)\) is a positive function. Second, the authors consider the following classes: (i) \textit{Poisson manifolds.} They recall the results of \textit{A. Lichnerowicz} [J. Math. Pures Appl., IX. Sér. 57, 453--488 (1978; Zbl 0407.53025] for a Poisson manifold \((M,\Lambda)\), which relate the Lichnerowicz-Poisson cohomology (LP-cohomology, in short) and the LJ-cohomology. For \(\mathfrak{g}\) a Lie algebra of compact type, these together with the computation of the LP-cohomology of \(\mathfrak{g}^*\), due to \textit{V. L. Ginzburg} and \textit{A. Weinstein} [J. Am. Math. Soc. 5, No. 2, 445--453 (1992; Zbl 0766.58018)], show that \[ H^\bullet_{LJ}(\mathfrak{g}^*)=(H^\bullet(\mathfrak{g})\otimes\text{Inv})\oplus (H^{\bullet-1}(\mathfrak{g})\otimes\text{Inv}), \] where Inv denotes the algebra of Casimirs. (ii) \textit{Contact manifolds.} For a contact manifold \(M\), they show that its LJ-cohomology is given by: \[ H^\bullet_{LJ}(M)=H^\bullet_{dR}(M)\oplus H^{\bullet-1}_{dR}(M). \] (iii) \textit{Locally conformal symplectic manifolds.} Let \((M,\Omega)\) be a locally conformal symplectic manifold with Lee 1-form \(\omega\). Under some finiteness assumptions, they show that the relation between the LJ-cohomology and the de Rham cohomology, is the same as in the symplectic case. (iv) \textit{Unit sphere in a Lie algebra.} Let \(\mathfrak{g}\) be a Lie algebra with a scalar product \(\langle\cdot,\cdot\rangle\). They show that the unit sphere in \(\mathfrak…
reviewed by: Rui Loja Fernandes
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 1886801
zbMATH Keywords: Jacobi manifolds
zbMATH Keywords: LJ-cohomology
MaRDI profile type: Publication
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full work available at URL: s0393-0440(02)00056-6
OpenAlex ID: W2051244734
description: scientific article; zbMATH DE number 1886801