An inverse image functor for Lie algebroids.
type: Item
instance of: scholarly article
title: An inverse image functor for Lie algebroids.
zbMATH Open document ID: 1040.32008
DOI: 10.1016/S0021-8693(03)00284-9
author: Sophie Chemla
published in: Journal of Algebra
publication date: 2003-11-19T00:00:00Z
review text: The author generalizes to Lie algebroids many results from the theory of \({\mathcal D}\)-modules. Let \(X\) be a complex analytic manifold, \({\mathcal O}_X\) the sheaf of holomorphic functions on \(X\), and \(\Theta_X\) the \({\mathcal O}_X\)-module of holomorphic vector fields on \(X\). A complex Lie algebroid over \(X\) is a pair \(({\mathcal L}_X,\omega)\) when \({\mathcal L}_X\) is a locally-free \({\mathcal O}_X\)-module of finite constant rank and a sheaf of \(\mathbb{C}\)-Lie algebras and \(\omega:{\mathcal L}_X\to \Theta_X\) an \({\mathcal O}_X\)-linear morphism of sheaves of \(\mathbb{C}\)-Lie algebras such that a natural compatibility condition holds. An important example are the Poisson manifolds. If \(X\) is a Poisson manifold, \((X,\Omega^1_X)\) is a Lie algebroid. The sheaf \({\mathcal D}({\mathcal L}_X)\) of generalized differential operators is constructed in a similar way to the sheaf \({\mathcal D}_X({\mathcal O}_X)\) of differential operators on \(X\) (and when \({\mathcal L}_X= \Theta_X\) and \(\omega=\) identity \({\mathcal D}({\mathcal L}_X)\) is just \({\mathcal D}_X({\mathcal O}_X)\); when \(X\) reduces to a point and \({\mathcal L}_X\) is a Lie algebra \({\mathcal D}({\mathcal L}_X)\) is the enveloping algebra of \({\mathcal L}_X\)). The author constructs a Koszul resolution of \({\mathcal O}_X\) considered as a left \({\mathcal D}({\mathcal L}_X)\) module, and an equivalence of categories between complex of left \({\mathcal D}({\mathcal L})\)-modules and complex of right \({\mathcal D}({\mathcal L}_X)\)-modules in a natural way. The definition of Lie algebroid morphisms given by the author in [\textit{S. Chemla}, Manusc. Math. 87, No. 2, 199--223 (1995; Zbl 0999.17009) and Math. Z. 232, No. 2, 367--388 (1999; Zbl 0933.32015)] coincide with that given previously by \textit{R. Almeida} and \textit{A. Kumpera} [An. Acad. Brasil, Cienc. 53, 247--250 (1981; Zbl 0484.58033)]. If \(\Phi= (f,F)\) is a Lie algebraic morphism from \(({\mathcal L}_X…
reviewed by: Gheorghe Gussi
Mathematics Subject Classification ID: 32C38
zbMATH DE Number: 2005902
zbMATH Keywords: Lie algebroid
zbMATH Keywords: inverse image functor
Wikidata QID: Q115340492
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2005902