Differential cohomology of \(C^\infty ({\mathfrak G}^*)\)
type: Item
instance of: scholarly article
title: Differential cohomology of \(C^\infty ({\mathfrak G}^*)\)
zbMATH Open document ID: 0684.17007
DOI: 10.1016/0393-0440(87)90002-7
published in: Journal of Geometry and Physics
publication date: 1987-01-01T00:00:00Z
review text: Let \({\mathfrak g}\) be a finite dimensional Lie algebra. It is well known that the dual \({\mathfrak g}^*\) of \({\mathfrak g}\) carries a Poisson structure, and consequently on the vector space N of all smooth functions on \({\mathfrak g}^*\) we can introduce a Lie algebra structure. Motivated by the problem of existence and classification of differential *-products on \({\mathfrak g}^*\), the author investigates the Chevalley cohomology of N, i.e. the differentiable cohomology of the Lie algebra N with coefficients in N with respect to the adjoint representation. We shall denote it by \(H^*_{diff}(N,\partial)\). He expresses \(H^ 1_{diff}(N,\partial)\) and \(H^ 2_{diff}(N,\partial)\) in terms of \(H^ 1({\mathfrak g},N)\) and \(H^ 2({\mathfrak g},N)\) respectively, where \(H^*({\mathfrak g},N)\) denotes the cohomology of the Lie algebra \({\mathfrak g}\) with coefficients in N with respect to the natural representation of \({\mathfrak g}\) in N. The author obtains the same results when replacing N by the vector space of all polynomial functions on \({\mathfrak g}^*.\) The methods applied in the paper represent a nontrivial modification of the methods used for symplectic manifolds by \textit{S. Gutt} [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 33, 1-31 (1980; Zbl 0476.53021)].
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 4120360
zbMATH Keywords: Poisson manifold
zbMATH Keywords: dual of Lie algebra
zbMATH Keywords: smooth functions
zbMATH Keywords: Chevalley cohomology
zbMATH Keywords: differentiable cohomology
zbMATH Keywords: polynomial functions
Wikidata QID: Q115363171
author: D. Arnal
reviewed by: Jiří Vanžura
MaRDI profile type: Publication
full work available at URL: 0393-0440(87)90002-7
OpenAlex ID: W2063551085
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description: scientific article; zbMATH DE number 4120360