Jets de fonctions differentiables sur le dual d'un groupe de Lie nilpotent. (Jets of differentiable functions on the dual of a nilpotent Lie group)
type: Item
instance of: scholarly article
title: Jets de fonctions differentiables sur le dual d'un groupe de Lie nilpotent. (Jets of differentiable functions on the dual of a nilpotent Lie group)
zbMATH Open document ID: 0595.43008
DOI: 10.1007/BF01388913
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143458
review text: Let G be a simply connected nilpotent Lie group. In the setting of Alain Connes' non-commutative differential geometry, it is natural to take the convolution algebra \({\mathcal S}={\mathcal S}(G)\subset L^ 1(G)\) of Schwartz functions on G as the algebra of differentiable functions on the dual G{\^{\ }}. In this article, we define the notion of a ''Taylor series'' of such a function at a point (E,\(\pi)\) of G{\^{\ }}. More precisely, let M be the annihilator of E in \({\mathcal S}\). We show that the powers of M are closed ideals of \({\mathcal S}\), and we define \({\mathcal S}/M^{\infty}=\lim_{\leftarrow}{\mathcal S}/M^ n\) to be the algebra of jets of differentiable functions at the point (E,\(\pi)\). We show that \({\mathcal S}/M^{\infty}\) is Morita-equialent, in a suitable sense, to an algebra of ''non-commutative formal power series'' which is a projective limit of finite dimensional algebras over \({\mathbb{C}}\); we use a theorem of Gabriel on the unicity of the ''sober'' algebra attached to a finite abelian category. We give explicitly the formula for the Taylor series expansion of a function \(\phi\in {\mathcal S}\) at the trivial representation.
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 58A20
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 43A95
zbMATH DE Number: 3957910
zbMATH Keywords: nilpotent Lie group
zbMATH Keywords: non-commutative differential geometry
zbMATH Keywords: Schwartz functions
zbMATH Keywords: convolution algebra
zbMATH Keywords: Taylor series
zbMATH Keywords: jets of differentiable functions
zbMATH Keywords: non-commutative formal power series
zbMATH Keywords: sober algebra
zbMATH Keywords: trivial representation
author: Fokko du Cloux
MaRDI profile type: Publication
cites work: Q4065692
cites work: Q4173774
cites work: Q5539801
cites work: On a connection between nilpotent groups and oscillatory integrals associated to singularities
OpenAlex ID: W2116148660
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description: scientific article; zbMATH DE number 3957910