The symmetry groups of linear partial differential equations and representation theory. I
type: Item
instance of: scholarly article
title: The symmetry groups of linear partial differential equations and representation theory. I
zbMATH Open document ID: 0845.35020
author: Mark Craddock
published in: Journal of Differential Equations
publication date: 1996-09-11T00:00:00Z
review text: This paper represents a novel approach to the symmetry theory of linear partial differential equations connecting the classical Lie theory with analytic representation theory. It is generally accepted that Lie symmetry groups of partial differential equations are local Lie groups. Whilst generally true for nonlinear equations, the present paper shows that for suitably interpreted solution spaces of linear equations the Lie group may actually be global. The Lie representation is the action of the symmetry group on the (linear) space of solutions. It is necessary in general to deal with a space of distributions, the action of the differential operator and symmetry algebra being via the Fourier transform. Equations of the form \(P(D) u- u_t= 0\) on \(\mathbb{R}^n \times \mathbb{R}\) possess at least the \((2n+ 1)\)-dimensional Heisenberg group as a global symmetry group and for a large class of such equations it is found that the Lie representation is intertwined with a standard representation using a fundamental solution. In particular for the Schrödinger equation \(iu_t= u_{xx}\) the Lie representation is irreducible and unitary and associated with a representation of \(H_3\times SL(2, \mathbb{R})\) due to \textit{K. I. Gross} [Am. J. Math. 93, 398-428 (1971; Zbl 0229.22025)], and similar results are shown to hold for the heat and a Fokker-Planck equation.
Mathematics Subject Classification ID: 35G05
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 729103
zbMATH Keywords: Lie symmetry groups
zbMATH Keywords: Heisenberg group
reviewed by: Christopher Athorne
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1034
OpenAlex ID: W2009817733
DOI: 10.1006/JDEQ.1995.1034
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description: scientific article; zbMATH DE number 729103