The symmetry groups of linear partial differential equations and representation theory. I
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.
scientific article; zbMATH DE number 729103