Lowest weight representations of the Schrödinger algebra and generalized heat/Schrödinger equations
type: Item
instance of: scholarly article
title: Lowest weight representations of the Schrödinger algebra and generalized heat/Schrödinger equations
zbMATH Open document ID: 0884.22009
DOI: 10.1016/S0034-4877(97)88001-9
author: Ch. Mrugalla
author: V. K. Dobrev
author: H.-D. Doebner
published in: Reports on Mathematical Physics
publication date: 1998-04-01T00:00:00Z
review text: The representation theory of the Schrödinger algebra \(S(n)\) in \((n+1)\)-dimensional space-time, treated as an example of a non-semisimple algebra (Barut, Rączka, 1980), is developed in order to construct differential equations, invariant with respect to the central extension \(\widehat S(n)\) of \(S(n)\) and called generalized heat/Schrödinger equations. In analogy with the semisimple case, basic informations about lowest weight modules (LWM), in particular Verma modules, over \(\widehat S(n)\) of \(S(n)\) are presented and some structural results in this area are obtained. The Schrödinger algebra \(S(1)\) and its central extension \(\widehat S(1)\) are studied in detail. The singular vectors of \(\widehat S(1)\) are constructed, and an analogue of the Shapovalov form (1972) is introduced. On this basis, a classification of the irreducible LWM over \(\widehat S(1)\) is given. The vector field realization of the Schrödinger algebra (Hagen, 1972; Barut, Wu, 1981) is used to obtain a polynomial realization of the irreducible LWM, with the help of which an infinite hierarchy of generalized heat/Schrödinger equations is obtained. It is shown that in some cases the irreducible LWM can be obtained as solution spaces of these differential equations. In such a way a well-known procedure (Dobrev, 1988) is extended to the non-semisimple situation under consideration.
reviewed by: A. A. Bogush
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 81Q60
zbMATH DE Number: 1091045
zbMATH Keywords: non-semisimple Lie algebra
zbMATH Keywords: representation theory
zbMATH Keywords: Schrödinger algebra
zbMATH Keywords: generalized heat/Schrödinger equations
zbMATH Keywords: lowest weight modules
zbMATH Keywords: Verma modules
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1091045