Lowest weight representations of the Schrödinger algebra and generalized heat/Schrödinger equations
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.
scientific article; zbMATH DE number 1091045
scientific article