Lowest weight representations of the Schrödinger algebra and generalized heat/Schrödinger equations

• Published in: Reports on mathematical physics 1997-04, Vol.39 (2), p.201-218
• Main Authors: Dobrev, V.K., Doebner, H.-D., Mrugalla, Ch
• Format: Article
• Language: English
• ISSN: 0034-4877; 1879-0674
• DOI: 10.1016/S0034-4877(97)88001-9

The present paper contains two interrelated developments. First, the basic elements of the theory of lowest weight modules, in particular, of Verma modules, over certain non-semisimple Lie algebras are developed in analogy with the semisimple case. This is done on the example of the (central extension of the) Schrödinger algebra in ( n + 1)-dimensional space-time. In more detail is considered the Schrödinger algebra S and its central extension S ̂ in the case n = 1. In particular, there are constructed the singular vectors of S ̂ and the Shapovalov form. The classification of the irreducible lowest weight modules over S ̂ is given. The second development is the proposal of an infinite hierarchy of differential equations, invariant with respect to S ̂ , which are called generalized heat/Schrödinger equations. The ordinary heat/Schrödinger equation is the first member of this hierarchy. These equations are obtained using a vector field realization of S ̂ which provides a polynomial basis realization of the irreducible lowest modules. In some cases the irreducible lowest weight modules are obtained as solution spaces of these differential equations.