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

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 Ŝ in the case n = 1. In particular, there are constructed the singular vectors of Ŝ and the Shapovalov form. The classification of the irreducible lowest weight modules over Ŝ is given. The second development is the proposal of an infinite hierarchy of differential equations, invariant with respect to Ŝ, 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 Ŝ 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.