A q-Schrodinger algebra, its lowest-weight representations and generalized q-deformed heat/Schrodinger equations

We construct a q-deformation (S) over cap(q) of the centrally-extended Schrodinger algebra (S) over cap and an algebraic representation theory through lowest-weight representations. We use Verma modules over (S) over cap(q), calculate their singular vectors and factorize the Verma modules by submodules built on the singular vectors. We also give a realization of (S) over cap(q) with q-difference operators and obtain a polynomial realization of the lowest-weight representations and an infinite family of q-difference equations which may be called generalized q-deformed heat/Schrodinger equations. We also apply our methods to the on-shell q-Schrodinger algebra proposed by Floreanini and Vinet.