Electrical Lie algebras, the Schrödinger algebras and their representations

Abstract

In this paper, we show the relations between the electrical Lie algebras and the Schrödinger algebras. In particular, we show that the electrical Lie algebra of type F4 is isomorphic to the Schrödinger algebra in (5+1)-dimensional spacetime, which consequently shows that the dimension conjecture for eF4 holds. Then we classify all simple highest weight modules, all simple Whittaker modules at nonzero level and all simple U(H)-free modules for the Schrödinger algebra in (n+1) dimensional spacetime.