Central Extensions of the Heisenberg Algebra

Abstract. We study the non-trivial central extensions CEHeis of the Heisenberg algebra Heis recently constructed in [1]. We prove that a real form of CEHeis is one the fifteen classified real four–dimensional solvable Lie algebras. We also show that CEHeis can be realized (i) as a sub–Lie–algebra of the Schroedinger algebra and (ii) in terms of two independent copies of the canonical commutation relations (CCR). This gives a natural family of unitary representations of CEHeis and allows an explicit determination of the associated group by exponentiation. In contrast with Heis, the group law for CEHeis is given by nonlinear (quadratic) functions of the coordinates.