The Heisenberg Group
The Euclidean space Rn has been one of the most important spaces on which mathematics is studied. It is a Riemannian manifold, and it is a group with respect to the usual addition and scalar multiplication of points in Rn. In this chapter we introduce the Heisenberg group H1, which is a noncommutative group of which the underlying manifold is R3. In subsequent chapters we study partial differential equations related to the Laplacians on the Heisenberg group. For the sake of simple notation, we look at the one-dimensional Heisenberg group H1 only. Extensions to higher-dimensional Heisenberg groups are routine. The contents in this chapter and the following chapters are based on the paper . The Heisenberg group and its connections with quantum mechanics and other branches of mathematics can be found in [4, 18, 30, 35], among others.