On pseudo-Heisenberg type Lie algebras

A special class of pseudo-Heisenberg type Lie algebras was introduced by A. Kaplan in 1980 to study hypoelliptic partial differential operators and their fundamental solutions. In the present talk, we will explain how the pseudo-Heisenberg type Lie algebras are related to the Clifford algebras and their representations. The name "pseudo-Heisenberg" is attached due to the presence of a natural non-positive definite scalar product. The classical Heisenberg algebra is the simplest example in this construction. The Heisenberg type Lie algebras admit rational structural constants, which leads to the existence of lattices on the corresponding Lie groups according to the Malcev theorem. The factor of H-type Lie groups by the lattices gives rise to a chain of examples of nilmanifolds that are isospectral but non-diffeomorphic. In the talk, we will explain the construction of the Heisenberg type Lie algebras and give examples. We also will discuss the classification of the constructed Lie algebras and their group of automorphisms.