From Clifford algebras to Heisenberg type Lie algebras

As it is well known, the Clifford algebras have numerous applications. In the present talk, we will explain how the Clifford algebras and their representation can build two-step nilpotent Lie algebras. They received the name Heisenberg type Lie algebras, due to the fact that the classical Heisenberg algebra is the simplest example in this construction.

A special class of Heisenberg type Lie algebras was introduced by A. Kaplan in 1980 to study hypoelliptic partial differential operators and their fundamental solutions. The Heisenberg type Lie algebras admit rational structural constants, that lead to the existence of lattices on the corresponding Lie groups according to the Malcev theorem. The factor of Heisenberg 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.