Heisenberg type Lie algebras

KAPLAN, AROLDO; TIRABOSCHI, ALEJANDRO

Cartagena de Indias

Workshop; CIMPA school ?Algebraic structures, their representation and applications in geometry and non-associative models?.; 2012

CIMPA

Since their introduction by A. Kaplan some fifteen years ago, generalized Heisenberg groups, also known as groups of Heisenberg type or H-type groups, have provided a framework in which to construct interesting examples in geometry and analysis. The corresponding Lie algebras are 2-step nilpotent and regular (or fat). We will give an introduction to H-type algebras, fat algebras and we will show the classification of the real 2-step nilpotent Lie algebras with center of dimension two, in particular fat algebras. Once this is done, we will give examples of fat Lie algebras that are not of H-type; in fact, we find several different families of them. It follows from Adam?s theorem on frames on spheres that for any fat algebra there is an H-type algebra with the same dimension. That these were, in some sense, the most symmetric, and we will show that maximality of some automorphisms groups of fat algebras is related to how close is the algebra to being of Heisenberg type. For example, at least when the dimension of the center is two, dimAut(n) is maximal if and only if n is type H