On pseudo H-type Lie algebras (parte I)

Abstract: In this mini course we will introduce and study 2-step nilpotent Lie algebras, closely related to the Clifford algeras. The pseudo H-type Lie algebras, are generalisations of the Heisenberg type Lie algebras, introduced by Aroldo Kaplan at 1980 for the study of hypoelliptic operators. The mini course will include the following topics. 1. Definition of pseudo H-type Lie algebras. We give equivalent definitions of the pseudo H-type Lie algebras, that originated from the composition of quadratic forms, the representation of the Clifford algebras and isometric properties of the adjoint operator on the Lie algebra. 2. Relation of pseudo H-type Lie algebras and representations of Clifford algebras. Representations of Clifford algebras can be used for the construction of the Lie algebras if and only if the representation space can be endowed with a non degenerate bi-linear symmetric form, making the representation map skew symmetric with respect to this form. We will explain main difficulties of finding such a non-degenerate bi-linear symmetric form and provide several examples of the construction of the pseudo H-type Lie algebras from the Clifford algebra representations, including those that were introduced by A. Kaplan. 3. We will show the method of construction of a special basis for the pseudo H-type Lie algebras, such that the structural constants of the Lie algebras are always 0, 1 or -1. We show that the Bott periodicity of the Clifford algebras are naturally inherited by the pseudo H-type Lie algebras. It allows to reduce the construction to some basic cases. 4. We will show that infinite number of Clifford algebras leads to the infinite number of pseudo H-type Lie algebras. Moreover, the isomorphism of Clifford algebras is not automatically transmitted to the isomorphism of the Lie algebras. We will provide a complete classification of pseudo H-type Lie algebras. 5. The last topic is the description of automorphism groups of the pseudo H-type Lie algebras. It is not a closed topic still, nevertheless, I will inform on some achieved results.