Multicontact mappings on Hessenberg manifolds

TL;DR

This thesis explores multicontact structures on Hessenberg manifolds, providing explicit polynomial bases for simple Lie algebras and demonstrating rigidity properties of these structures in certain cases.

Contribution

It introduces explicit polynomial bases for split simple Lie algebras and studies the rigidity of multicontact structures on Hessenberg manifolds.

Findings

Explicit polynomial bases for split simple Lie algebras are constructed.

Rigidity of multicontact structures is proved for a large subclass of Hessenberg manifolds.

Abstract

This thesis was inspired by work of M. Cowling, F. De Mari, A. Koranyi and M. Reimann, who studied multicontact structures for the homogeneous manifolds G/P, where G is a semisimple Lie group and P is the minimal parabolic subgroup of G. The multicontact structure here arises naturally by the nilpotent component N of the Iwasawa decomposition of G, which is an open and dense subset of G/P. In the thesis two problems are addressed. The first one concerns the representation of simple Lie algebras in terms of polynomial algebras. More precisely, a polynomial basis for split simple Lie algebras is explicitly given in a suitable choice of coordinates. The second problem investigates the multicontact structure on Hessenberg manifolds. These manifolds can be locally viewed as quotient of the nilpotent component in the Iwasawa decomposition of semisimple Lie groups. Then the multicontact structure is inherited by the one in the Iwasawa nilpotent case. Rigidity with respect to this structure is studied and proved for a large subclass of Hessenberg manifolds.