Algebraic constructions in the enveloping algebra of Lie algebras and their applications to quantum integrable systems

Project level: Masters, PhD

In recent years, commutant, coalgebra and partial Casimir allowed to build various structures inside the enveloping algebra of Lie algebras, not necessarily semi simple. In particular they allowed to construct polynomial algebras which have interesting features from point of view of representations. They have also provided new integrable and even superintegrable Hamiltonians. Those new descriptions are likely to play a broader role in different contexts in mathematical physics, and among those in regard of dynamical symmetries and subalgebras chain symmetries from nuclear physics. The purpose of this project is to provide further insights into those constructions. In addition, those constructions allow to provide further insight into enveloping algebras which is of interest beyond mathematical physics.