Compact Lie Groups, Generalised Euler Angles, and Applications

This is mainly a review of an intense 15-year long collaboration between the authors on explicit realisations of compact Lie groups and their applications. Starting with an elementary example, we will illustrate the main idea at the foundation of the generalisation of the Euler parametrisation of SU(2) to any compact Lie group. Based on this, we will provide a very detailed reconstruction of the possible Euler parametrisation associated with the so-called symmetric embedding. Then, we will recall how such constructions are related to the Dyson integrals, providing a geometrical interpretation of the latter, at least in certain cases. This includes a short review on the main properties of simple Lie groups, algebras, and their representations. Finally, we will conclude with some applications to nuclear physics and to measure theory in infinite dimensions and discuss some open questions.