KMLIEGROUPS · Infinite-dimensional Lie theory and Kac-Moody groups

Lie theory was created at the end of the 19th century, and rapidly became a central chapter of contemporary mathematics. Finite-dimensional Lie groups and Lie algebras were extensively studied for more than a century, and are well understood. In attempting to generalise the finite-dimensional objects, one can roughly distinguish two general approaches, one ""analytic"" (keeping the smooth manifold structure of Lie groups) and the other “algebraic” (which is best represented by the algebraic constructions of Kac-Moody groups).Although intensively studied, Kac–Moody groups and Lie groups beyond the affine case remain mysterious to a large extent, and many questions concerning their structure remain open. In my Ph.-D. thesis, I established several structure results concerning topological Kac–Moody groups of indefinite type, and part of this research project carries on this study further. The main goal of this research project is to get a better understanding of Kac–Moody groups beyond the affine case, from both the analytic and algebraic approaches, and to try to construct “concrete realisations” of these groups (at least for hyperbolic types), by studying the unitary representations of their Lie algebra. More precisely, my method would be to try to construct certain “concrete” representations of a distinguished class of Lie algebras that include all symmetrisable Kac–Moody algebras