USask PIMS Geometry and Physics (GAP) Seminar: Abid Ali

The question of integrality for semi-simple algebraic groups over the field of rational numbers was esablished by Chevalley in the 1950s as part of his work on associating affine group schemes with groups over integers. For infinite-dimensional Kac-Moody groups, it remains an open problem. To state this problem more precisely, let be a symmetrizable Kac-Moody algebra over , be an integral highest weight -module and be a -form of . Let be an associated minimal representation-theoretic Kac-Moody group and let be its integral subgroup. Suppose is the Chevalley subgroup of , that is, the the subgroup that stabilizes the lattice in . The integrality for is to determine if . We will discuss some progress on this problem, which we made in a joint work with Lisa Carbone, Dongwen Liu, and Scott H. Murray. Our results have various applications, including the integrality of subgroups of the unipotent subgroup of that are generated by commuting real root groups.