PIMS Geometry, Algebra and Physics Seminar

Date/Location

Title and Abstract

Thursday, October 12th

4:30-6:00PM CST

(Hybrid)

THORV 124

Strong Integrality of Inversion Subgroups of Kac-Moody Groups

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 \(\mathfrak{g}\) be a symmetrizable Kac-Moody algebra over \(\mathbb{Q}\), \(V\) be an integral highest weight \(\mathfrak{g}\)-module and \(V_{\mathbb{Z}}\) be a \(\mathbb{Z}\)-form of \(V\). Let \(G=G(\mathbb{Q})\) be an associated minimal representation-theoretic Kac-Moody group and let \(G(\mathbb{Z})\) be its integral subgroup. Suppose \(\Gamma(\mathbb{Z})\) is the Chevalley subgroup of \(G\), that is, the the subgroup that stabilizes the lattice \(V_{\mathbb{Z}}\) in \(V\). The integrality for \(G\) is to determine if \(G(\mathbb{Z})=\Gamma(Z)\). 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 \(U\) of \(G\) that are generated by commuting real root groups.

Wednesday, November 22nd

4-5:30pm CST

(Hybrid)

PHYSICS 128

Geometric and Algebraic Reduction of Multisymplectic Manifolds

A symplectic Hamiltonian manifold consists of a Lie group action on a symplectic manifold, together with the additional structure of a moment map, which encodes the group action in terms of the assignment of Hamiltonian vector fields. In special cases, the moment map determines a smooth submanifold to which the Lie group action restricts and the resulting quotient inherits the structure of symplectic manifold. In every case, it is possible to construct a reduced Poisson algebra that plays the role of the space of smooth functions on the reduced symplectic manifold.

In this talk, we will discuss an adaptation of these ideas to the multisymplectic setting. Specifically, we will exhibit a geometric reduction procedure for multisymplectic manifolds in the presence of a Hamiltonian action, an algebraic reduction procedure for the associated L-infinity algebras of classical observables, and a comparison of these two construction. This is joint work with Antonio Miti and Leonid Ryvkin.