Strong integrality of inversion subgroups of Kac-Moody groups

Abstract: The question of integrality for semi-simple algebraic groups over the field of rational numbers was established 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 integrable 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 subgroup that stabilizes the lattice $V_{\mathbb Z}$ in $V$. The integrality for $G$ is to determine if $G(\mathbb{Z})=\Gamma(\mathbb{Z})$. 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.

commutative algebragroup theoryrings and algebrasrepresentation theory

Audience: researchers in the topic

Comments: Hybrid delivery (in person on University of Saskatchewan campus and via Zoom).

PIMS Geometry / Algebra / Physics (GAP) Seminar