Seminar: Real elements in groups of type $F_4$

Abstract

Let $G$ be a group (resp. an algebraic group defined over a field $k$). For the latter case, let $G(k)$ denote the group $k$-rational points of $G$. An element $g \in G$ (resp. $G(k)$) is called real (resp. $k$-real) if there exists $h\in G$ (resp. $G(k)$) such that $hgh^{-1}=g^{-1}$. An element $g\in G$ (resp. $G(k)$) is said to be strongly real (resp. strongly $k$-real) if there exists $h\in G$ (resp. $G(k)$) such that $hgh^{-1}=g^{-1}$ and $h^2=1$. An exceptional algebraic group of type $F_4$ over a field $k$, is defined as the automorphism group of an Albert algebra over $k$. In this talk we prove that in a compact connected Lie group of type $F_4$,
every element is strongly real. We also describe the structure of $k$-real elements in algebraic groups of type $F_4$ defined over an arbitrary field $k$.

Date and Time : 03-04-2017 16:00

Venue : Seminar Hall, 219 @ CET

Speaker : Anirban Bose, IMSc