Lie algebras and incidence geometry

Roberts, Kieran (2012). Lie algebras and incidence geometry. University of Birmingham. Ph.D.

Preview

Roberts12PhD.pdf
PDF
Download (685kB)

Abstract

An element $\char{cmti10}{0x78}$ of a Lie algebra $\char{cmmi10}{0x4c}$ over the field $\char{cmmi10}{0x46}$ is extremal if [ $\char{cmti10}{0x78}$ , [ $\char{cmti10}{0x78}$ , $\char{cmmi10}{0x4c}$ ]] $\subseteq$ $\char{cmmi10}{0x46}$ $\char{cmti10}{0x78}$ . One can define the extremal geometry of $\char{cmmi10}{0x4c}$ whose points $\char{cmsy10}{0x45}$ are the projective points of extremal elements and lines $\char{cmsy10}{0x46}$ are projective lines all of whose points belong to $\char{cmsy10}{0x45}$ . We prove that any finite dimensional simple Lie algebra $\char{cmmi10}{0x4c}$ is a classical Lie algebra of type A $_n$ if it satisfies the following properties: $\char{cmmi10}{0x4c}$ contains no elements $\char{cmti10}{0x78}$ such that [ $\char{cmti10}{0x78}$ , [ $\char{cmti10}{0x78}$ , $\char{cmmi10}{0x4c}$ ]] = 0, $\char{cmmi10}{0x4c}$ is generated by its extremal elements and the extremal geometry $\char{cmsy10}{0x45}$ of $\char{cmmi10}{0x4c}$ is a root shadow space of type A $_{n,(1,n)}$ .