# Lie algebras and incidence geometry

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

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## 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$$_{n,(1,n)}$$.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Shpectorov Prof., SergeyUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: Engineering and Physical Sciences Research Council, Other
Other Funders: The University of Birmingham
Subjects: Q Science > QA Mathematics
URI: http://etheses.bham.ac.uk/id/eprint/3483

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