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Lie algebras and incidence geometry

Roberts, Kieran (2012)
Ph.D. thesis, University of Birmingham.

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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:Ph.D. thesis.
Supervisor(s):Shpectorov, Sergey (Prof.)
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:3483
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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