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.

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