Costantini, Mauro (1989) On the lattice automorphisms of certain algebraic groups. PhD thesis, University of Warwick.

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Abstract

In the first chapter we give an introduction, and a survey of known results, which we shall use throughout the dissertation.

In the second chapter we first prove that every projectivity of a connected reductive non-abelian algebraic group G over K = Fp is strictly index-preserving (Theorem 2.1.6.). Then we prove that every autoprojectivity of G induces an automorphism of the building canonically associated to O. Furthermore we show how certain autoprojectivities of G act on the Weyl group of G and on the Dynkin diagram of G.

In the third chapter we restrict our attention to simple algebraic groups over K. We prove that if G is a simple algebraic group over K of rank at least 2, then the problem whether every autoprojectivity of G is induced by an automorphism, is reduced to the problem whether every autoprojectivity of G fixing every parabolic subgroup of G is the identity. Namely, if we let

Γ(G) – {φε Aut L(G) I Pφ = P for every parabolic subgroup P of G} , we have

Aut L(G) = Γ (Aut G)*,

where (Aut G)* is the group of all autoprojectivities of G induced by an automorphism (Theorem 3.4.9. and Corollary 3.4.15.).

In Chapter 4 we prove that actually Γ = {1} if G has rank at least 3 and p ≠ 2 (Theorem 4.6.5.), while in Chapter 5 we prove the same result , with different arguments, for the case of rank 1 (Corollary 5.2.6.) and 2, type A₂ excluded (Corollary 5.3.8.) (for groups of rank 1 we impose no restrictions on p).

Finally, in Chapter 6 we show that for the groups of type A₂ Theorem 4.6.5. does not hold. For this purpose we construct a non-trivial subgroup of the group Γ(SL₃(F₂₃)) (Corollary 6.4.15.).

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Automorphisms, Lattice theory, Affine algebraic groups
Official Date:	September 1989
Dates:	Date Event September 1989 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Carter, Roger W. (Roger William)
Sponsors:	Consiglio nazionale delle ricerche (Italy)
Format of File:	pdf
Extent:	iii, 135 leaves : illustrations
Language:	eng

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