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Finite and Locally Finite Groups pp 45–70Cite as

Algebraic Groups

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Part of the book series: NATO ASI Series ((ASIC,volume 471))

Abstract

The notes discuss material on the theory of algebraic groups which is essential for a detailed study of the subgroup structure of algebraic groups, finite groups of Lie type, and certain locally finite groups.

The first section covers the general theory of algebraic groups, starting from the definition of algebraic variety. The interaction of various layers of structures (the coordinate ring, associated Lie algebra, Zariski topology) is discussed. Key points of the basic theory are mentioned (Jordan decomposition, reductive and semisimple groups, big cell, commutator relations), progressing towards the classification of simple algebraic groups.

The second section concerns the subgroup theory of simple algebraic groups. This begins with a study of subsystem groups and parabolic subgroups. Included is a brief discussion of modules occurring within parabolic subgroups and connections with unipotent classes. The section concludes with a discussion of the determination of the maximal closed connected subgroups of simple algebraic groups.

The final section is an overview of the representation theory of simple algebraic groups. This begins with a discussion of high weight modules and proceeds to the parametrization of irreducible modules by high weights. Included is a discussion of Weyl modules and consequences for extension theory. The section on tensor product theorems includes results of Steinberg, Borel-Tits, and a recent result of the author on abstract homomorphisms. The section concludes with applications to the theory of locally finite groups.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, Oregon, 97403, USA

    G. M. Seitz

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  1. G. M. Seitz
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Manchester, Manchester, UK

    B. Hartley

  2. Department of Mathematics, University of Oregon, Eugene, Oregon, USA

    G. M. Seitz

  3. UMIST, Manchester, UK

    A. V. Borovik  & R. M. Bryant  & 

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© 1995 Springer Science+Business Media Dordrecht

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Seitz, G.M. (1995). Algebraic Groups. In: Hartley, B., Seitz, G.M., Borovik, A.V., Bryant, R.M. (eds) Finite and Locally Finite Groups. NATO ASI Series, vol 471. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0329-9_2

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  • DOI: https://doi.org/10.1007/978-94-011-0329-9_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4145-4

  • Online ISBN: 978-94-011-0329-9

  • eBook Packages: Springer Book Archive

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