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Algebraic Groups

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Part of the Springer Series in Soviet Mathematics book series (SSSOV)

Abstract

The definition of an algebraic group is similar to that of a Lie group, except that differentiable manifolds are replaced by algebraic varieties and differentiable maps by morphisms of algebraic varieties. In this book we will only consider the algebraic groups whose underlying varieties are affine ones. They are called “affine” or “linear” algebraic groups. The difference between arbitrary groups and affine ones is quite essential from the point of view of algebraic geometry and almost indiscernible from the group-theoretical points of view, since the commutator group of any irreducible algebraic group is an affine algebraic group. Besides, the general linear groups and any of their algebraic subgroups are affine algebraic groups. Therefore the affine algebraic groups are the most interesting ones for the Lie group theory. We will simply call them algebraic groups.

Keywords

  • Algebraic Group
  • Algebraic Variety
  • Borel Subgroup
  • Algebraic Linear Group
  • Semisimple Element

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1990 Springer-Verlag Berlin Heidelberg

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Onishchik, A.L., Vinberg, E.B. (1990). Algebraic Groups. In: Lie Groups and Algebraic Groups. Springer Series in Soviet Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-74334-4_3

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  • DOI: https://doi.org/10.1007/978-3-642-74334-4_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-74336-8

  • Online ISBN: 978-3-642-74334-4

  • eBook Packages: Springer Book Archive