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Classical Lie Groups

  • Joachim HilgertEmail author
  • Karl-Hermann Neeb
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter, we apply the general theory to classical matrix groups such as \(\mathop {\mathrm {GL}}\nolimits _{n}({\mathbb{C}}), \mathop {\rm SL}\nolimits _{n}({\mathbb{C}}), \mathop {\rm SO}\nolimits _{n}({\mathbb{C}}), \mathop {\rm Sp}\nolimits _{2n}({\mathbb{C}})\), and some of their real forms to provide explicit structural and topological information. We will start with compact real forms, i.e., \(\mathop {\rm U{}}\nolimits _{n}({\mathbb{K}})\) and \(\mathop {\rm SU}\nolimits _{n}({\mathbb{K}})\), where \({\mathbb{K}}\) is ℝ, ℂ, or ℍ, since many results can be reduced to compact groups.

References

  1. [GW09]
    Goodman, R., and N. R. Wallach, “Symmetry, Representations, and Invariants,” Springer-Verlag, New York, 2009. Google Scholar
  2. [Gr01]
    Grove, L. C., “Classical Groups and Geometric Algebra,” Amer. Math. Soc., Providence, 2001 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of PaderbornPaderbornGermany
  2. 2.Department of MathematicsFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

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