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Lie Groups, Lie Algebras, and Representations

  • Brian C. Hall
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 267)

Abstract

An important concept in physics is that of symmetry, whether it be rotational symmetry for many physical systems or Lorentz symmetry in relativistic systems. In many cases, the group of symmetries of a system is a continuous group, that is, a group that is parameterized by one or more real parameters. More precisely, the symmetry group is often a Lie group, that is, a smooth manifold endowed with a group structure in such a way that operations of inversion and group multiplication are smooth. The tangent space at the identity in a Lie group has a natural “bracket” operation that makes the tangent space into a Lie algebra. The Lie algebra of a Lie group encodes many of the properties of the Lie group, and yet the Lie algebra is easier to work with because it is a linear space.

Keywords

Universal Cover Unitary Representation Quotient Group Projective Representation Closed Normal Subgroup 
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.

References

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    B.C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics, vol. 222 (Springer, New York, 2003)Google Scholar
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    N. Jacobson, Lie Algebras (Dover Publications, New York, 1979)Google Scholar
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    N.R. Wallach, Real Reductive Groups I (Academic, San Diego, 1988)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Brian C. Hall
    • 1
  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

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