Lie Groups pp 67-70 | Cite as

Extension of Scalars

  • Daniel Bump
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)


We will be interested in complex representations of both real and complex Lie algebras. There is an important distinction to be made. If \(\mathfrak{g}\) is a real Lie algebra, then a complex representation is an \(\mathbb{R}\)-linear homomorphism \(\mathfrak{g}\longrightarrow \mathrm{End}(V )\), where V is a complex vector space. On the other hand, if \(\mathfrak{g}\) is a complex Lie algebra, we require that the homomorphism be \(\mathbb{C}\)-linear. The reader should note that we ask more of a complex representation of a complex Lie algebra than we do of a complex representation of a real Lie algebra.

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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