Lie Groups pp 67-70 | Cite as

# Extension of Scalars

Chapter

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## Abstract

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