Extension of Scalars

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.KeywordsComplex Vector SpaceComplex RepresentationHomomorphismImportant DistinctionReal Vector SubspaceThese 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.