Abstract
We will be interested in complex representations of both real and complex Lie algebras. There is an important distinction to be made. If g is a real Lie algebra, then a complex representation is an ℝ-linear homomorphism g → End(V), where V is a complex vector space. On the other hand, if g is a complex Lie algebra, we require that the homomorphism be (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.
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