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Representations of a Lie algebra

DEFINITION 2.1   Let $ k$ be a field. A finite dimensional representation of a Lie algebra $ \mathfrak{g}$ over $ k$ is a Lie algebra homomorphism

$\displaystyle \rho : \mathfrak{g}\to M_n(k).
$

Note: The full matrix algebra $ M_n(k)$ , when regarded as a Lie algebra equipped with the commutator product, is commonly denoted as $ \mathfrak{gl}_n(k)$ .

EXAMPLE 2.2   Let $ k$ be a field. Let $ \mathfrak{g}$ be a finite dimensional Lie algebra over $ k$ . We then have an adjoint representation

$\displaystyle \mathfrak{g}\ni X \mapsto \operatorname{ad}(X)=(Y\mapsto [X,Y])
\in \operatorname{End}_{k-\operatorname{linear}}(\mathfrak{g}).
$



2007-12-19