MathGloss

Let g\mathfrak g be a Lie algebra. Then the adjoint map ad:ggl(g)\text{ad}:\mathfrak g \to \mathfrak{gl}(\mathfrak g) is given by adX(Y)=[X,Y].\text{ad}_X(Y) = [X,Y]. Because the adjoint map is a Lie algebra homomorphism, it is a representation of g\mathfrak g, and therefore we may refer to ad\text{ad} as the adjoint representation of g\mathfrak g.

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