Let g\mathfrak gg be a Lie algebra. Then the adjoint map ad:g→gl(g)\text{ad}:\mathfrak g \to \mathfrak{gl}(\mathfrak g)ad:g→gl(g) is given by adX(Y)=[X,Y].\text{ad}_X(Y) = [X,Y].adX(Y)=[X,Y]. Because the adjoint map is a Lie algebra homomorphism, it is a representation of g\mathfrak gg, and therefore we may refer to ad\text{ad}ad as the adjoint representation of g\mathfrak gg.
Wikidata ID: Q4379157