Adjoint representation of a Lie algebra

In mathematics, the adjoint endomorphism or adjoint action is a homomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras.

Given an element x of a Lie algebra ${\displaystyle {\mathfrak {g}}}$, one defines the adjoint action of x on ${\displaystyle {\mathfrak {g}}}$ as the map

for all y in ${\displaystyle {\mathfrak {g}}}$.

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