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In mathematics, the adjoint representation of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is G L {\displaystyle GL} , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix g {\displaystyle g} to an endomorphism of the vector space of all linear transformations of R n {\displaystyle \mathbb {R} ^{n}} defined by: x ↦ g x g − 1 {\displaystyle x\mapsto gxg^{-1}}.
For any Lie group, this natural representation is obtained by linearizing the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields.