Adjoint representation

In mathematics, the adjoint representation (or adjoint action) 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, in the case where G is the Lie group of invertible matrices of size n, GL(n), the Lie algebra is the vector space of all (not necessarily invertible) n-by-n matrices. So in this case the adjoint representation is the vector space of n-by-n matrices , and any element g in GL(n) acts as a linear transformation of this vector space given by conjugation: .