Editing Adjoint representation (section)

== Examples ==
*If ''G'' is [[abelian group|abelian]] of dimension ''n'', the adjoint representation of ''G'' is the trivial ''n''-dimensional representation.
*If ''G'' is a [[matrix Lie group]] (i.e. a closed subgroup of <math>\mathrm{GL}(n, \Complex)</math>), then its Lie algebra is an algebra of ''n''×''n'' matrices with the commutator for a Lie bracket (i.e. a subalgebra of <math>\mathfrak{gl}_n(\Complex)</math>). In this case, the adjoint map is given by Ad<sub>''g''</sub>(''x'') = ''gxg''<sup>−1</sup>.
*If ''G'' is [[SL2(R)|SL(2, '''R''')]] (real 2×2 matrices with [[determinant]] 1), the Lie algebra of ''G'' consists of real 2×2 matrices with [[trace (linear algebra)|trace]] 0.  The representation is equivalent to that given by the action of ''G'' by linear substitution on the space of binary (i.e., 2 variable) [[quadratic form]]s.