Adjoint Representation

FORMAL DEFINITION

Let ''G'' be a Lie Group and let

\mathfrak g

be its Lie Algebra (which we identify with ''T''_''e''''G'', the Tangent Space to the Identity Element in ''G''). Define a map Ψ : ''G'' → Aut(''G'') by
:

\Psi_g(h) = ghg^{-1}.\,

For each ''g'' in ''G'', Ψ_''g'' is an Automorphism of ''G''. It follows that the Derivative of Ψ_''g'' at the identity is an automorphism of the Lie algebra

\mathfrak g

. We denote this map by Ad_''g'':
:

\mathrm{Ad}_g\colon \mathfrak g 	o \mathfrak g.

To say that Ad_''g'' is an Lie algebra automorphism is to say that Ad_''g'' is a Linear Transformation of

\mathfrak g

that preserves the Lie Bracket . The map
:

\mathrm{Ad}\colon G 	o \mathrm{Aut}(\mathfrak g)

which sends ''g'' to Ad_''g'' is called the adjoint representation of ''G''. This is indeed a Representation of ''G'' since

\mathrm{Aut}(\mathfrak g)

is a Lie Subgroup of

\mathrm{GL}(\mathfrak g)

and the above adjoint map is a Lie group homomorphism. The dimension of the adjoint representation is the same as the dimension of the group ''G''.

Adjoint representation of a Lie algebra

One may always pass from a representation of a Lie group ''G'' to a Representation Of Its Lie Algebra by taking the Derivative At The Identity . Taking the derivative of the adjoint map
:

\mathrm{Ad}\colon G 	o \mathrm{Aut}(\mathfrak g)

gives the adjoint representation of the Lie algebra

\mathfrak g

:
:

\mathrm{ad}\colon \mathfrak g 	o \mathrm{Der}(\mathfrak g).

Here

\mathrm{Der}(\mathfrak g)

is the Lie algebra of

\mathrm{Aut}(\mathfrak g)

which may be identified with the Derivation Algebra of

\mathfrak g

. The adjoint representation of a Lie algebra is related in a fundamental way to the structure of that algebra. In particular, one can show that
:

\mathrm{ad}_x(y) =

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for all

x,y \in \mathfrak g

. For more information see: '' Adjoint Representation Of A Lie Algebra ''.

EXAMPLES

If ''G'' is 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 $\mathrm{GL}_n(\mathbb C)$ ), then its Lie algebra is an algebra of ''n''×''n'' matrices with the commutator for a Lie bracket (i.e. a subalgebra of $\mathfrak{gl}_n(\mathbb C)$ ). In this case, the adjoint map is given by Ad_''g''(''x'') = ''gxg''⁻¹.

If ''G'' is SL₂(R) (real 2×2 matrices with Determinant 1), the Lie algebra of ''G'' consists of real 2×2 matrices with 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.

PROPERTIES

The following table summarizes the properties of the various maps mentioned in the definition

	Align	center <math>\mathrm{Ad}\colon G o \mathrm{Aut}(\mathfrak g)</math>
	Align	center <math>\mathrm{Ad}_g\colon \mathfrak g o \mathfrak g</math>
	Valign	top Lie group homomorphism:
	Valign	top Lie algebra automorphism:

\mathrm{Ad}_g = [\mathrm{Ad}_g(x),\mathrm{Ad}_g(y)

	Align	center <math>\mathrm{ad}\colon \mathfrak g o \mathrm{Der}(\mathfrak g)</math>
	Align	center <math>\mathrm{ad}_x\colon \mathfrak g o \mathfrak g</math>
	Valign	top Lie algebra homomorphism:
	Valign	top Lie algebra derivation:

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	CATEGORIES ABOUT ADJOINT REPRESENTATION

	representation theory of lie groups

Information About

Adjoint Representation