Adjoint representation

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In mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its own Lie algebra. This representation is the linearized version of the action of G on itself by conjugation.

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[edit] Formal definition

Let G be a Lie group and let \mathfrak g be its Lie algebra (which we identify with TeG, the tangent space to the identity element in G). Define a map

\Psi : G \to \mathrm{Aut}(G)\,

by the equation Ψ(g) = Ψg for all g in G, where Aut(G) is the automorphism group of G and the automorphism Ψg is defined by

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

for all h in G. It follows that the derivative of Ψg at the identity is an automorphism of the Lie algebra \mathfrak g. We denote this map by Adg:

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

To say that Adg is a Lie algebra automorphism is to say that Adg is a linear transformation of \mathfrak g that preserves the Lie bracket. The map

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

which sends g to Adg 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.

[edit] 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 \to \mathrm{Aut}(\mathfrak g)

gives the adjoint representation of the Lie algebra \mathfrak g:

\mathrm{ad}\colon \mathfrak g \to \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) = [x,y]\,

for all x,y \in \mathfrak g. For more information see: adjoint representation of a Lie algebra.

[edit] 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 GL(n,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 Adg(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 forms.

[edit] Properties

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

\Psi\colon G \to \mathrm{Aut}(G)\, \Psi_g\colon G \to G\,
Lie group homomorphism:
  • Ψgh = ΨgΨh
 Lie group automorphism:
  • Ψg(ab) = Ψg(a)Ψg(b)
  • (\Psi_g)^{-1} = \Psi_{g^{-1}}
\mathrm{Ad}\colon G \to \mathrm{Aut}(\mathfrak g) \mathrm{Ad}_g\colon \mathfrak g \to \mathfrak g
Lie group homomorphism:
  • Adgh = AdgAdh
 Lie algebra automorphism:
  • Adg is linear
  • (\mathrm{Ad}_g)^{-1} = \mathrm{Ad}_{g^{-1}}
  • Adg[x,y] = [Adg(x),Adg(y)]
\mathrm{ad}\colon \mathfrak g \to \mathrm{Der}(\mathfrak g) \mathrm{ad}_x\colon \mathfrak g \to \mathfrak g
Lie algebra homomorphism:
  • ad is linear
  • ad[x,y] = [adx,ady]
 Lie algebra derivation:
  • adx is linear
  • adx[y,z] = [adx(y),z] + [y,adx(z)]

The image of G under the adjoint representation is denoted by AdG. If G is connected, the kernel of the adjoint representation coincides with the kernel of Ψ which is just the center of G. Therefore the adjoint representation of a connected Lie group G is faithful if and only if G is centerless. More generally, if G is not connected, then the kernel of the adjoint map is the centralizer of the identity component G0 of G. By the first isomorphism theorem we have

\mathrm{Ad}_G \cong G/C_G(G_0).

[edit] Roots of a semisimple Lie group

If G is semisimple, the non-zero weights of the adjoint representation form a root system. To see how this works, consider the case G=SLn(R). We can take the group of diagonal matrices diag(t1,...,tn) as our maximal torus T. Conjugation by an element of T sends

\begin{bmatrix} a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\\ \end{bmatrix} \mapsto \begin{bmatrix} a_{11}&t_1t_2^{-1}a_{12}&\cdots&t_1t_n^{-1}a_{1n}\\ t_2t_1^{-1}a_{21}&a_{22}&\cdots&t_2t_n^{-1}a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ t_nt_1^{-1}a_{n1}&t_nt_2^{-1}a_{n2}&\cdots&a_{nn}\\ \end{bmatrix}.

Thus, T acts trivially on the diagonal part of the Lie algebra of G and with eigenvectors titj-1 on the various off-diagonal entries. The roots of G are the weights diag(t1,...,tn)→titj-1. This accounts for the standard description of the root system of G=SLn(R) as the set of vectors of the form ei−ej.

[edit] Variants and analogues

The adjoint representation can also be defined for algebraic groups over any field.

The co-adjoint representation is the contragredient representation of the adjoint representation. Alexandre Kirillov observed that the orbit of any vector in a co-adjoint representation is a symplectic manifold. According to the philosophy in representation theory known as the orbit method (see also the Kirillov character formula), the irreducible representations of a Lie group G should be indexed in some way by its co-adjoint orbits. This relationship is closest in the case of nilpotent Lie groups.

[edit] References

  • Fulton, William; Joe Harris (1991). Representation Theory: A First Course. New York: Springer. ISBN 0-387-97495-4. 
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