Adjoint representation

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.

Formal definition

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

$\Psi :G\to \mathrm {Aut} (G)\,$

by the equation $\Psi (g)=\Psi _{g}$ for all g in G, where $\mathrm {Aut} (G)$ is the automorphism group of G and the automorphism $\Psi _{g}$ is defined by

$\Psi _{g}(h)=ghg^{-1}\,$

for all h in G. It follows that the derivative of Ψ_g at the identity is an automorphism of the Lie algebra ${\mathfrak {g}}$.

$d\Psi _{x}:T_{x}G\to T_{\Psi (x)}\mathrm {Aut} (G)$

$d\Psi _{e}:T_{e}G\to T_{\Psi (e)=e}\mathrm {Aut} (G)$

$d(\Psi _{g})_{x}:T_{x}G\to T_{\Psi _{g}(x)}(G)$

$d(\Psi _{g})_{e}:T_{e}G\to T_{\Psi _{g}(e)=e}(G)$

We denote this map by Ad_g:

$\mathrm {Ad} _{g}\colon {\mathfrak {g}}\to {\mathfrak {g}}.$

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

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

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

Main article: 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 ${\mathfrak {g}}$:

$d(\mathrm {Ad} )_{x}:T_{x}(G)\to T_{Ad(x)}(\mathrm {Aut} ({\mathfrak {g}}))$

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

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 $x,y\in {\mathfrak {g}}$.

$\mathrm {ad} _{x}(y)=d(\mathrm {Ad} _{e})_{x}(y)$

$=Lim_{\epsilon \to 0}{\frac {(I+\epsilon x)y(I+\epsilon x)^{-1}-y}{\epsilon }}$

$=Lim_{\epsilon \to 0}{\frac {(I+\epsilon x)y(I-\epsilon x+(\epsilon x)^{2}+O(\epsilon ^{3}))-y}{\epsilon }}$

$=Lim_{\epsilon \to 0}{\frac {((I+\epsilon x)yI-(I+\epsilon x)y\epsilon x+(I+\epsilon x)y(\epsilon x)^{2}+O(\epsilon ^{3}))-y}{\epsilon }}$

$=Lim_{\epsilon \to 0}{\frac {(IyI+\epsilon xyI-Iy\epsilon x-\epsilon xy\epsilon x+Iy(\epsilon x)^{2}+\epsilon xy(\epsilon x)^{2}+O(\epsilon ^{3}))-y}{\epsilon }}$

$=Lim_{\epsilon \to 0}{\frac {y+xy\epsilon -yx\epsilon -xyx\epsilon ^{2}+yx^{2}\epsilon ^{2}+xyx^{2}\epsilon ^{2}+O(\epsilon ^{3})-y}{\epsilon }}$

$=Lim_{\epsilon \to 0}xy-yx-xyx\epsilon +yx^{2}\epsilon +xyx^{2}\epsilon +O(\epsilon ^{2})$

$=[x,y]$

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 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 forms.

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: $\Psi _{gh}=\Psi _{g}\Psi _{h}$	Lie group automorphism: $\Psi _{g}(ab)=\Psi _{g}(a)\Psi _{g}(b)$ $(\Psi _{g})^{-1}=\Psi _{g^{-1}}$
$\mathrm {Ad} \colon G\to \mathrm {Aut} ({\mathfrak {g}})$	$\mathrm {Ad} _{g}\colon {\mathfrak {g}}\to {\mathfrak {g}}$
Lie group homomorphism: $\mathrm {Ad} _{gh}=\mathrm {Ad} _{g}\mathrm {Ad} _{h}$	Lie algebra automorphism: $\mathrm {Ad} _{g}$ is linear $(\mathrm {Ad} _{g})^{-1}=\mathrm {Ad} _{g^{-1}}$ $\mathrm {Ad} _{g}[x,y]=[\mathrm {Ad} g(x),\mathrm {Ad} g(y)]$
$\mathrm {ad} \colon {\mathfrak {g}}\to \mathrm {Der} ({\mathfrak {g}})$	$\mathrm {ad} _{x}\colon {\mathfrak {g}}\to {\mathfrak {g}}$
Lie algebra homomorphism: $\mathrm {ad}$ is linear $\mathrm {ad} _{[x,y]}=[\mathrm {ad} x,\mathrm {ad} y]$	Lie algebra derivation: $\mathrm {ad} _{x}$ is linear $\mathrm {ad} _{x}[y,z]=[\mathrm {ad} x(y),z]+[y,\mathrm {ad} x(z)]$

The image of G under the adjoint representation is denoted by Ad_G. 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

$\mathrm {Ad} _{G}\cong G/C_{G}(G_{0}).$

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=SL_n(R). We can take the group of diagonal matrices diag(t₁,...,t_n) as our maximal torus T. Conjugation by an element of T sends

${\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\\\end{bmatrix}}\mapsto {\begin{bmatrix}a_{11}&t_{1}t_{2}^{-1}a_{12}&\cdots &t_{1}t_{n}^{-1}a_{1n}\\t_{2}t_{1}^{-1}a_{21}&a_{22}&\cdots &t_{2}t_{n}^{-1}a_{2n}\\\vdots &\vdots &\ddots &\vdots \\t_{n}t_{1}^{-1}a_{n1}&t_{n}t_{2}^{-1}a_{n2}&\cdots &a_{nn}\\\end{bmatrix}}.$

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

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.

References

• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.