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 $x$ , and any element g in GL(n) acts as a linear transformation of this vector space given by conjugation: $x\mapsto gxg^{-1}$ .

For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields.

Definition

Let G be a Lie group, and let

\Psi :G\to \operatorname {Aut} (G)

be the mapping $g \mapsto Ψ g$ , with Aut(G) the automorphism group of G and $Ψ g : G \to G$ given by the inner automorphism

\Psi _{g}(h)=ghg^{{-1}}~.

This Ψ is an example of a Lie group homomorphism.

For each g in G, define $Ad g$ to be the derivative of $Ψ g$ at the origin:

\operatorname {Ad}_{g}=(d\Psi _{g})_{e}:T_{e}G\rightarrow T_{e}G

where $d$ is the differential and ${\mathfrak {g}}=T_{e}G$ is the tangent space at the origin $e$ ( $e$ being the identity element of the group $G$ ). Since $\Psi _{g}$ is a Lie group automorphism, Ad_g is a Lie algebra automorphism; i.e., an invertible linear transformation of ${\mathfrak {g}}$ to itself that preserves the Lie bracket. The map

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

is a group representation called the adjoint representation of G. (Here, $\mathrm {Aut} ({\mathfrak {g}})$ is a closed^[1] Lie subgroup of $\mathrm {GL} ({\mathfrak {g}})$ and the above adjoint map is a Lie group homomorphism.)

If G is an (immersed) Lie subgroup of the general linear group $\mathrm {GL} _{n}(\mathbb {C} )$ , then, since the exponential map is the matrix exponential: $\operatorname {exp} (X)=e^{X}$ , taking the derivative of $\Psi _{g}(\operatorname {exp} (tX))=ge^{tX}g^{-1}$ at t = 0, one gets: for g in G and X in ${\mathfrak {g}}$ ,

\operatorname {Ad} _{g}(X)=gXg^{-1}

where on the right we have the products of matrices.

Derivative of Ad

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}})

at the identity element gives the adjoint representation of the Lie algebra ${\mathfrak {g}}$ :

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

where $\mathrm {Der} ({\mathfrak {g}})$ is the Lie algebra of $\mathrm {Aut} ({\mathfrak {g}})$ which may be identified with the derivation algebra of ${\mathfrak {g}}$ . One can show that

\mathrm {ad} _{x}(y)=[x,y]\,

for all $x,y\in {\mathfrak {g}}$ .^[2] Thus, $\mathrm {ad} _{x}$ coincides with the same one defined in #Adjoint representation of a Lie algebra below.

Ad and ad are related through the exponential map: Specifically, Ad_exp(x) = exp(ad_x) for all x in the Lie algebra.^[3] It is a consequence of the general result relating Lie group and Lie algebra homomorphisms via the exponential map.^[4]

The upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector $x$ in the algebra ${\mathfrak {g}}$ generates a vector field $X$ in the group $G$ . Similarly, the adjoint map $ad x y = [x, y]$ of vectors in ${\mathfrak {g}}$ is homomorphic to the Lie derivative $L X Y = [X, Y]$ of vector fields on the group $G$ considered as a manifold.

Further see the derivative of the exponential map.

Adjoint representation of a Lie algebra

Let ${\mathfrak {g}}$ be a Lie algebra over a field $k$ . Given an element $x$ of a Lie algebra ${\mathfrak {g}}$ , one defines the adjoint action of $x$ on ${\mathfrak {g}}$ as the map

\operatorname {ad}_{x}:{\mathfrak {g}}\to {\mathfrak {g}}\qquad {\text{with}}\qquad \operatorname {ad}_{x}(y)=[x,y]

for all $y$ in ${\mathfrak {g}}$ . It is called the adjoint endomorphism or adjoint action. Then there is the linear mapping

\operatorname {ad}:{\mathfrak {g}}\to \operatorname {End}({\mathfrak {g}})

given by $x \mapsto ad x$ . Within End $({\mathfrak {g}})$ , the Lie bracket is, by definition, given by the commutator of the two operators:

[\operatorname {ad}_{x},\operatorname {ad}_{y}]=\operatorname {ad}_{x}\circ \operatorname {ad}_{y}-\operatorname {ad}_{y}\circ \operatorname {ad}_{x}

where $\circ$ denotes composition of linear maps. Using the above definition of the Lie bracket, the Jacobi identity

[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0

takes the form

\left([\operatorname {ad}_{x},\operatorname {ad}_{y}]\right)(z)=\left(\operatorname {ad}_{{[x,y]}}\right)(z)

where $x$ , $y$ , and $z$ are arbitrary elements of ${\mathfrak {g}}$ .

This last identity says that ad is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets. Hence, ad is a representation of a Lie algebra and is called the adjoint representation of the algebra ${\mathfrak {g}}$ .

If ${\mathfrak {g}}$ is finite-dimensional, then End $({\mathfrak {g}})$ is isomorphic to ${\mathfrak {gl}}({\mathfrak {g}})$ , the Lie algebra of the general linear group over the vector space ${\mathfrak {g}}$ and if a basis for it is chosen, the composition corresponds to matrix multiplication.

In a more module-theoretic language, the construction simply says that ${\mathfrak {g}}$ is a module over itself.

The kernel of ad is the center of ${\mathfrak {g}}$ . Next, we consider the image of ad. Recall that a derivation on a Lie algebra is a linear map $\delta :{\mathfrak {g}}\rightarrow {\mathfrak {g}}$ that obeys the Leibniz' law, that is,

\delta ([x,y])=[\delta (x),y]+[x,\delta (y)]

for all $x$ and $y$ in the algebra.

That ad_x is a derivation is a consequence of the Jacobi identity. This implies that the image of ${\mathfrak {g}}$ under ad is a subalgebra of Der $({\mathfrak {g}})$ , the space of all derivations of ${\mathfrak {g}}$ .

When ${\mathfrak {g}}=\operatorname {Lie} (G)$ is the Lie algebra of a Lie group G, ad is the differential of Ad at the identity element of G.

Structure constants

The explicit matrix elements of the adjoint representation are given by the structure constants of the algebra. That is, let {eⁱ} be a set of basis vectors for the algebra, with

[e^{i},e^{j}]=\sum _{k}{c^{{ij}}}_{k}e^{k}.

Then the matrix elements for ad_eⁱ are given by

{\left[\operatorname {ad}_{{e^{i}}}\right]_{k}}^{j}={c^{{ij}}}_{k}~.

Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).

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 SL(2, 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/Z_{G}(G_{0}).

Given a finite-dimensional real Lie algebra ${\mathfrak {g}}$ , by Lie's third theorem, there is a connected Lie group $\operatorname {Int} ({\mathfrak {g}})$ whose Lie algebra is the image of the adjoint representation of ${\mathfrak {g}}$ (i.e., $\operatorname {Lie} (\operatorname {Int} ({\mathfrak {g}}))=\operatorname {ad} ({\mathfrak {g}})$ .) It is called the adjoint group of ${\mathfrak {g}}$ .

Now, if ${\mathfrak {g}}$ is the Lie algebra of a connected Lie group G, then $\operatorname {Int} ({\mathfrak {g}})$ is the image of the adjoint representation of G: $\operatorname {Int} ({\mathfrak {g}})=\operatorname {Ad} (G)$ .

Roots of a semisimple Lie group

If G is semisimple, the non-zero weights of the adjoint representation form a root system.^[5] (In general, one needs to pass to the complexification of the Lie algebra before proceeding.) 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.

Example SL(2, R)

Let us compute the root system for one of the simplest cases of Lie Groups. Let us consider the group SL(2, R) of two dimensional matrices with determinant 1. This consists of the set of matrices of the form:

{\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}

with a, b, c, d real and ad − bc = 1.

A maximal compact connected abelian Lie subgroup, or maximal torus T, is given by the subset of all matrices of the form

{\begin{bmatrix}t_{1}&0\\0&t_{2}\\\end{bmatrix}}={\begin{bmatrix}t_{1}&0\\0&1/t_{1}\\\end{bmatrix}}={\begin{bmatrix}\exp(\theta )&0\\0&\exp(-\theta )\\\end{bmatrix}}

with $t_{1}t_{2}=1$ . The Lie algebra of the maximal torus is the Cartan subalgebra consisting of the matrices

{\begin{bmatrix}\theta &0\\0&-\theta \\\end{bmatrix}}=\theta {\begin{bmatrix}1&0\\0&0\\\end{bmatrix}}-\theta {\begin{bmatrix}0&0\\0&1\\\end{bmatrix}}=\theta (e_{1}-e_{2}).

If we conjugate an element of SL(2, R) by an element of the maximal torus we obtain

{\begin{bmatrix}t_{1}&0\\0&1/t_{1}\\\end{bmatrix}}{\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}{\begin{bmatrix}1/t_{1}&0\\0&t_{1}\\\end{bmatrix}}={\begin{bmatrix}at_{1}&bt_{1}\\c/t_{1}&d/t_{1}\\\end{bmatrix}}{\begin{bmatrix}1/t_{1}&0\\0&t_{1}\\\end{bmatrix}}={\begin{bmatrix}a&bt_{1}^{2}\\ct_{1}^{-2}&d\\\end{bmatrix}}

The matrices

{\begin{bmatrix}1&0\\0&0\\\end{bmatrix}}{\begin{bmatrix}0&0\\0&1\\\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\begin{bmatrix}0&0\\1&0\\\end{bmatrix}}

are then 'eigenvectors' of the conjugation operation with eigenvalues $1,1,t_{1}^{2},t_{1}^{-2}$ . The function Λ which gives $t_{1}^{2}$ is a multiplicative character, or homomorphism from the group's torus to the underlying field R. The function λ giving θ is a weight of the Lie Algebra with weight space given by the span of the matrices.

It is satisfying to show the multiplicativity of the character and the linearity of the weight. It can further be proved that the differential of Λ can be used to create a weight. It is also educational to consider the case of SL(3, R).

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.

Notes

↑ The condition that a linear map is a Lie algebra homomorphism is a closed condition.
↑ This is shown as follows:
${\begin{aligned}\mathrm {ad} _{x}(y)&=d(\mathrm {Ad} _{x})_{e}(y)\\&=\lim _{\varepsilon \to 0}{\frac {(I+\varepsilon x)y(I+\varepsilon x)^{-1}-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}{\frac {(I+\varepsilon x)y(I-\varepsilon x+(\varepsilon x)^{2}+O(\varepsilon ^{3}))-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}{\frac {((I+\varepsilon x)yI-(I+\varepsilon x)y\varepsilon x+(I+\varepsilon x)y(\varepsilon x)^{2}+O(\varepsilon ^{3}))-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}{\frac {(IyI+\varepsilon xyI-Iy\varepsilon x-\varepsilon xy\varepsilon x+Iy(\varepsilon x)^{2}+\varepsilon xy(\varepsilon x)^{2}+O(\varepsilon ^{3}))-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}{\frac {y+xy\varepsilon -yx\varepsilon -xyx\varepsilon ^{2}+yx^{2}\varepsilon ^{2}+xyx^{2}\varepsilon ^{2}+O(\varepsilon ^{3})-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}xy-yx-xyx\varepsilon +yx^{2}\varepsilon +xyx^{2}\varepsilon +O(\varepsilon ^{2})\\&=[x,y]\end{aligned}}$
↑ Hall 2015 Proposition 3.35
↑ Hall 2015 Theorem 3.28
↑ Hall 2015 Section 7.3

References

Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666 .

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[1] The condition that a linear map is a Lie algebra homomorphism is a closed condition.

[2] This is shown as follows:
${\begin{aligned}\mathrm {ad} _{x}(y)&=d(\mathrm {Ad} _{x})_{e}(y)\\&=\lim _{\varepsilon \to 0}{\frac {(I+\varepsilon x)y(I+\varepsilon x)^{-1}-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}{\frac {(I+\varepsilon x)y(I-\varepsilon x+(\varepsilon x)^{2}+O(\varepsilon ^{3}))-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}{\frac {((I+\varepsilon x)yI-(I+\varepsilon x)y\varepsilon x+(I+\varepsilon x)y(\varepsilon x)^{2}+O(\varepsilon ^{3}))-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}{\frac {(IyI+\varepsilon xyI-Iy\varepsilon x-\varepsilon xy\varepsilon x+Iy(\varepsilon x)^{2}+\varepsilon xy(\varepsilon x)^{2}+O(\varepsilon ^{3}))-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}{\frac {y+xy\varepsilon -yx\varepsilon -xyx\varepsilon ^{2}+yx^{2}\varepsilon ^{2}+xyx^{2}\varepsilon ^{2}+O(\varepsilon ^{3})-y}{\varepsilon }}\\&=\lim _{\varepsilon \to 0}xy-yx-xyx\varepsilon +yx^{2}\varepsilon +xyx^{2}\varepsilon +O(\varepsilon ^{2})\\&=[x,y]\end{aligned}}$

[3] Hall 2015 Proposition 3.35

[4] Hall 2015 Theorem 3.28

[5] Hall 2015 Section 7.3