Adjoint representation of a Lie group

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

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

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

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

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

We denote this map by Adg:

${\displaystyle d(\Psi _{g})_{e}=\mathrm {Ad} _{g}\colon {\mathfrak {g}}\to {\mathfrak {g}}.}$

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

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

which sends g to Adg is called the adjoint representation of G. This is indeed a representation of G since ${\displaystyle \mathrm {Aut} ({\mathfrak {g}})}$ is a Lie subgroup of ${\displaystyle \mathrm {GL} ({\mathfrak {g}})}$ 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

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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

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

gives the adjoint representation of the Lie algebra ${\displaystyle {\mathfrak {g}}}$:

${\displaystyle d(\mathrm {Ad} )_{x}:T_{x}(G)\to T_{Ad(x)}(\mathrm {Aut} ({\mathfrak {g}}))}$
${\displaystyle \mathrm {ad} \colon {\mathfrak {g}}\to \mathrm {Der} ({\mathfrak {g}}).}$

Here ${\displaystyle \mathrm {Der} ({\mathfrak {g}})}$ is the Lie algebra of ${\displaystyle \mathrm {Aut} ({\mathfrak {g}})}$ which may be identified with the derivation algebra of ${\displaystyle {\mathfrak {g}}}$. 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

${\displaystyle \mathrm {ad} _{x}(y)=[x,y]\,}$
{\displaystyle {\begin{aligned}\mathrm {ad} _{x}(y)&=d(\mathrm {Ad} _{e})_{x}(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}}}

Properties

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

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

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

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, ..., tn) as our maximal torus T. Conjugation by an element of T sends

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

Thus, T acts trivially on the diagonal part of the Lie algebra of G and with eigenvectors 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 eiej.

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:

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

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

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

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

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

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

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

The matrices

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

are then 'eigenvectors' of the conjugation operation with eigenvalues ${\displaystyle 1,1,t_{1}^{2},t_{1}^{2}}$. The function Λ which gives ${\displaystyle t_{1}^{2}}$ 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.