Adjoint representation

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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 xdisplaystyle x, and any element g in GL(n) acts as a linear transformation of this vector space given by conjugation: x↦gxg−1displaystyle xmapsto 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.

Contents

• 
  1 Definition
  
  • 
    1.1 Derivative of Ad
    
  
  


• 
  2 Adjoint representation of a Lie algebra
  


• 
  3 Structure constants
  


• 
  4 Examples
  


• 
  5 Properties
  


• 
  6 Roots of a semisimple Lie group
  
  • 
    6.1 Example SL(2, R)
    
  
  


• 
  7 Variants and analogues
  


• 
  8 Notes
  


• 
  9 References
  

Definition

See also: Representation theory and Lie group § The Lie algebra associated with a Lie group

Let G be a Lie group, and let

Ψ:G→Aut⁡(G)displaystyle Psi :Gto operatorname Aut (G)

be the mapping g ↦ Ψ_g,
with Aut(G) the automorphism group of G and Ψ_g: G → G given by the inner automorphism

Ψg(h)=ghg−1 .displaystyle 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:

Adg=(dΨg)e:TeG→TeGdisplaystyle operatorname Ad _g=(dPsi _g)_e:T_eGrightarrow T_eG

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

Ad:G→Aut(g),g↦Adgdisplaystyle mathrm Ad colon Gto mathrm Aut (mathfrak g),,gmapsto mathrm Ad _g

is a group representation called the adjoint representation of G. (Here, Aut(g)displaystyle mathrm Aut (mathfrak g) is a closed^[1]Lie subgroup of GL(g)displaystyle 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 GLn(C)displaystyle mathrm GL _n(mathbb C ), then, since the exponential map is the matrix exponential: exp⁡(X)=eXdisplaystyle operatorname exp (X)=e^X, taking the derivative of Ψg(exp⁡(tX))=getXg−1displaystyle Psi _g(operatorname exp (tX))=ge^tXg^-1 at t = 0, one gets: for g in G and X in gdisplaystyle mathfrak g,

Adg⁡(X)=gXg−1displaystyle 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

Ad:G→Aut(g)displaystyle mathrm Ad colon Gto mathrm Aut (mathfrak g)

at the identity element gives the adjoint representation of the Lie algebra gdisplaystyle mathfrak g:

ad:g→Der(g)displaystyle mathrm ad colon mathfrak gto mathrm Der (mathfrak g)

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

adx(y)=[x,y]displaystyle mathrm ad _x(y)=[x,y],

for all x,y∈gdisplaystyle x,yin mathfrak g.^[2] Thus, adxdisplaystyle 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 gdisplaystyle mathfrak g generates a vector field X in the group G. Similarly, the adjoint map ad_xy = [x,y] of vectors in gdisplaystyle mathfrak g is homomorphic to the Lie derivative L_XY = [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 gdisplaystyle mathfrak g be a Lie algebra over a field k. Given an element x of a Lie algebra gdisplaystyle mathfrak g, one defines the adjoint action of x on gdisplaystyle mathfrak g as the map

adx:g→gwithadx⁡(y)=[x,y]displaystyle operatorname ad _x:mathfrak gto mathfrak gqquad textwithqquad operatorname ad _x(y)=[x,y]

for all y in gdisplaystyle mathfrak g. It is called the adjoint endomorphism or adjoint action. Then there is the linear mapping

ad:g→End⁡(g)displaystyle operatorname ad :mathfrak gto operatorname End (mathfrak g)

given by x ↦ ad_x. Within End(g)displaystyle (mathfrak g), the Lie bracket is, by definition, given by the commutator of the two operators:

[adx,ady]=adx∘ady−ady∘adxdisplaystyle [operatorname ad _x,operatorname ad _y]=operatorname ad _xcirc operatorname ad _y-operatorname ad _ycirc operatorname ad _x

where ∘displaystyle 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]]=0displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0

takes the form

([adx,ady])(z)=(ad[x,y])(z)displaystyle 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 gdisplaystyle 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 gdisplaystyle mathfrak g.

If gdisplaystyle mathfrak g is finite-dimensional, then End(g)displaystyle (mathfrak g) is isomorphic to gl(g)displaystyle mathfrak gl(mathfrak g), the Lie algebra of the general linear group over the vector space gdisplaystyle 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 gdisplaystyle mathfrak g is a module over itself.

The kernel of ad is the center of gdisplaystyle mathfrak g. Next, we consider the image of ad. Recall that a derivation on a Lie algebra is a linear map δ:g→gdisplaystyle delta :mathfrak grightarrow mathfrak g that obeys the Leibniz' law, that is,

δ([x,y])=[δ(x),y]+[x,δ(y)]displaystyle 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 gdisplaystyle mathfrak g under ad is
a subalgebra of Der(g)displaystyle (mathfrak g), the space of all derivations of gdisplaystyle mathfrak g.

When g=Lie⁡(G)displaystyle 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

[ei,ej]=∑kcijkek.displaystyle [e^i,e^j]=sum _kc^ij_ke^k.

Then the matrix elements for
ad_eⁱ
are given by

[adei]kj=cijk .displaystyle left[operatorname ad _e^iright]_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 gln(C)displaystyle 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

Ψ:G→Aut(G)displaystyle Psi colon Gto mathrm Aut (G),	Ψg:G→Gdisplaystyle Psi _gcolon Gto G,
Lie group homomorphism: Ψgh=ΨgΨhdisplaystyle Psi _gh=Psi _gPsi _h	Lie group automorphism: Ψg(ab)=Ψg(a)Ψg(b)displaystyle Psi _g(ab)=Psi _g(a)Psi _g(b) (Ψg)−1=Ψg−1displaystyle (Psi _g)^-1=Psi _g^-1
Ad:G→Aut(g)displaystyle mathrm Ad colon Gto mathrm Aut (mathfrak g)	Adg:g→gdisplaystyle mathrm Ad _gcolon mathfrak gto mathfrak g
Lie group homomorphism: Adgh=AdgAdhdisplaystyle mathrm Ad _gh=mathrm Ad _gmathrm Ad _h	Lie algebra automorphism: Adgdisplaystyle mathrm Ad _g is linear (Adg)−1=Adg−1displaystyle (mathrm Ad _g)^-1=mathrm Ad _g^-1 Adg[x,y]=[Adgx,Adgy]displaystyle mathrm Ad _g[x,y]=[mathrm Ad _gx,mathrm Ad _gy]
ad:g→Der(g)displaystyle mathrm ad colon mathfrak gto mathrm Der (mathfrak g)	adx:g→gdisplaystyle mathrm ad _xcolon mathfrak gto mathfrak g
Lie algebra homomorphism: addisplaystyle mathrm ad is linear ad[x,y]=[adx,ady]displaystyle mathrm ad _[x,y]=[mathrm ad _x,mathrm ad _y]	Lie algebra derivation: adxdisplaystyle mathrm ad _x is linear adx[y,z]=[adxy,z]+[y,adxz]displaystyle mathrm ad _x[y,z]=[mathrm ad _xy,z]+[y,mathrm ad _xz]

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

Ad(G)≅G/ZG(G0).displaystyle mathrm Ad (G)cong G/Z_G(G_0).

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

Now, if gdisplaystyle mathfrak g is the Lie algebra of a connected Lie group G, then Int⁡(g)displaystyle operatorname Int (mathfrak g) is the image of the adjoint representation of G: Int⁡(g)=Ad⁡(G)displaystyle 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

[a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯ann]↦[a11t1t2−1a12⋯t1tn−1a1nt2t1−1a21a22⋯t2tn−1a2n⋮⋮⋱⋮tnt1−1an1tnt2−1an2⋯ann].displaystyle beginbmatrixa_11&a_12&cdots &a_1n\a_21&a_22&cdots &a_2n\vdots &vdots &ddots &vdots \a_n1&a_n2&cdots &a_nn\endbmatrixmapsto beginbmatrixa_11&t_1t_2^-1a_12&cdots &t_1t_n^-1a_1n\t_2t_1^-1a_21&a_22&cdots &t_2t_n^-1a_2n\vdots &vdots &ddots &vdots \t_nt_1^-1a_n1&t_nt_2^-1a_n2&cdots &a_nn\endbmatrix.

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:

[abcd]displaystyle beginbmatrixa&b\c&d\endbmatrix

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

[t100t2]=[t1001/t1]=[exp⁡(θ)00exp⁡(−θ)]displaystyle beginbmatrixt_1&0\0&t_2\endbmatrix=beginbmatrixt_1&0\0&1/t_1\endbmatrix=beginbmatrixexp(theta )&0\0&exp(-theta )\endbmatrix

with t1t2=1displaystyle t_1t_2=1. The Lie algebra of the maximal torus is the Cartan subalgebra consisting of the matrices

[θ00−θ]=θ[1000]−θ[0001]=θ(e1−e2).displaystyle beginbmatrixtheta &0\0&-theta \endbmatrix=theta beginbmatrix1&0\0&0\endbmatrix-theta beginbmatrix0&0\0&1\endbmatrix=theta (e_1-e_2).

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

[t1001/t1][abcd][1/t100t1]=[at1bt1c/t1d/t1][1/t100t1]=[abt12ct1−2d]displaystyle beginbmatrixt_1&0\0&1/t_1\endbmatrixbeginbmatrixa&b\c&d\endbmatrixbeginbmatrix1/t_1&0\0&t_1\endbmatrix=beginbmatrixat_1&bt_1\c/t_1&d/t_1\endbmatrixbeginbmatrix1/t_1&0\0&t_1\endbmatrix=beginbmatrixa&bt_1^2\ct_1^-2&d\endbmatrix

The matrices

[1000][0001][0100][0010]displaystyle beginbmatrix1&0\0&0\endbmatrixbeginbmatrix0&0\0&1\endbmatrixbeginbmatrix0&1\0&0\endbmatrixbeginbmatrix0&0\1&0\endbmatrix

are then 'eigenvectors' of the conjugation operation with eigenvalues 1,1,t12,t1−2displaystyle 1,1,t_1^2,t_1^-2. The function Λ which gives t12displaystyle 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

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


2. 
   ^
   This is shown as follows:
   adx(y)=d(Adx)e(y)=limε→0(I+εx)y(I+εx)−1−yε=limε→0(I+εx)y(I−εx+(εx)2+O(ε3))−yε=limε→0((I+εx)yI−(I+εx)yεx+(I+εx)y(εx)2+O(ε3))−yε=limε→0(IyI+εxyI−Iyεx−εxyεx+Iy(εx)2+εxy(εx)2+O(ε3))−yε=limε→0y+xyε−yxε−xyxε2+yx2ε2+xyx2ε2+O(ε3)−yε=limε→0xy−yx−xyxε+yx2ε+xyx2ε+O(ε2)=[x,y]displaystyle beginalignedmathrm ad _x(y)&=d(mathrm Ad _x)_e(y)\&=lim _varepsilon to 0frac (I+varepsilon x)y(I+varepsilon x)^-1-yvarepsilon \&=lim _varepsilon to 0frac (I+varepsilon x)y(I-varepsilon x+(varepsilon x)^2+O(varepsilon ^3))-yvarepsilon \&=lim _varepsilon to 0frac ((I+varepsilon x)yI-(I+varepsilon x)yvarepsilon x+(I+varepsilon x)y(varepsilon x)^2+O(varepsilon ^3))-yvarepsilon \&=lim _varepsilon to 0frac (IyI+varepsilon xyI-Iyvarepsilon x-varepsilon xyvarepsilon x+Iy(varepsilon x)^2+varepsilon xy(varepsilon x)^2+O(varepsilon ^3))-yvarepsilon \&=lim _varepsilon to 0frac y+xyvarepsilon -yxvarepsilon -xyxvarepsilon ^2+yx^2varepsilon ^2+xyx^2varepsilon ^2+O(varepsilon ^3)-yvarepsilon \&=lim _varepsilon to 0xy-yx-xyxvarepsilon +yx^2varepsilon +xyx^2varepsilon +O(varepsilon ^2)\&=[x,y]endaligned
   


3. 
   ^ Hall 2015 Proposition 3.35
   


4. 
   ^ Hall 2015 Theorem 3.28
   


5. 
   ^ 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 .