Lie groups andLie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups inphysics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
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Inmathematics, theadjoint representation (oradjoint action) of aLie groupG is a way of representing the elements of the group aslinear transformations of the group'sLie algebra, considered as avector space. For example, ifG is${\displaystyle \mathrm{G}\mathrm{L}(n,\mathbb{R})}$, the Lie group of realn-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertiblen-by-n matrix${\displaystyle g}$ to anendomorphism of the vector space of all linear transformations of${\displaystyle {\mathbb{R}}^n}$ defined by:${\displaystyle x\mapsto gx{g}^{-1}}$.

For any Lie group, this naturalrepresentation is obtained by linearizing (i.e. taking thedifferential of) theaction ofG on itself byconjugation. The adjoint representation can be defined forlinear algebraic groups over arbitraryfields.

LetG be aLie group, and let

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

be the mappingg ↦ Ψ_g, with Aut(G) theautomorphism group ofG andΨ_g:G →G given by theinner automorphism (conjugation)

${\displaystyle {\mathrm{\Psi }}_g(h)=gh{g}^{-1}.}$

This Ψ is a group homomorphism (it is aLie group homomorphism if${\displaystyle G}$ is connected^{[1][citation needed]}).

For eachg inG, defineAd_g to be thederivative ofΨ_g at the origin:

${\displaystyle {\mathrm{Ad}}_g=(d{\mathrm{\Psi }}_g{)}_e:T_{e}G\to T_{e}G}$

whered is the differential and${\displaystyle \mathfrak{g}=T_{e}G}$ is thetangent space at the origine (e being the identity element of the groupG). Since${\displaystyle {\mathrm{\Psi }}_g}$ is a Lie group automorphism, Ad_g is aLie algebra automorphism; i.e., an invertiblelinear transformation of${\displaystyle \mathfrak{g}}$ to itself that preserves theLie bracket. Moreover, since${\displaystyle g\mapsto {\mathrm{\Psi }}_g}$ is a group homomorphism,${\displaystyle g\mapsto {\mathrm{Ad}}_g}$ too is a group homomorphism.^[2] Hence, the map

${\displaystyle \mathrm{A}\mathrm{d}:G\to \mathrm{A}\mathrm{u}\mathrm{t}(\mathfrak{g}),g\mapsto {\mathrm{A}\mathrm{d}}_g}$

is agroup representation called theadjoint representation ofG.

IfG is animmersed Lie subgroup of the general linear group${\displaystyle {\mathrm{G}\mathrm{L}}_n(\mathbb{C})}$ (called immersely linear Lie group), then the Lie algebra${\displaystyle \mathfrak{g}}$ consists of matrices and theexponential map is the matrix exponential${\displaystyle \exp (X)=e^X}$ for matricesX with small operator norms. We will compute the derivative of${\displaystyle {\mathrm{\Psi }}_g}$ at${\displaystyle e}$. Forg inG and smallX in${\displaystyle \mathfrak{g}}$, the curve${\displaystyle t\to \exp (tX)}$ has derivative${\displaystyle X}$ att = 0, one then gets:

${\displaystyle {\mathrm{Ad}}_g(X)=(d{\mathrm{\Psi }}_g{)}_e(X)=({\mathrm{\Psi }}_g\circ \exp (tX){)}^{\prime }(0)=(g\exp (tX)g^{-1}{)}^{\prime }(0)=gX{g}^{-1}}$

where on the right we have the products of matrices. If${\displaystyle G\subset {\mathrm{G}\mathrm{L}}_n(\mathbb{C})}$ is a closed subgroup (that is,G is a matrix Lie group), then this formula is valid for allg inG and allX in${\displaystyle \mathfrak{g}}$.

Succinctly, an adjoint representation is anisotropy representation associated to the conjugation action ofG around the identity element ofG.

One may always pass from a representation of a Lie groupG to arepresentation of its Lie algebra by taking the derivative at the identity.

Taking the derivative of the adjoint map

${\displaystyle \mathrm{A}\mathrm{d}:G\to \mathrm{A}\mathrm{u}\mathrm{t}(\mathfrak{g})}$

at the identity element gives theadjoint representation of the Lie algebra${\displaystyle \mathfrak{g}=\mathrm{Lie}(G)}$ ofG:

where${\displaystyle \mathrm{D}\mathrm{e}\mathrm{r}(\mathfrak{g})=\mathrm{Lie}(\mathrm{Aut}(\mathfrak{g}))}$ is the Lie algebra of${\displaystyle \mathrm{A}\mathrm{u}\mathrm{t}(\mathfrak{g})}$ which may be identified with thederivation algebra of${\displaystyle \mathfrak{g}}$. One can show that

${\displaystyle {\mathrm{a}\mathrm{d}}_x(y)=[x,y]}$

for all${\displaystyle x,y\in \mathfrak{g}}$, where the right hand side is given (induced) by theLie bracket of vector fields. Indeed,^[3] recall that, viewing${\displaystyle \mathfrak{g}}$ as the Lie algebra of left-invariant vector fields onG, the bracket on${\displaystyle \mathfrak{g}}$ is given as:^[4] for left-invariant vector fieldsX,Y,

${\displaystyle [X,Y]=\underset{t\to 0}{lim}\frac{1}{t}(d{\phi }_{-t}(Y)-Y)}$

where${\displaystyle {\phi }_t:G\to G}$ denotes theflow generated byX. As it turns out,${\displaystyle {\phi }_t(g)=g{\phi }_t(e)}$, roughly because both sides satisfy the same ODE defining the flow. That is,${\displaystyle {\phi }_t=R_{{\phi }_t(e)}}$ where${\displaystyle R_h}$ denotes the right multiplication by${\displaystyle h\in G}$. On the other hand, since${\displaystyle {\mathrm{\Psi }}_g=R_{g^{-1}}\circ L_g}$, by thechain rule,

${\displaystyle {\mathrm{Ad}}_g(Y)=d(R_{g^{-1}}\circ L_g)(Y)=d{R}_{g^{-1}}(d{L}_g(Y))=d{R}_{g^{-1}}(Y)}$

asY is left-invariant. Hence,

${\displaystyle [X,Y]=\underset{t\to 0}{lim}\frac{1}{t}({\mathrm{Ad}}_{{\phi }_t(e)}(Y)-Y)}$,

which is what was needed to show.

Thus,${\displaystyle {\mathrm{a}\mathrm{d}}_x}$ coincides with the same one defined in§ Adjoint representation of a Lie algebra below. Ad and ad are related through theexponential map: Specifically, Ad_exp(x) = exp(ad_x) for allx in the Lie algebra.^[5] It is a consequence of the general result relating Lie group and Lie algebra homomorphisms via the exponential map.^[6]

IfG is an immersely linear Lie group, then the above computation simplifies: indeed, as noted early,${\displaystyle {\mathrm{Ad}}_g(Y)=gY{g}^{-1}}$ and thus with${\displaystyle g=e^{tX}}$,

${\displaystyle {\mathrm{Ad}}_{e^{tX}}(Y)=e^{tX}Y{e}^{-tX}}$.

Taking the derivative of this at${\displaystyle t=0}$, we have:

${\displaystyle {\mathrm{ad}}_{X}Y=XY-YX}$.

The general case can also be deduced from the linear case: indeed, let${\displaystyle G^{\prime }}$ be an immersely linear Lie group having the same Lie algebra as that ofG. Then the derivative of Ad at the identity element forG and that forG' coincide; hence, without loss of generality,G can be assumed to beG'.

The upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vectorx in the algebra${\displaystyle \mathfrak{g}}$ generates avector fieldX in the groupG. Similarly, the adjoint mapad_xy = [x,y] of vectors in${\displaystyle \mathfrak{g}}$ is homomorphic^{[clarification needed]} to theLie derivativeL_XY = [X,Y] of vector fields on the groupG considered as amanifold.

Further see thederivative of the exponential map.

Let${\displaystyle \mathfrak{g}}$ be a Lie algebra over some field. Given an elementx of a Lie algebra${\displaystyle \mathfrak{g}}$, one defines the adjoint action ofx on${\displaystyle \mathfrak{g}}$ as the map

${\displaystyle {\mathrm{ad}}_x:\mathfrak{g}\to \mathfrak{g}\text{with}{\mathrm{ad}}_x(y)=[x,y]}$

for ally in${\displaystyle \mathfrak{g}}$. It is called theadjoint endomorphism oradjoint action. (${\displaystyle {\mathrm{ad}}_x}$ is also often denoted as${\displaystyle \mathrm{ad}(x)}$.) Since a bracket is bilinear, this determines thelinear mapping

${\displaystyle \mathrm{ad}:\mathfrak{g}\to \mathfrak{g}\mathfrak{l}(\mathfrak{g})=(\mathrm{End}(\mathfrak{g}),[,])}$

given byx ↦ ad_x. Within End${\displaystyle (\mathfrak{g})}$, the bracket is, by definition, given by the commutator of the two operators:

${\displaystyle [T,S]=T\circ S-S\circ T}$

where${\displaystyle \circ }$ denotes composition of linear maps. Using the above definition of the bracket, theJacobi identity

${\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0}$

takes the form

${\displaystyle \left([{\mathrm{ad}}_x,{\mathrm{ad}}_y]\right)(z)=\left({\mathrm{ad}}_{[x,y]}\right)(z)}$

wherex,y, andz are arbitrary elements of${\displaystyle \mathfrak{g}}$.

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

If${\displaystyle \mathfrak{g}}$ is finite-dimensional and a basis for it is chosen, then${\displaystyle \mathfrak{g}\mathfrak{l}(\mathfrak{g})}$ is the Lie algebra of square matrices and the composition corresponds tomatrix multiplication.

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

The kernel of ad is thecenter of${\displaystyle \mathfrak{g}}$ (that's just rephrasing the definition). On the other hand, for each elementz in${\displaystyle \mathfrak{g}}$, the linear mapping${\displaystyle \delta ={\mathrm{ad}}_z}$ obeys theLeibniz' law:

${\displaystyle \delta ([x,y])=[\delta (x),y]+[x,\delta (y)]}$

for allx andy in the algebra (the restatement of the Jacobi identity). That is to say, ad_z is aderivation and the image of${\displaystyle \mathfrak{g}}$ under ad is a subalgebra of Der${\displaystyle (\mathfrak{g})}$, the space of all derivations of${\displaystyle \mathfrak{g}}$.

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

There is the following formula similar to theLeibniz formula: for scalars${\displaystyle \alpha ,\beta }$ and Lie algebra elements${\displaystyle x,y,z}$,

${\displaystyle ({\mathrm{ad}}_x-\alpha -\beta {)}^n[y,z]=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})\left[({\mathrm{ad}}_x-\alpha {)}^{i}y,({\mathrm{ad}}_x-\beta {)}^{n-i}z\right].}$

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

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

Then the matrix elements for ad_eⁱare given by

${\displaystyle {{\left[{\mathrm{ad}}_{e^i}\right]}_k}^j={c^{ij}}_k.}$

Thus, for example, the adjoint representation ofsu(2) is the defining representation ofso(3).

• IfG isabelian of dimensionn, the adjoint representation ofG is the trivialn-dimensional representation.
• IfG is amatrix Lie group (i.e. a closed subgroup of${\displaystyle \mathrm{G}\mathrm{L}(n,\mathbb{C})}$), then its Lie algebra is an algebra ofn×n matrices with the commutator for a Lie bracket (i.e. a subalgebra of${\displaystyle {\mathfrak{g}\mathfrak{l}}_n(\mathbb{C})}$). In this case, the adjoint map is given by Ad_g(x) =gxg⁻¹.
• IfG isSL(2,R) (real 2×2 matrices withdeterminant 1), the Lie algebra ofG consists of real 2×2 matrices withtrace 0. The representation is equivalent to that given by the action ofG by linear substitution on the space of binary (i.e., 2 variable)quadratic forms.

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

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

Theimage ofG under the adjoint representation is denoted by Ad(G). IfG isconnected, thekernel of the adjoint representation coincides with the kernel of Ψ which is just thecenter ofG. Therefore, the adjoint representation of a connected Lie groupG isfaithful if and only ifG is centerless. More generally, ifG is not connected, then the kernel of the adjoint map is thecentralizer of theidentity componentG₀ ofG. By thefirst isomorphism theorem we have

${\displaystyle \mathrm{A}\mathrm{d}(G)\cong G/Z_G(G_0).}$

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

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

IfG issemisimple, the non-zeroweights of the adjoint representation form aroot system.^[7] (In general, one needs to pass to the complexification of the Lie algebra before proceeding.) To see how this works, consider the caseG = SL(n,R). We can take the group of diagonal matrices diag(t₁, ..., t_n) as ourmaximal torusT. Conjugation by an element ofT sends

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

When computing the root system for one of the simplest cases of Lie Groups, the group SL(2,R) of two dimensional matrices with determinant 1 consists of the set of matrices of the form:

witha,b,c,d real andad − bc = 1.

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

with${\displaystyle t_1{t}_2=1}$. The Lie algebra of the maximal torus is the Cartan subalgebra consisting of the matrices

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

The matrices

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).

The adjoint representation can also be defined foralgebraic groups over any field.^{[clarification needed]}

Theco-adjoint representation is thecontragredient representation of the adjoint representation.Alexandre Kirillov observed that theorbit of any vector in a co-adjoint representation is asymplectic manifold. According to the philosophy inrepresentation theory known as theorbit method (see also theKirillov character formula), the irreducible representations of a Lie groupG should be indexed in some way by its co-adjoint orbits. This relationship is closest in the case ofnilpotent Lie groups.

• Adjoint bundle

• 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.
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996).Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley-Interscience.ISBN 978-0-471-15733-5.
• Hall, Brian C. (2015),Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,ISBN 978-3319134666.

• Representation theory of Lie groups
• Lie groups

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