Editing Adjoint representation

{{Short description|Mathematical term}}
{{Redirect|Adjoint map|the term in functional analysis|adjoint operator}}
{{Lie groups |Algebras}}
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 map|linear transformations]] of the group's [[Lie algebra]], considered as a [[vector space]]. For example, if ''G'' is <math>\mathrm{GL}(n, \mathbb{R})</math>, the Lie group of real [[invertible matrix|''n''-by-''n'' invertible matrices]], then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix <math> g </math> to an [[endomorphism]] of the vector space of all linear transformations of <math>\mathbb{R}^n</math> defined by: <math> x \mapsto g x g^{-1} </math>.

For any Lie group, this natural [[group representation|representation]] is obtained by linearizing (i.e. taking the [[Differential of a function|differential]] of) the [[Group action (mathematics)|action]] of ''G'' on itself by [[conjugation (group theory)|conjugation]]. The adjoint representation can be defined for [[linear algebraic group]]s over arbitrary [[field (mathematics)|fields]].

==Definition==
{{see also|Representation theory|Lie group#The Lie algebra associated with a Lie group}}
Let ''G'' be a [[Lie group]], and let
:<math>\Psi: G \to \operatorname{Aut}(G)</math>
be the mapping {{math|''g'' ↦ Ψ<sub>''g''</sub>}},  
with Aut(''G'') the [[automorphism group]] of ''G'' and {{math|Ψ<sub>''g''</sub>: ''G'' → ''G''}} given by the [[inner automorphism]] (conjugation)
:<math>\Psi_g(h)= ghg^{-1}~.</math>
This Ψ is a [[Lie_group#Homomorphisms_and_isomorphisms|Lie group homomorphism]].

For each ''g'' in ''G'', define {{math|Ad<sub>''g''</sub>}} to be the [[tangent space|derivative]] of {{math|Ψ<sub>''g''</sub>}} at the origin:
:<math>\operatorname{Ad}_g = (d\Psi_g)_e : T_eG \rightarrow T_eG</math>
where  {{mvar|d}} is the differential and <math>\mathfrak{g} = T_e G</math> is the [[tangent space]] at the origin  {{mvar|e}} ({{mvar|e}} being the identity element of the group  {{mvar|G}}). Since <math>\Psi_g</math> is a Lie group automorphism, Ad<sub>''g''</sub> is a [[Lie algebra automorphism]]; i.e., an invertible [[linear transformation]] of <math>\mathfrak g</math> to itself that preserves the [[Lie algebra#Definitions|Lie bracket]]. Moreover, since <math>g \mapsto \Psi_g</math> is a group homomorphism, <math>g \mapsto \operatorname{Ad}_g</math> too is a group homomorphism.<ref>Indeed, by the [[chain rule]], <math>\operatorname{Ad}_{gh} = d (\Psi_{gh})_e = d (\Psi_g \circ \Psi_h)_e = d (\Psi_g)_e \circ d (\Psi_h)_e = \operatorname{Ad}_g \circ \operatorname{Ad}_h.</math></ref> Hence, the map
:<math>\mathrm{Ad}\colon G \to \mathrm{Aut}(\mathfrak g), \, g \mapsto \mathrm{Ad}_g</math>
is a [[group representation]] called the '''adjoint representation''' of ''G''.

If ''G'' is an [[Lie_group–Lie_algebra_correspondence#Homomorphisms|immersed Lie subgroup]] of the general linear group <math>\mathrm{GL}_n(\mathbb{C})</math> (called immersely linear Lie group), then the Lie algebra <math>\mathfrak{g}</math> consists of matrices and the [[exponential map (Lie theory)|exponential map]] is the matrix exponential <math>\operatorname{exp}(X) = e^X</math> for matrices ''X'' with small operator norms. We will compute the derivative of  <math>\Psi_g</math> at <math>e</math>. For ''g'' in ''G'' and small ''X'' in <math>\mathfrak{g}</math>, the curve <math>t\to \exp(tX)</math> has derivative <math>X</math> at ''t'' = 0, one then gets:
:<math>\operatorname{Ad}_g(X) = (d\Psi_g)_e (X)=(\Psi_g\circ \exp(tX))'(0)=(g\exp(tX)g^{-1})'(0)=gX g^{-1}</math>
where on the right we have the products of matrices. If <math>G \subset \mathrm{GL}_n(\mathbb{C})</math> is a closed subgroup (that is, ''G'' is a matrix Lie group), then this formula is valid for all ''g'' in ''G'' and all ''X'' in <math>\mathfrak g</math>.

Succinctly, an adjoint representation is an [[isotropy representation]] associated to the conjugation action of ''G'' around the identity element of ''G''.

===Derivative of Ad===
One may always pass from a representation of a Lie group ''G'' to a [[representation of a Lie algebra|representation of its Lie algebra]] by taking the derivative at the identity.

Taking the derivative of the adjoint map
:<math>\mathrm{Ad} : G \to \mathrm{Aut}(\mathfrak g)</math>
at the identity element gives the '''adjoint representation''' of the Lie algebra <math>\mathfrak g = \operatorname{Lie}(G)</math> of ''G'':
:<math>\begin{align}
\mathrm{ad} : & \, \mathfrak g \to \mathrm{Der}(\mathfrak g) \\
& \,x \mapsto \operatorname{ad}_x = d(\operatorname{Ad})_e(x)
\end{align}</math>
where <math>\mathrm{Der}(\mathfrak g) = \operatorname{Lie}(\operatorname{Aut}(\mathfrak{g}))</math> is the Lie algebra of <math>\mathrm{Aut}(\mathfrak g)</math> which may be identified with the [[differential algebra#Lie algebra|derivation algebra]] of <math>\mathfrak g</math>. One can show that
:<math>\mathrm{ad}_x(y) = [x,y]\,</math>
for all <math>x,y \in \mathfrak g</math>, where the right hand side is given (induced) by the [[Lie bracket of vector fields]]. Indeed,<ref>{{harvnb|Kobayashi|Nomizu|1996|loc=page 41}}</ref> recall that, viewing <math>\mathfrak{g}</math> as the Lie algebra of left-invariant vector fields on ''G'', the bracket on <math>\mathfrak g</math> is given as:<ref>{{harvnb|Kobayashi|Nomizu|1996|loc=Proposition 1.9.}}</ref> for left-invariant vector fields ''X'', ''Y'',
:<math>[X, Y] = \lim_{t \to 0} {1 \over t}(d \varphi_{-t}(Y) - Y)</math>
where <math>\varphi_t: G \to G</math> denotes the [[flow (mathematics)|flow]] generated by ''X''. As it turns out, <math>\varphi_t(g) = g\varphi_t(e)</math>, roughly because both sides satisfy the same ODE defining the flow. That is, <math>\varphi_t = R_{\varphi_t(e)}</math> where <math>R_h</math> denotes the right multiplication by <math>h \in G</math>. On the other hand, since <math>\Psi_g = R_{g^{-1}} \circ L_g</math>, by the [[chain rule]],
:<math>\operatorname{Ad}_g(Y) = d (R_{g^{-1}} \circ L_g)(Y) = d R_{g^{-1}} (d L_g(Y)) = d R_{g^{-1}}(Y)</math>
as ''Y'' is left-invariant. Hence,
:<math>[X, Y] = \lim_{t \to 0} {1 \over t}(\operatorname{Ad}_{\varphi_t(e)}(Y) - Y)</math>,
which is what was needed to show.

Thus, <math>\mathrm{ad}_x</math> coincides with the same one defined in {{section link||Adjoint representation of a Lie algebra}} below. Ad and ad are related through the [[exponential map (Lie theory)|exponential map]]: Specifically, Ad<sub>exp(''x'')</sub> = exp(ad<sub>''x''</sub>) for all ''x'' in the Lie algebra.<ref>{{harvnb|Hall|2015}} Proposition 3.35</ref> It is a consequence of the general result relating Lie group and Lie algebra homomorphisms via the exponential map.<ref>{{harvnb|Hall|2015}} Theorem 3.28</ref>

If ''G'' is an immersely linear Lie group, then the above computation simplifies: indeed, as noted early, <math>\operatorname{Ad}_g(Y) = gYg^{-1}</math> and thus with <math>g = e^{tX}</math>,
:<math>\operatorname{Ad}_{e^{tX}}(Y) = e^{tX} Y e^{-tX}</math>.
Taking the derivative of this at <math>t = 0</math>, we have:
:<math>\operatorname{ad}_X Y = XY - YX</math>.
The general case can also be deduced from the linear case: indeed, let <math>G'</math> be an immersely linear Lie group having the same Lie algebra as that of ''G''. Then the derivative of Ad at the identity element for ''G'' and that for ''G{{'}}'' coincide; hence, without loss of generality, ''G'' can be assumed to be ''G{{'}}''.

The upper-case/lower-case notation is used extensively in the literature.  Thus, for example, a vector {{mvar|x}}    in the algebra <math>\mathfrak{g}</math> generates a [[vector field]] {{mvar|X}}    in the group  {{mvar|G}}.  Similarly, the adjoint map {{math|ad<sub>x</sub>y {{=}} [''x'',''y'']}} of vectors in <math>\mathfrak{g}</math> is homomorphic{{clarify|map is homomorphic??|date=December 2018}} to the [[Lie derivative]] {{math|L<sub>''X''</sub>''Y'' {{=}} [''X'',''Y'']}} of vector fields on the group  {{mvar|G}} considered as a [[manifold]].

Further see the [[derivative of the exponential map]].

== Adjoint representation of a Lie algebra ==
Let <math>\mathfrak{g}</math> be a Lie algebra over some field. Given an element {{mvar|x}} of a Lie algebra <math>\mathfrak{g}</math>, one defines the adjoint action of {{mvar|x}} on <math>\mathfrak{g}</math> as the map 
:<math>\operatorname{ad}_x : \mathfrak{g} \to \mathfrak{g} \qquad\text{with}\qquad \operatorname{ad}_x (y) = [x, y]</math>
for all {{mvar|y}} in <math>\mathfrak{g}</math>. It is called the '''adjoint endomorphism''' or '''adjoint action'''. (<math>\operatorname{ad}_x</math> is also often denoted as <math>\operatorname{ad}(x)</math>.) Since a bracket is bilinear, this determines the [[linear map|linear mapping]]
:<math>\operatorname{ad}:\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}) = (\operatorname{End}(\mathfrak{g}), [\;,\;])</math>
given by {{math|''x'' ↦ ad<sub>''x''</sub>}}. Within End<math>(\mathfrak{g})</math>, the bracket is, by definition, given by the commutator of the two operators:
:<math>[T, S] = T \circ S - S \circ T</math>
where <math>\circ</math> denotes composition of linear maps. Using the above definition of the bracket, the [[Jacobi identity]]
:<math>[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0</math> 
takes the form 
:<math>\left([\operatorname{ad}_x, \operatorname{ad}_y]\right)(z) = \left(\operatorname{ad}_{[x, y]}\right)(z)</math>
where {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} are arbitrary elements of <math>\mathfrak{g}</math>.

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 <math>\mathfrak{g}</math>.

If <math>\mathfrak{g}</math> is finite-dimensional and a basis for it is chosen, then <math>\mathfrak{gl}(\mathfrak{g})</math> is the Lie algebra of square matrices and the composition corresponds to [[matrix multiplication]].

In a more module-theoretic language, the construction says that <math>\mathfrak{g}</math> is a module over itself.

The kernel of ad is the [[center (algebra)|center]] of <math>\mathfrak{g}</math> (that's just rephrasing the definition). On the other hand, for each element {{mvar|z}} in <math>\mathfrak{g}</math>, the linear mapping <math>\delta = \operatorname{ad}_z</math> obeys the [[general Leibniz rule|Leibniz' law]]:
:<math>\delta ([x, y]) = [\delta(x),y] + [x, \delta(y)]</math>
for all {{mvar|x}} and {{mvar|y}} in the algebra (the restatement of the Jacobi identity). That is to say, ad<sub>''z''</sub> is a [[Lie algebra extension#Derivations|derivation]] and the image of <math>\mathfrak{g}</math> under ad is a subalgebra of Der<math>(\mathfrak{g})</math>, the space of all derivations of <math>\mathfrak{g}</math>.

When <math>\mathfrak{g} = \operatorname{Lie}(G)</math> is the Lie algebra of a Lie group ''G'', [[#Derivative of Ad|ad is the differential of Ad]] at the identity element of ''G''.

There is the following formula similar to the [[General Leibniz rule|Leibniz formula]]: for scalars <math>\alpha, \beta</math> and Lie algebra elements <math>x, y, z</math>,
:<math>(\operatorname{ad}_x - \alpha - \beta)^n [y, z] = \sum_{i = 0}^n \binom{n}{i} \left[(\operatorname{ad}_x - \alpha)^i y, (\operatorname{ad}_x - \beta)^{n - i} z\right].</math>

== Structure constants ==

The explicit matrix elements of the adjoint representation are given by the [[structure constants]] of the algebra. That is, let {e<sup>i</sup>} be a set of [[basis vectors]] for the algebra, with 
:<math>[e^i,e^j]=\sum_k{c^{ij}}_k e^k.</math>
Then the matrix elements for 
ad<sub>e<sup>i</sup></sub>
are given by
:<math>{\left[ \operatorname{ad}_{e^i}\right]_k}^j = {c^{ij}}_k ~. </math>

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

== Examples ==
*If ''G'' is [[abelian group|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 <math>\mathrm{GL}(n, \Complex)</math>), then its Lie algebra is an algebra of ''n''×''n'' matrices with the commutator for a Lie bracket (i.e. a subalgebra of <math>\mathfrak{gl}_n(\Complex)</math>). In this case, the adjoint map is given by Ad<sub>''g''</sub>(''x'') = ''gxg''<sup>−1</sup>.
*If ''G'' is [[SL2(R)|SL(2, '''R''')]] (real 2×2 matrices with [[determinant]] 1), the Lie algebra of ''G'' consists of real 2×2 matrices with [[trace (linear algebra)|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 form]]s.

==Properties==

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

{| class=wikitable
|-
! <math>\Psi\colon G \to \operatorname{Aut}(G)\,</math>
! <math>\Psi_g\colon G \to G\,</math>
|- style="vertical-align:top;"
| Lie group homomorphism:
* <math>\Psi_{gh} = \Psi_g\Psi_h</math>
* <math>(\Psi_g)^{-1} = \Psi_{g^{-1}}</math>
| Lie group automorphism:
* <math>\Psi_g(ab) = \Psi_g(a)\Psi_g(b)</math>
|-
! <math>\operatorname{Ad}\colon G \to \operatorname{Aut}(\mathfrak{g})</math>
! <math>\operatorname{Ad}_g\colon \mathfrak{g} \to \mathfrak{g}</math>
|- style="vertical-align:top;"
| Lie group homomorphism:
* <math>\operatorname{Ad}_{gh} = \operatorname{Ad}_g\operatorname{Ad}_h</math>
* <math>\left(\operatorname{Ad}_g\right)^{-1} = \operatorname{Ad}_{g^{-1}}</math>
| Lie algebra automorphism:
* <math>\operatorname{Ad}_g</math> is linear
* <math>\operatorname{Ad}_g[x,y] = [\operatorname{Ad}_g x,\operatorname{Ad}_g y]</math>
|-
! <math>\operatorname{ad}\colon \mathfrak g \to \operatorname{Der}(\mathfrak g)</math>
! <math>\operatorname{ad}_x\colon \mathfrak g \to \mathfrak g</math>
|- style="vertical-align:top;"
| Lie algebra homomorphism:
* <math>\operatorname{ad}</math> is linear
* <math>\operatorname{ad}_{[x,y]} = [\operatorname{ad}_x, \operatorname{ad}_y]</math>
| Lie algebra derivation:
* <math>\operatorname{ad}_x</math> is linear
* <math>\operatorname{ad}_x[y, z] = [\operatorname{ad}_x y, z] + [y, \operatorname{ad}_x z]</math>
|}

The [[image (mathematics)|image]] of ''G'' under the adjoint representation is denoted by Ad(''G''). If ''G'' is [[connected space|connected]], the [[kernel (group theory)|kernel]] of the adjoint representation coincides with the kernel of Ψ which is just the [[center (group theory)|center]] of ''G''. Therefore, the adjoint representation of a connected Lie group ''G'' is [[faithful representation|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]] ''G''<sub>0</sub> of ''G''. By the [[first isomorphism theorem]] we have
:<math>\mathrm{Ad}(G) \cong G/Z_G(G_0).</math>

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

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

== Roots of a semisimple Lie group ==
If ''G'' is [[semisimple group|semisimple]], the non-zero [[weight (representation theory)|weights]] of the adjoint representation form a [[root system]].<ref>{{harvnb|Hall|2015}} Section 7.3</ref> (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(''t''<sub>1</sub>,&nbsp;...,&nbsp;''t''<sub>''n''</sub>) as our [[maximal torus]] ''T''.  Conjugation by an element of ''T'' sends

:<math>\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_1t_2^{-1}a_{12}&\cdots&t_1t_n^{-1}a_{1n}\\
t_2t_1^{-1}a_{21}&a_{22}&\cdots&t_2t_n^{-1}a_{2n}\\
\vdots&\vdots&\ddots&\vdots\\
t_nt_1^{-1}a_{n1}&t_nt_2^{-1}a_{n2}&\cdots&a_{nn}\\
\end{bmatrix}.
</math>

Thus, ''T'' acts trivially on the diagonal part of the Lie algebra of ''G'' and with eigenvectors ''t''<sub>''i''</sub>''t''<sub>''j''</sub><sup>&minus;1</sup> on the various off-diagonal entries.  The roots of ''G'' are the weights diag(''t''<sub>1</sub>, ..., ''t''<sub>''n''</sub>) → ''t''<sub>''i''</sub>''t''<sub>''j''</sub><sup>&minus;1</sup>.  This accounts for the standard description of the root system of ''G''&nbsp;=&nbsp;SL<sub>''n''</sub>('''R''') as the set of vectors of the form ''e<sub>i</sub>''−''e<sub>j</sub>''.

=== Example SL(2, R) ===
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:

: <math>\begin{bmatrix}
a & b\\
c & d\\
\end{bmatrix} </math>

with ''a'', ''b'', ''c'', ''d'' real and ''ad''&nbsp;&minus;&nbsp;''bc''&nbsp;=&nbsp;1.

A maximal compact connected abelian Lie subgroup, or maximal torus ''T'', is given by the subset of all matrices of the form

: <math>\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} </math>

with <math> t_1 t_2 = 1 </math>. The Lie algebra of the maximal torus is the Cartan subalgebra consisting of the matrices

: <math>
\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).
</math>

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

: <math>
\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}
a t_1 & b t_1 \\
c / t_1 & d / t_1\\
\end{bmatrix}
\begin{bmatrix}
1 / t_1 & 0\\
0       & t_1\\
\end{bmatrix}
=
\begin{bmatrix}
a       & b t_1^2\\
c t_1^{-2} & d\\
\end{bmatrix}
</math>

The matrices

: <math>
\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}
</math>

are then 'eigenvectors' of the conjugation operation with eigenvalues <math>1,1,t_1^2, t_1^{-2}</math>. The function Λ which gives <math>t_1^2</math> 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 group]]s over any field.<!-- even for an elliptic curve? -->{{clarify|give a definition|date=November 2018}}

The '''[[Coadjoint representation|co-adjoint representation]]''' is the [[contragredient representation]] of the adjoint representation.  [[Alexandre Kirillov]] observed that the [[orbit (group theory)|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 group]]s.

==See also==

* {{annotated link|Adjoint bundle}}

==Notes==

{{reflist}}

==References==
*{{Fulton-Harris}}
*{{cite book |last1=Kobayashi|first1= Shoshichi  |last2=Nomizu|first2= Katsumi | title = Foundations of Differential Geometry, Vol. 1 | publisher=Wiley-Interscience | year=1996|edition=New |isbn = 978-0-471-15733-5|title-link=Foundations of Differential Geometry }}
* {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition= 2nd|series=Graduate Texts in Mathematics|volume=222 |publisher=Springer|year=2015|isbn=978-3319134666}}.

{{DEFAULTSORT:Adjoint Representation Of A Lie Group}}
[[Category:Representation theory of Lie groups]]
[[Category:Lie groups]]