Difference between revisions of "Adjoint representation of a Lie algebra"

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{{Lie groups |Algebras}}
{{Lie groups |Algebras}}
 
{{mergeto|Adjoint representation of a Lie group|date=December 2014}}
In [[mathematics]], the '''adjoint endomorphism''' or '''adjoint action''' is a [[homomorphism]] of [[Lie algebra]]s that plays a fundamental role in the development of the theory of [[Lie algebras]].
In [[mathematics]], the '''adjoint endomorphism''' or '''adjoint action''' is a [[homomorphism]] of [[Lie algebra]]s that plays a fundamental role in the development of the theory of [[Lie algebras]].


Given an element ''x'' of a Lie algebra <math>\mathfrak{g}</math>, one defines the adjoint action of ''x'' on <math>\mathfrak{g}</math> as the map <math>\operatorname{ad}_x :\mathfrak{g}\to \mathfrak{g}</math> with
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>
:<math>\operatorname{ad}_x (y) = [x,y]</math>
for all {{mvar|y}} in <math>\mathfrak{g}</math>.
 
for all ''y'' in <math>\mathfrak{g}</math>.


The concept generates the [[adjoint representation of a Lie group]] <math>\operatorname{Ad}</math>. In fact, <math>\operatorname{ad}</math> is precisely the differential of <math>\operatorname{Ad}</math> at the identity element of the group.
The concept generates the [[adjoint representation of a Lie group]] <big>Ad</big>. In fact, <big>ad</big> is the differential of <big>Ad</big> at the identity element of the group.


== Adjoint representation ==
== Adjoint representation ==


Let <math>\mathfrak{g}</math> be a Lie algebra over a field ''k''. Then the [[linear map|linear mapping]]
Let <math>\mathfrak{g}</math> be a Lie algebra over a field {{mvar|k}}. Then the [[linear map|linear mapping]]
:<math>\operatorname{ad}:\mathfrak{g} \to \operatorname{End}(\mathfrak{g})</math>
:<math>\operatorname{ad}:\mathfrak{g} \to \operatorname{End}(\mathfrak{g})</math>
given by <math>x\mapsto \operatorname{ad}_x</math> is a [[representation of a Lie algebra]] and is called the '''adjoint representation''' of the algebra. (Its image actually lies in <math>\operatorname{Der}(\mathfrak{g})</math>. See below.)
given by {{math|''x'' ↦ ad<sub>''x''</sub>}} is a [[representation of a Lie algebra]] and is called the '''adjoint representation''' of the algebra. (Its image actually lies in   Der<math>(\mathfrak{g})</math>. See below.)


Within <math>\operatorname{End}(\mathfrak{g})</math>, the [[Lie bracket]] is, by definition, given by the commutator of the two operators:
Within End<math>(\mathfrak{g})</math>, the [[Lie bracket]] is, by definition, given by the commutator of the two operators:
:<math>[\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_x \circ \operatorname{ad}_y - \operatorname{ad}_y \circ \operatorname{ad}_x</math>
:<math>[\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_x \circ \operatorname{ad}_y - \operatorname{ad}_y \circ \operatorname{ad}_x</math>
where <math>\circ</math> denotes composition of linear maps. If <math>\mathfrak{g}</math> is finite-dimensional, then <math>\operatorname{End}(\mathfrak{g})</math> is isomorphic to <math>\mathfrak{gl}(\mathfrak{g})</math>, the Lie algebra of the [[general linear group]] over the vector space <math>\mathfrak{g}</math> and if a basis for it is chosen, the composition corresponds to [[matrix multiplication]].
where denotes composition of linear maps.  
 
If <math>\mathfrak{g}</math> is finite-dimensional, then End<math>(\mathfrak{g})</math> is isomorphic to <math>\mathfrak{gl}(\mathfrak{g})</math>, the Lie algebra of the [[general linear group]] over the vector space <math>\mathfrak{g}</math> and if a basis for it is chosen, the composition corresponds to [[matrix multiplication]].


Using the above definition of the Lie bracket, the [[Jacobi identity]]
Using the above definition of the Lie bracket, the [[Jacobi identity]]
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takes the form  
takes the form  
:<math>\left([\operatorname{ad}_x,\operatorname{ad}_y]\right)(z) = \left(\operatorname{ad}_{[x,y]}\right)(z)</math>
:<math>\left([\operatorname{ad}_x,\operatorname{ad}_y]\right)(z) = \left(\operatorname{ad}_{[x,y]}\right)(z)</math>
where ''x'', ''y'', and ''z'' are arbitrary elements of <math>\mathfrak{g}</math>.
where {{mvar|x}}, {{mvar|y}}, and {{mvar|z}} are arbitrary elements of <math>\mathfrak{g}</math>.


This last identity says that ''ad'' really is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets.
This last identity says that <big>ad</big> really is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets.


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


The kernel of <math>\operatorname{ad}</math> is, by definition, the center of <math>\mathfrak{g}</math>. Next, we consider the image of <math>\operatorname{ad}</math>. Recall that a '''[[derivation (abstract algebra)|derivation]]''' on a Lie algebra is a [[linear map]] <math>\delta:\mathfrak{g}\rightarrow \mathfrak{g}</math> that obeys the [[General Leibniz rule|Leibniz' law]], that is,
The kernel of <big>ad</big> is, by definition, the center of <math>\mathfrak{g}</math>. Next, we consider the image of <big>ad</big>. Recall that a '''[[derivation (abstract algebra)|derivation]]''' on a Lie algebra is a [[linear map]] <math>\delta:\mathfrak{g}\rightarrow \mathfrak{g}</math> that obeys the [[General Leibniz rule|Leibniz' law]], that is,
 
:<math>\delta ([x,y]) = [\delta(x),y] + [x, \delta(y)]</math>
:<math>\delta ([x,y]) = [\delta(x),y] + [x, \delta(y)]</math>
for all ''x'' and ''y'' in the algebra.
for all {{mvar|x}} and {{mvar|y}} in the algebra.


That ad<sub>x</sub> is a derivation is a consequence of the Jacobi identity.  This implies that the image of <math>\mathfrak{g}</math> under ''ad'' is a subalgebra of <math>\operatorname{Der}(\mathfrak{g})</math>, the space of all derivations of <math>\mathfrak{g}</math>.
That ad<sub>''x''</sub> is a derivation is a consequence of the Jacobi identity.  This implies that 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>.


== Structure constants ==
== Structure constants ==
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ad<sub>e<sup>i</sup></sub>
ad<sub>e<sup>i</sup></sub>
are given by
are given by
:<math>{\left[ \operatorname{ad}_{e^i}\right]_k}^j = {c^{ij}}_k. </math>
:<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 rep of so(3).
Thus, for example, the adjoint representation of '''su(2)''' is the defining rep of '''so(3)'''.


== Relation to Ad ==
== Relation to <big>Ad</big> ==


Ad and ad are related through the [[exponential map]]; crudely, Ad = exp ad, where Ad is the [[adjoint representation]] for a [[Lie group]].
Ad and ad are related through the [[exponential map (Lie theory)|exponential map]]: crudely, Ad = exp ad, where Ad is the [[adjoint representation]] for a [[Lie group]].


To be precise, let ''G'' be a Lie group, and let <math>\Psi:G\rightarrow \operatorname{Aut} (G)</math> be the mapping <math>g\mapsto \Psi_g</math> with <math>\Psi_g:G\to G</math> given by the [[inner automorphism]]  
To be more precise, let {{mvar|G}}  be a Lie group, and let {{math: ''G'' → Aut(''G'')}} be the mapping {{math|''g'' ↦ Ψ<sub>''g''</sub>}}, 
:<math>\Psi_g(h)= ghg^{-1}.</math>
with {{math|Ψ<sub>''g''</sub>: ''G'' → ''G''}} given by the [[inner automorphism]]  
It is an example of a Lie group map. Define <math>\operatorname{Ad}_g</math> to be the [[tangent space|derivative]] of <math>\Psi_g</math> at the origin:
:<math>\Psi_g(h)= ghg^{-1}~.</math>
It is an example of a Lie group map. 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>
:<math>\operatorname{Ad}_g = (d\Psi_g)_e : T_eG \rightarrow T_eG</math>
where ''d'' is the differential and ''T''<sub>e</sub>G is the [[tangent space]] at the origin ''e'' (''e'' is the identity element of the group ''G'').
where {{mvar|d}} is the differential and {{math|''T''<sub>''e''</sub>''G''}} is the [[tangent space]] at the origin {{mvar|e}} ({{mvar|e}} being the identity element of the group {{mvar|G}}).


The Lie algebra of ''G'' is <math>\mathfrak{g} = T_e G</math>.  Since <math>\operatorname{Ad}_g\in\operatorname{Aut}(\mathfrak{g})</math>, <math>\operatorname{Ad}:g\mapsto \operatorname{Ad}_g</math> is a map from ''G'' to Aut(''T''<sub>e</sub>''G'') which will have a derivative from ''T''<sub>e</sub>''G'' to End(''T''<sub>e</sub>''G'') (the Lie algebra of Aut(''V'') is End(''V'')).
The Lie algebra of {{mvar|G}} is <math>\mathfrak{g}</math> = {{math|''T''<sub>''e''</sub> ''G''}}.  Since Ad<sub>''g''</sub> ∈ Aut<math>(\mathfrak{g})</math>, &nbsp; {{math| Ad: ''g'' ↦ Ad<sub>''g''</sub>}} is a map from {{mvar|G}} to {{math|Aut(''T''<sub>e</sub>''G'')}} which will have a derivative from {{math|''T''<sub>e</sub>''G''}} to {{math|End(''T''<sub>e</sub>''G'')}} (the Lie algebra of {{math|Aut(''V'')}} being {{math|End(''V'')}}).


Then we have  
Then we have  
:<math>\operatorname{ad} = d(\operatorname{Ad})_e:T_eG\rightarrow \operatorname{End} (T_eG).</math>
:<math>\operatorname{ad} = d(\operatorname{Ad})_e:T_eG\rightarrow \operatorname{End} (T_eG).</math>


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


== References ==
== References ==

Latest revision as of 00:32, 16 December 2014

Template:Lie groups Template:Mergeto In mathematics, the adjoint endomorphism or adjoint action is a homomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras.

Given an element Template:Mvar of a Lie algebra , one defines the adjoint action of Template:Mvar on as the map

for all Template:Mvar in .

The concept generates the adjoint representation of a Lie group Ad. In fact, ad is the differential of Ad at the identity element of the group.

Adjoint representation

Let be a Lie algebra over a field Template:Mvar. Then the linear mapping

given by x ↦ adx is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Its image actually lies in Der. See below.)

Within End, the Lie bracket is, by definition, given by the commutator of the two operators:

where ○ denotes composition of linear maps.

If is finite-dimensional, then End is isomorphic to , the Lie algebra of the general linear group over the vector space and if a basis for it is chosen, the composition corresponds to matrix multiplication.

Using the above definition of the Lie bracket, the Jacobi identity

takes the form

where Template:Mvar, Template:Mvar, and Template:Mvar are arbitrary elements of .

This last identity says that ad really is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets.

In a more module-theoretic language, the construction simply says that is a module over itself.

The kernel of ad is, by definition, the center of . Next, we consider the image of ad. Recall that a derivation on a Lie algebra is a linear map that obeys the Leibniz' law, that is,

for all Template:Mvar and Template:Mvar in the algebra.

That adx is a derivation is a consequence of the Jacobi identity. This implies that the image of under ad is a subalgebra of Der, the space of all derivations of .

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

Then the matrix elements for adei are given by

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

Relation to Ad

Ad and ad are related through the exponential map: crudely, Ad = exp ad, where Ad is the adjoint representation for a Lie group.

To be more precise, let Template:Mvar be a Lie group, and let Ψ: G → Aut(G) be the mapping g ↦ Ψg, with Ψg: GG given by the inner automorphism

It is an example of a Lie group map. Define Adg to be the derivative of Ψg at the origin:

where Template:Mvar is the differential and TeG is the tangent space at the origin Template:Mvar (Template:Mvar being the identity element of the group Template:Mvar).

The Lie algebra of Template:Mvar is = Te G. Since Adg ∈ Aut,   Ad: g ↦ Adg is a map from Template:Mvar to Aut(TeG) which will have a derivative from TeG to End(TeG) (the Lie algebra of Aut(V) being End(V)).

Then we have

The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector Template:Mvar in the algebra generates a vector field Template:Mvar in the group Template:Mvar. Similarly, the adjoint map adxy = [x,y] of vectors in is homomorphic to the Lie derivative LXY = [X,Y] of vector fields on the group Template:Mvar considered as a manifold.

Further see the derivative of the exponential map.

References