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单词 InvariantDifferentialForm
释义

invariant differential form


1 Lie Groups

Let G be a Lie group and gG.

Let Lg:GG and Rg:GG be the functions of left and right multiplication by g (respectively). Let Cg:GG be the function of conjugationMathworldPlanetmath by g, i.e. Cg(h):=ghg-1.

A differential k-form (http://planetmath.org/DifferentialForms) ω on G is said to be

  • left invariant if Lg*ω=ω for every gG, where Lg* is the pullbackPlanetmathPlanetmath induced Lg.

  • right invariant if Rg*ω=ω for every gG, where Rg* is the pullback induced Rg.

  • invariant or bi-invariant if it is both left invariant and right invariant.

  • adjointMathworldPlanetmath invariant if Cg*ω=ω for every gG, where Cg* is the pullback induced by Cg.

Much like left invariant vector fields (http://planetmath.org/LieGroup), left invariant forms are uniquely determined by their values in Te(G), the tangent space at the identity elementMathworldPlanetmath eG, i.e. a left invairant form ω is uniquely determined by the values

we(X1,,Xk),X1,,XkTe(G)

This means that left invariant forms are uniquely determined by their values on the Lie algebra of G.

Under this setting, the space ΩLk(G) of left invariant k-forms can be identified with Hom(Λk𝔤,), the space of homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from the k-th exterior power of 𝔤 to , where 𝔤 denotes the Lie algebra of G.

- Let Ωk(G) be the space of k-forms in G. The exterior derivativeMathworldPlanetmath d:Ωk(G)Ωk+1(G) takes left invariant forms to left invariant forms. Moreover, the formulaMathworldPlanetmathPlanetmath for exterior derivative for left invariant forms simplifies to

dω(X0,,Xk)=i<j(-1)i+jω([Xi,Xj],X0,,Xi^,,Xj^,,Xk)

where ωΩk(G) and X0,,Xk are left invariant vector fields in G.

Hence, the exterior derivative induces a map d:ΩLk(G)ΩLk+1(G) and (ΩL*(G),d) forms a chain complexMathworldPlanetmath. Thus, we can talk about the cohomology groups of left invariant forms.

Similar results hold for right invariant forms.

2 Manifolds

Suppose a Lie group G acts smoothly (http://planetmath.org/ManifoldMathworldPlanetmath) on a differential manifold M and let

(g,x)tg(x),gG,xM

denote the action of G.

A differential k-form ω in M is said to be invariant if tg*ω=ω for every gG, where tg* denotes the pullback induced by tg.

This definition reduces to the previous ones when we take M as the group G itself and when the action is

  • the action of G on itself by left multiplication.

  • the action of G on itself by right multiplication.

  • the action of G×G on G defined by t(g,h)(k):=gkh-1.

  • the action of G on itself by conjugation.

3 Compact Lie Group Actions

We now consider actions of a compact Lie group G on a manifold M. Let Ωk(M) the space of k-forms in M and ΩGk(M) the space of invariant k-forms in M. Let μ be the Haar measure of G.

From each k-form in M we can construct an invariant form by taking on its ””. Following this idea we define a map J:Ωk(M)ΩGk(M) by

J(ω)(X1,,Xk):=1μ(G)Gtg*ω(X1,,Xk)𝑑μ(g)

where ωΩk(M) and X1,,Xk are vector fields of M.

The image of the map J is indeed in ΩGk(M) since for every hG:

th(J(ω))(X1,,Xk)=J(ω)((th)*X1,,(th)*Xk)
=1μ(G)Gω((tg)*(th)*X1,,(tg)*(th)*Xk)𝑑μ(g)
=1μ(G)Gω((tgh)*X1,,(tgh)*Xk)𝑑μ(g)
=1μ(G)Gω((tg)*X1,,(tg)*Xk)𝑑μ(g)
=J(ω)(X1,,Xk)

Moreover, J is the identityPlanetmathPlanetmathPlanetmathPlanetmath for invariant k-forms. Suppose ωΩGk(M), then

J(ω)(X1,,Xk)=1μ(G)Gtg*(ω)(X1,,Xk)𝑑μ(g)
=1μ(G)Gω(X1,,Xk)𝑑μ(g)
=ω(X1,,Xk)

Theorem - The map J is a chain map, i.e. dJ=Jd, where d is the exterior derivative of a form.

From the previous observations we can see that the exterior derivative takes invariant forms to invariant forms, inducing a map d:ΩGk(M)ΩGk+1(M). Hence, (ΩG*(M),d) is a chain complex and we can talk about the cohomology groups of invariant forms in M.

4 Cohomology of Manifolds

Let G be a compact Lie group that acts smoothly on a manifold M (again, with the action denoted by tg).

Since tg is a diffeomorphism of M it induces an automorphism tg* on the cohomology groups Hk(M;). Hence, G acts as a group of automorphisms on Hk(M;). Let Hk(M;)G be the fixed pointPlanetmathPlanetmath set of this action.

Theorem - The inclusion I:ΩGk(M)Ωk(M) induces an isomorphismMathworldPlanetmathPlanetmath

\\xymatrixI*:Hk(ΩG(M))\\ar[r]&Hk(M;)G

If in G is connected, then tg and the identity 1M are homotopic, tg1M, for every gG. This implies that the induced automorphisms are the same, i.e. tg*=Id, where Id is the identity on Hk(M;). Hence, the fixed point set is the whole Hk(M;) and there is an isomorphism

\\xymatrixI*:Hk(ΩG(M))\\ar[r]&Hk(M;)

Thus, the cohomology groups of a manifold where a compact connected Lie group acts are just the cohomology groups defined by the invariant forms on M. This means we can ”forget” the whole of differential forms in M and regard only those who are invariant.

Titleinvariant differential form
Canonical nameInvariantDifferentialForm
Date of creation2013-03-22 17:48:31
Last modified on2013-03-22 17:48:31
Ownerasteroid (17536)
Last modified byasteroid (17536)
Numerical id25
Authorasteroid (17536)
Entry typeDefinition
Classificationmsc 58A10
Classificationmsc 57T10
Classificationmsc 57S15
Classificationmsc 22E30
Classificationmsc 22E15
Synonyminvariant form
Synonymbi-invariant form
Synonymbi-invariant differential form
Related topicCohomologyOfCompactConnectedLieGroups
Definesleft invariant differential form
Definesleft invariant form
Definesright invariant differential form
Definesright invariant form
Definesadjoint invariant form
Definesadjoint invariant differential form
Defineschain complex of invariant forms
DefinescohomologyMathworldPlanetmathPlanetmath of manifolds with a Lie group action
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