invariant differential form
1 Lie Groups
Let be a Lie group and .
A differential -form (http://planetmath.org/DifferentialForms) on is said to be
right invariant if for every , where is the pullback induced .
invariant or bi-invariant if it is both left invariant and right invariant.
adjoint invariant if for every , where is the pullback induced by .
Much like left invariant vector fields (http://planetmath.org/LieGroup), left invariant forms are uniquely determined by their values in , the tangent space at the identity element , i.e. a left invairant form is uniquely determined by the values
This means that left invariant forms are uniquely determined by their values on the Lie algebra of .
where and are left invariant vector fields in .
Similar results hold for right invariant forms.
Suppose a Lie group acts smoothly (http://planetmath.org/Manifold) on a differential manifold and let
denote the action of .
A differential -form in is said to be invariant if for every , where denotes the pullback induced by .
This definition reduces to the previous ones when we take as the group itself and when the action is
the action of on itself by left multiplication.
the action of on itself by right multiplication.
the action of on defined by .
the action of on itself by conjugation.
3 Compact Lie Group Actions
We now consider actions of a compact Lie group on a manifold . Let the space of -forms in and the space of invariant -forms in . Let be the Haar measure of .
From each -form in we can construct an invariant form by taking on its ””. Following this idea we define a map by
where and are vector fields of .
The image of the map is indeed in since for every :
Moreover, is the identity for invariant -forms. Suppose , then
From the previous observations we can see that the exterior derivative takes invariant forms to invariant forms, inducing a map . Hence, is a chain complex and we can talk about the cohomology groups of invariant forms in .
4 Cohomology of Manifolds
Let be a compact Lie group that acts smoothly on a manifold (again, with the action denoted by ).
If in is connected, then and the identity are homotopic, , for every . This implies that the induced automorphisms are the same, i.e. , where is the identity on . Hence, the fixed point set is the whole and there is an isomorphism
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 . This means we can ”forget” the whole of differential forms in and regard only those who are invariant.
|Title||invariant differential form|
|Date of creation||2013-03-22 17:48:31|
|Last modified on||2013-03-22 17:48:31|
|Last modified by||asteroid (17536)|
|Synonym||bi-invariant differential form|
|Defines||left invariant differential form|
|Defines||left invariant form|
|Defines||right invariant differential form|
|Defines||right invariant form|
|Defines||adjoint invariant form|
|Defines||adjoint invariant differential form|
|Defines||chain complex of invariant forms|
|Defines||cohomology of manifolds with a Lie group action|