cohomology of compact connected Lie groups
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This entry aims to describe some properties of the cohomology (http://planetmath.org/DeRhamCohomology) of compact (http://planetmath.org/Compact) connected (http://planetmath.org/ConnectedSpace) Lie groups. It turns out that this type of Lie groups admit a somewhat simplified cohomology theory, compared with singular cohomology or de Rham cohomology. This simplified theory then allows one to observe some of the imposed on the structure of the cohomology groups of compact connected Lie groups.
The construction and results presented in the entry on invariant differential forms are assumed and will be the of what we are about to describe.
1 Cohomology of invariant forms
Let be a compact connected Lie group. Given a differential manifold and smooth action of in , let denote the space of the -invariant differential forms (http://planetmath.org/InvariantDifferentialForm) in . It is known (see this entry (http://planetmath.org/InvariantDifferentialForm)) that forms a chain complex and the cohomology groups of this complex are isomorphic to the cohomology groups of , i.e.
We now regard the manifold as a compact connected Lie group . There are several smooth actions on that are worth to be considered, such as:
The action of on itself by left multiplication.
The action of on itself by right multiplication.
The action of on itself by conjugation.
The action of on given by
Thus, by the previous remark, the cohomology of a compact connected Lie group restricts to the cohomology of left invariant forms, right invariant forms, adjoint invariant forms or bi-invariant forms (http://planetmath.org/InvariantDifferentialForm). Moreover, the cohomology of is the cohomology of differential forms invariant under any smooth action of any compact connect Lie group on .
1.1 Left invariant forms
Since left invariant forms are uniquely determined by their values in , the tangent space (http://planetmath.org/TangentSpace) at the identity, they allow a simpler characterization of the cohomology of compact connected Lie groups.
Proposition - Let be a compact connected Lie group and the chain complex of alternating forms on with coboundary operator given by
Then, the cohomology groups of are isomorphic to the cohomology groups of the complex .
1.2 Bi-invariant forms
Two sided invariance is just the same as left invariance adjoint invariance. Hence, every bi-invariant form can be seen as an alternating form on wich is also adjoint invariant. Thus, just like the previous proposition, the cohomology groups of are just the cohomology groups of the complex of adjoint invariant alternating forms on , with coboundary operator
This can be further improved. The following important theorem will be the key to more specific results.
Theorem 1 - Let be a compact connected Lie group. Let be a multilinear -form on . Then is adjoint invariant if and only if the following equality holds:
Proposition - Every adjoint invariant alternating form on is closed (http://planetmath.org/ClosedDifferentialForm).
Proof: Let , if and if . Then
where the sum is zero by Theorem 1.
Corollary 1 - Let be a compact connected Lie group. The cohomology groups are isomorphic to the vector space of adjoint invariant alternating -forms on .
2 Relations between cohomology groups
Let be a compact connected Lie group. In the proofs of the following results we are always using the fact stated in Corollary 1, that is exactly the space of adjoint invariant alternating -forms on . Let denote the subspace (http://planetmath.org/VectorSubspace) of generated by elements of the form , with .
Proof: It follows easily form the fact that adjoint invariant alternating -forms on are precisely those who satisfy for all (Theorem 1).
Proof: Let be an adjoint invariant -form on . We have that
where the last step comes from Theorem 1. Now, as by the previous Proposition, .
Let be the space of adjoint invariant symmetric bilinear forms on and the space of adjoint invariant alternating -forms on .
Theorem 2 - Suppose . The function defined by
is well defined and bijective.
One can always assure the existence of nonzero adjoint invariant symmetric bilinear forms on . This can be achieved by taking an inner product on and defining, for ,
where is the Haar measure of and is the function of conjugation by . Hence we can conclude that
Corollary - Suppose is nontrivial. Then
|Title||cohomology of compact connected Lie groups|
|Date of creation||2013-03-22 17:49:47|
|Last modified on||2013-03-22 17:49:47|
|Last modified by||asteroid (17536)|