Equivariant differential form explained
In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map
\alpha:ak{g}\to\Omega*(M)
from the Lie algebra
ak{g}=\operatorname{Lie}(G)
to the space of
differential forms on
M that are equivariant; i.e.,
\alpha(\operatorname{Ad}(g)X)=g\alpha(X).
In other words, an equivariant differential form is an invariant element of
[1] C[ak{g}] ⊗ \Omega*(M)=\operatorname{Sym}(ak{g}*) ⊗ \Omega*(M).
For an equivariant differential form
, the
equivariant exterior derivative
of
is defined by
(dak{g}\alpha)(X)=d(\alpha(X))-
(\alpha(X))
where
d is the usual exterior derivative and
is the
interior product by the
fundamental vector field generated by
X.It is easy to see
(use the fact the Lie derivative of
along
is zero) and one then puts
=\kerdak{g}/\operatorname{im}dak{g},
which is called the
equivariant cohomology of
M (which coincides with the ordinary equivariant cohomology defined in terms of
Borel construction.) The definition is due to H. Cartan. The notion has an application to the
equivariant index theory.
-closed or
-exact forms are called
equivariantly closed or
equivariantly exact.
The integral of an equivariantly closed form may be evaluated from its restriction to the fixed point by means of the localization formula.
Notes and References
- Proof: with
, we have:
\operatorname{Mor}G(ak{g},V)=\operatorname{Mor}(ak{g},V)G=(\operatorname{Mor}(ak{g},C) ⊗ V)G.
