In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra.
Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.
A Lie-algebra-valued differential
, is a smooth section of the bundle
exterior power.
The wedge product of ordinary, real-valued differential forms is defined using multiplication of real numbers.
For a pair of Lie algebra–valued differential forms, the wedge product can be defined similarly, but substituting the bilinear Lie bracket operation, to obtain another Lie algebra–valued form.
's are tangent vectors.
The notation is meant to indicate both operations involved.
are Lie-algebra-valued one forms, then one has The operation
can also be defined as the bilinear operation on
Some authors have used the notation
, which resembles a commutator, is justified by the fact that if the Lie algebra
is a matrix algebra then
are wedge products formed using the matrix multiplication on
be a Lie algebra homomorphism.
-valued form on a manifold, then
-valued form on the same manifold obtained by applying
is a multilinear functional on
f ( φ , η )
when is a multilinear map,
amounts to giving an action of
ρ = ad
, the adjoint representation.
-valued one-form (for example, a connection form),
be a smooth principal bundle with structure group
via adjoint representation and so one can form the associated bundle: Any
-valued forms on the base space of
are in a natural one-to-one correspondence with any tensorial forms on
of adjoint type.