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.
, 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.
, their wedge product
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
η ∧ ω ∈
are wedge products formed using the matrix multiplication on
be a Lie algebra homomorphism.
-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
, the adjoint representation.
-form, then one more commonly writes
α ⋅ φ = f ( α , φ )
Explicitly, With this notation, one has for example: Example: If
-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