Lie algebra-valued differential form


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.

Formal Definition

A Lie algebra-valued differential k-form on a manifold,, is a smooth section of the bundle, where is a Lie algebra, is the cotangent bundle of and Λk denotes the kth exterior power.

Wedge product

Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie algebra-valued forms can be composed with the bracket operation to obtain another Lie algebra-valued form. This operation, denoted by, is given by: for -valued p-form and -valued q-form
where vi's are tangent vectors. The notation is meant to indicate both operations involved. For example, if and are Lie algebra-valued one forms, then one has
The operation can also be defined as the bilinear operation on satisfying
for all and.
Some authors have used the notation instead of. The notation, which resembles a commutator, is justified by the fact that if the Lie algebra is a matrix algebra then is nothing but the graded commutator of and, i. e. if and then
where are wedge products formed using the matrix multiplication on.

Operations

Let be a Lie algebra homomorphism. If φ is a -valued form on a manifold, then f is an -valued form on the same manifold obtained by applying f to the values of φ:.
Similarly, if f is a multilinear functional on, then one puts
where q = q1 + … + qk and φi are -valued qi-forms. Moreover, given a vector space V, the same formula can be used to define the V-valued form when
is a multilinear map, φ is a -valued form and η is a V-valued form. Note that, when
giving f amounts to giving an action of on V; i.e., f determines the representation
and, conversely, any representation ρ determines f with the condition. For example, if , then we recover the definition of given above, with ρ = ad, the adjoint representation.
In general, if α is a -valued p-form and φ is a V-valued q-form, then one more commonly writes α⋅φ = f when Explicitly,
With this notation, one has for example:
Example: If ω is a -valued one-form, ρ a representation of on a vector space V and φ a V-valued zero-form, then

Forms with values in an adjoint bundle

Let P be a smooth principal bundle with structure group G and. G acts on via adjoint representation and so one can form the associated bundle:
Any -valued forms on the base space of P are in a natural one-to-one correspondence with any tensorial forms on P of adjoint type.