Vector-valued Differential Form - Lie Algebra-valued Forms

An important case of vector-valued differential forms are Lie algebra-valued forms. These are -valued forms where is 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.

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 is denoted by to indicate both operations involved, or often just . For example, if and are Lie algebra-valued one forms, then one has

With this operation the set of all Lie algebra-valued forms on a manifold M becomes a graded Lie superalgebra.

The operation can also be defined as the bilinear operation on satisfying by the formula

for all and .

The alternative 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 .