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. A Lie-algebra-valued differential -form on a manifold, , is a smooth section of the bundle , where is a Lie algebra, is the cotangent bundle of and denotes the exterior power. 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. For a -valued -form and a -valued -form , their wedge product is given by where the '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 . Let be a Lie algebra homomorphism. If is a -valued form on a manifold, then is an -valued form on the same manifold obtained by applying to the values of : . Similarly, if is a multilinear functional on , then one puts where and are -valued -forms. Moreover, given a vector space , the same formula can be used to define the -valued form when is a multilinear map, is a -valued form and is a -valued form. Note that, when giving amounts to giving an action of on ; i.e., determines the representation and, conversely, any representation determines with the condition . For example, if (the bracket of ), then we recover the definition of given above, with , the adjoint representation. (Note the relation between and above is thus like the relation between a bracket and .