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 formula_1-form on a manifold, formula_2, is a smooth section of the bundle formula_3, where formula_4 is a Lie algebra, formula_5 is the cotangent bundle of formula_2 and formula_7 denotes the formula_8 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. For a formula_4-valued formula_10-form formula_11 and a formula_4-valued formula_13-form formula_14, their wedge product formula_15 is given by where the formula_17's are tangent vectors. The notation is meant to indicate both operations involved. For example, if formula_11 and formula_14 are Lie-algebra-valued one forms, then one has The operation formula_15 can also be defined as the bilinear operation on formula_22 satisfying for all formula_24 and formula_25. Some authors have used the notation formula_26  instead of formula_15. The notation formula_26, which resembles a commutator, is justified by the fact that if the Lie algebra formula_29 is a matrix algebra then formula_15 is nothing but the graded commutator of formula_11 and formula_14, i. e. if formula_33 and formula_34 then where formula_36 are wedge products formed using the matrix multiplication on formula_29. Operations. Let formula_38 be a Lie algebra homomorphism. If formula_39 is a formula_4-valued form on a manifold, then formula_41 is an formula_42-valued form on the same manifold obtained by applying formula_43 to the values of formula_39: formula_45. Similarly, if formula_43 is a  multilinear functional on formula_47, then one puts where formula_49 and formula_50 are formula_4-valued formula_52-forms. Moreover, given a vector space formula_53, the same formula can be used to define the formula_53-valued form formula_55 when is a multilinear map, formula_39 is a formula_4-valued form and formula_14 is a formula_53-valued form. Note that, when giving formula_43 amounts to giving an action of formula_4 on formula_53; i.e., formula_43 determines the representation and, conversely, any representation formula_67 determines formula_43 with the condition formula_69. For example, if formula_70 (the bracket of formula_4), then we recover the definition of formula_72 given above , with formula_73, the adjoint representation. (Note the relation between formula_43 and formula_67 above is thus like the relation between a bracket and formula_76.) In general, if formula_77 is a formula_78-valued formula_10-form and formula_39 is a formula_53-valued formula_13-form, then one more commonly writes formula_83 when formula_84. Explicitly, With this notation, one has for example: Example: If formula_11 is a formula_4-valued one-form (for example, a connection form), formula_67 a representation of formula_4 on a vector space formula_53 and formula_39 a formula_53-valued zero-form, then Forms with values in an adjoint bundle. Let formula_95 be a  smooth principal bundle with structure group formula_96 and formula_97. formula_96 acts on formula_4 via adjoint representation and so one can form the associated bundle: Any formula_101-valued forms on the base space of formula_95 are in a natural one-to-one correspondence with any tensorial forms on formula_95 of adjoint type.