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
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
as well as in the theory of
Cartan connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...
s.
Formal definition
A Lie-algebra-valued differential
-form on a manifold,
, is a smooth
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
of the
bundle
Bundle or Bundling may refer to:
* Bundling (packaging), the process of using straps to bundle up items
Biology
* Bundle of His, a collection of heart muscle cells specialized for electrical conduction
* Bundle of Kent, an extra conduction pat ...
, where
is a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
,
is the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
of
and
denotes the
exterior power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
.
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
-valued
-form
and a
-valued
-form
, their wedge product