the exterior derivative ,
the space of contraction operators , where is a vector field on .
where the brackets on the right hand side denote the Lie bracket of vector fields.
Interpretation as a Lie Superalgebra
Since is a graded algebra, there is a natural grading on the space of linear operators on . Under this grading, the exterior derivative is degree , the Lie derivative operators are degree , and the contraction operators are degree .
where a plus sign is used if and are both of odd degree, and a minus sign is used otherwise. Equations of this form are called supercommutation relations and are usually written in the form
Graded derivations of
A degree linear operator on is a graded derivation if it satisfies the following property for any -form and any differential form :
All of the Calculus operators are graded derivations of .
|Date of creation||2013-03-22 15:35:39|
|Last modified on||2013-03-22 15:35:39|
|Last modified by||bci1 (20947)|
|Defines||Cartan’s magic formula|