Home > Exterior algebra
4 Differential forms
When dealing with differentiable manifolds, we define a "differential k-form" to be
a function that assigns to every point x of the manifold an element of the k-th exterior power of the cotangent space at x. (Or, what is the same thing: a function that assigns to every point x an anti-symmetric map from (TxM)k to the base field, where TxM is the tangent space at x).
Such a form is said to be differentiable if, when applied to k differentiable vector fields, the result is a differentiable function. The wedge product makes pointwise sense for differential forms. In the de Rham and Alexander-Spanier cohomology theories it is used to define a multiplication on the associated cohomology ring, in an analogous way to the cup product in singular cohomology.
5 Generalization
Given a commutative ring R and an R- module M, we can define the exterior algebra Λ(M) just as above, as a suitable quotient of the tensor algebra T(M). It will satisfy the analogous universal property.
6 Physical applications
Grassmann algebras have some important applications in physics where they are used to model various concepts related to fermions and supersymmetry.
See also: superspace, superalgebra, supergroup
7 Related topics
Multilinear algebra
Differential topology
