19 Facts About Exterior algebra

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.

Exterior algebra provides an algebraic setting in which to answer geometric questions.

For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules.

The exterior algebra contains objects that are not only k-blades, but sums of k-blades; such a sum is called a k-vector.

Definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions.

In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra.

The exterior algebra has many algebraic properties that make it a convenient tool in algebra itself.

The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces.

The exterior algebra is one example of a bialgebra, meaning that its dual space possesses a product, and this dual product is compatible with the exterior product.

Vectors in a 3-dimensional oriented vector space with a bilinear scalar product, the exterior algebra is closely related to the cross product and triple product.

Exterior algebra product is by construction alternating on elements of, which means that for all, by the above construction.

Exterior algebra product of a k-vector with a p-vector is a -vector, invoking bilinearity.

Any lingering doubt can be shaken by pondering the equalities ? = 1 ? and ? = v ? w, which follow from the definition of the coExterior algebra, as opposed to naive manipulations involving the tensor and wedge symbols.

Exterior algebra has notable applications in differential geometry, where it is used to define differential forms.

The exterior algebra of differential forms, equipped with the exterior derivative, is a cochain complex whose cohomology is called the de Rham cohomology of the underlying manifold and plays a vital role in the algebraic topology of differentiable manifolds.

In representation theory, the exterior algebra is one of the two fundamental Schur functors on the category of vector spaces, the other being the symmetric algebra.

The exterior algebra itself is then just a one-dimensional superspace: it is just the set of all of the points in the exterior algebra.

Exterior algebra is the main ingredient in the construction of the Koszul complex, a fundamental object in homological algebra.

Exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension.