alternating algebra | Ausdehnungslehre | extensions calculus | Grassmann algebra

Exterior algebra is the algebra of the wedge product, also called an alternating algebra or Grassmann algebra. The study of exterior algebra is also called Ausdehnungslehre or extensions calculus. Exterior algebras are graded algebras. In particular, the exterior algebra of a vector space is the direct sum over k in the natural numbers of the vector spaces of alternating differential k-forms on that vector space. The product on this algebra is then the wedge product of forms. The exterior algebra for a vector space V is constructed by forming monomials u, v⋀w, x⋀y⋀z, etc., where u, v, w, x, y, and z are vectors in V and ⋀ is wedge product. The sums formed from linear combinations of the monomials are the elements of an exterior algebra.

differential k-form | form envelope | group representation | symmetric group | vector space | wedge product