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Exterior algebra
Algebra of exterior/ wedge products / From Wikipedia, the free encyclopedia
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In mathematics, the exterior algebra or Grassmann algebra of a vector space is an associative algebra that contains
which has a product, called exterior product or wedge product and denoted with
, such that
for every vector
in
The exterior algebra is named after Hermann Grassmann,[3] and the names of the product come from the "wedge" symbol
and the fact that the product of two elements of
is "outside"
The wedge product of vectors
is called a blade of degree
or
-blade. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: The magnitude of a 2-blade
is the area of the parallelogram defined by
and
and, more generally, the magnitude of a
-blade is the (hyper)volume of the parallelotope defined by the constituent vectors. The alternating property that
implies a skew-symmetric property that
and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.
The full exterior algebra contains objects that are not themselves blades, but linear combinations of blades; a sum of blades of homogeneous degree is called a k-vector, while a more general sum of blades of arbitrary degree is called a multivector.[4] The linear span of the
-blades is called the
-th exterior power of
The exterior algebra is the direct sum of the
-th exterior powers of
and this makes the exterior algebra a graded algebra.
The exterior algebra is universal in the sense that every equation that relates elements of in the exterior algebra is also valid in every associative algebra that contains
and in which the square of every element of
is zero.
The definition of the exterior algebra can be extended for spaces built from vector spaces, such as vector fields and functions whose domain is a vector space. Moreover, the field of scalars may be any field (however for fields of characteristic two, the above condition must be replaced with
which is equivalent in other characteristics). More generally, the exterior algebra can be defined for modules over a commutative ring. In particular, the algebra of differential forms in
variables is an exterior algebra over the ring of the smooth functions in
variables.