Exterior algebra

Orientation defined by an ordered set of vectors.

Reversed orientation corresponds to negating the exterior product.

Geometric interpretation of grade n elements in a real exterior algebra for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-parallelotope, n-ellipsoid); with magnitude (hypervolume), and orientation defined by that of its (n − 1)-dimensional boundary and on which side the interior is.^[1][2]

In mathematics, the exterior algebra or Grassmann algebra of a vector space ${\displaystyle V}$ is an associative algebra that contains ${\displaystyle V,}$ which has a product, called exterior product or wedge product and denoted with ${\displaystyle \wedge }$, such that ${\displaystyle v\wedge v=0}$ for every vector ${\displaystyle v}$ in ${\displaystyle V.}$ The exterior algebra is named after Hermann Grassmann,^[3] and the names of the product come from the "wedge" symbol ${\displaystyle \wedge }$ and the fact that the product of two elements of ${\displaystyle V}$ are "outside" ${\displaystyle V.}$

The wedge product of ${\displaystyle k}$ vectors ${\displaystyle v_{1}\wedge v_{2}\wedge \dots \wedge v_{k}}$ is called a blade of degree ${\displaystyle k}$ or ${\displaystyle k}$-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 ${\displaystyle v\wedge w}$ is the area of the parallelogram defined by ${\displaystyle v}$ and ${\displaystyle w,}$ and, more generally, the magnitude of a ${\displaystyle k}$-blade is the (hyper)volume of the parallelotope defined by the constituent vectors. The alternating property that ${\displaystyle v\wedge v=0}$ implies a skew-symmetric property that ${\displaystyle v\wedge w=-w\wedge v,}$ 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 ${\displaystyle k}$ 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 ${\displaystyle k}$-blades is called the ${\displaystyle k}$-th exterior power of ${\displaystyle V.}$ The exterior algebra is the direct sum of the ${\displaystyle k}$-th exterior powers of ${\displaystyle V,}$ and this makes the exterior algebra a graded algebra.

The exterior algebra is universal in the sense that every equation that relates elements of ${\displaystyle V}$ in the exterior algebra is also valid in every associative algebra that contains ${\displaystyle V}$ and in which the square of every element of ${\displaystyle V}$ 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 ${\displaystyle v\wedge v=0}$ must be replaced with ${\displaystyle v\wedge w+w\wedge v=0,}$ 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 ${\displaystyle k}$ variables is an exterior algebra over the ring of the smooth functions in ${\displaystyle k}$ variables.