In mathematics, the exterior algebra of a vector space is a graded associative algebra
Elements in ∧nV are called -multivectors, and are given by a sum of -blades ("products" of elements of ); it is an abstraction of oriented lengths, areas, volumes and more generally oriented n-volumes for n ≥ 0.
The algebra product ∧ on ∧V is called the exterior product or wedge product; informally, it acts by taking the product of oriented volumes. Every element in ∧V is a sum of v1∧ ... ∧ vk where , and
Such an element is called a k-blade, which in the above figure corresponds to the parallelotope spanned by them.
This product satisfies the alternating property v∧v = 0 for all v ∈ V. This implies it is antisymmetric: u∧v = - v∧u. and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.
The exterior algebra is also called the Grassmann algebra, after Hermann Grassmann.
Formal definitions
There are several equivalent ways to define the exterior algebra of a vector space .
Definition as a quotient of the tensor algebra
Let V be a vector space over a field.
Recall the tensor algebra TV. As a vector space it is spanned by symbols v1⊗ ... ⊗vk for k ≥ 0 and vi ∈ V, and the only relations are those specifying these objects be linear in each variable vi.
∧V inherits the structure of a graded algebra from TV. An element of ∧V may be written (non-uniquely) as a finite sum v1∧ ... ∧ vk1+ w1∧ ... ∧ wk2 + ... + z1∧ ... ∧ zkn where each letter is an element of V, and ki, n ≥ 0. It is called a k-vector if all ki = k, in which case its rank is the minimum n for which it can be written in the above form. Rank one k-vectors v1∧ ... ∧ vk are also called k-blades.
Definition in terms of explicit basis
Let e1, ..., en be a basis of V.
Motivating examples
The first two examples assume a metric tensor field and an orientation; the third example does not assume either.
Areas in the plane
The Cartesian plane is a real vector space equipped with a basis consisting of a pair of unit vectors
with the orientation and with the metric
Suppose that
are a pair of given vectors in written in components. There is a unique parallelogram having v and w as two of its sides. The area of this parallelogram is given by the standard determinant formula:
Consider now the exterior product of v and w:
where the first step uses the distributive law for the exterior product, and the last uses the fact that the exterior product is an alternating map, and in particular (The fact that the exterior product is an alternating map also forces ) Note that the coefficient in this last expression is precisely the determinant of the matrix . The fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the signed area of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation.
The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if denotes the signed area of the parallelogram of which the pair of vectors v and w form two adjacent sides, then A must satisfy the following properties: for any real numbers r and s, since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram). , since the area of the degenerate parallelogram determined by v (i.e., a line segment) is zero. , since interchanging the roles of v and w reverses the orientation of the parallelogram. for any real number r, since adding a multiple of w to v affects neither the base nor the height of the parallelogram and consequently preserves its area. , since the area of the unit square is one.
With the exception of the last property, the exterior product of two vectors satisfies the same properties as the area. In a certain sense, the exterior product generalizes the final property by allowing the area of a parallelogram to be compared to that of any chosen parallelogram in a parallel plane (here, the one with sides e1 and e2). In other words, the exterior product provides a basis-independent formulation of area.