Exterior algebra

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, and the names of the product come from the "wedge" symbol ${\displaystyle \wedge }$ and the fact that the product of two elements of ${\displaystyle V}$ is "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. 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. 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.

The two-dimensional Euclidean vector space ${\displaystyle \mathbf {R} ^{2}}$ is a real vector space equipped with a basis consisting of a pair of orthogonal unit vectors $${\displaystyle \mathbf {e} _{1}={\begin{bmatrix}1\\0\end{bmatrix}},\quad \mathbf {e} _{2}={\begin{bmatrix}0\\1\end{bmatrix}}.}$$

Suppose that $${\displaystyle \mathbf {v} ={\begin{bmatrix}a\\b\end{bmatrix}}=a\,\mathbf {e} _{1}+b\,\mathbf {e} _{2},\quad \mathbf {w} ={\begin{bmatrix}c\\d\end{bmatrix}}=c\,\mathbf {e} _{1}+d\,\mathbf {e} _{2}}$$ are a pair of given vectors in ⁠${\displaystyle \mathbf {R} ^{2}}$⁠, written in components. There is a unique parallelogram having ${\displaystyle \mathbf {v} }$ and ${\displaystyle \mathbf {w} }$ as two of its sides. The area of this parallelogram is given by the standard determinant formula: $${\displaystyle {\text{Area}}=\left|\det {\begin{bmatrix}\mathbf {v} &\mathbf {w} \end{bmatrix}}\right|=\left|\det {\begin{bmatrix}a&c\\b&d\end{bmatrix}}\right|=\left|ad-bc\right|.}$$

Consider now the exterior product of ${\displaystyle \mathbf {v} }$ and ⁠${\displaystyle \mathbf {w} }$⁠: $${\displaystyle {\begin{aligned}\mathbf {v} \wedge \mathbf {w} &=(a\,\mathbf {e} _{1}+b\,\mathbf {e} _{2})\wedge (c\,\mathbf {e} _{1}+d\,\mathbf {e} _{2})\\&=ac\,\mathbf {e} _{1}\wedge \mathbf {e} _{1}+ad\,\mathbf {e} _{1}\wedge \mathbf {e} _{2}+bc\,\mathbf {e} _{2}\wedge \mathbf {e} _{1}+bd\,\mathbf {e} _{2}\wedge \mathbf {e} _{2}\\&=ad\,\mathbf {e} _{1}\wedge \mathbf {e} _{2}+bc\,\mathbf {e} _{2}\wedge \mathbf {e} _{1}\\&=\left(ad-bc\right)\mathbf {e} _{1}\wedge \mathbf {e} _{2},\end{aligned}}}$$ where the first step uses the distributive law for the exterior product. The second one uses the fact that the exterior product is an alternating map, i.e., ${\displaystyle \mathbf {e} _{1}\wedge \mathbf {e} _{1}=\mathbf {e} _{2}\wedge \mathbf {e} _{2}=0.}$ Being alternating also implies being anticommutative, ${\displaystyle \mathbf {e} _{2}\wedge \mathbf {e} _{1}=-(\mathbf {e} _{1}\wedge \mathbf {e} _{2})}$, which gives the last line. Note that the coefficient in this last expression is precisely the determinant of the matrix [v w]. 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.