Exterior algebra
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, 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 -blade is the area of the parallelogram defined by and and, more generally, the magnitude of a -blade is the volume of the parallelotope defined by the constituent vectors. Its bilinearity, expected from such a generalization of volume, and its 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 -vector, while a more general sum of blades of arbitrary degree is called a multivector. 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. 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.
Motivating examples
Areas in the plane
The two-dimensional Euclidean vector space is a real vector space equipped with a basis consisting of a pair of orthogonal unit vectorsSuppose that
are a pair of given vectors in, written in components. There is a unique parallelogram having and as two of its sides. The area of this parallelogram is given by the standard determinant formula:
Consider now the exterior product of and :
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.,
Being alternating also implies being anticommutative,, which gives the last line.
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.
- , since the area of the degenerate parallelogram determined by v 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. In other words, the exterior product provides a basis-independent formulation of area.
Cross and triple products
For vectors in, the exterior algebra is closely related to the cross product and triple product. Using the standard basis, the exterior product of a pair of vectorsand
is
where is the natural basis for the three-dimensional space. The coefficients above are the same as those in the usual definition of the cross product of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a bivector.
Bringing in a third vector
the exterior product of three vectors is
where is the basis vector for the one-dimensional space. The scalar coefficient is the triple product of the three vectors.
The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product can be interpreted as a vector which is perpendicular to both and and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the minors of the matrix with columns and. The triple product of,, and is geometrically a volume. Algebraically, it is the determinant of the matrix with columns,, and. The exterior product in three dimensions allows for similar interpretations. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions.
Formal definition
The exterior algebra of a vector space over a field is defined as the quotient algebra of the tensor algebra T, whereby the two-sided ideal generated by all elements of the form such that. Symbolically,
The exterior product of two elements of is defined by
Algebraic properties
Alternating product
The exterior product is by construction alternating on elements of, which means that for all by the above construction. It follows that the product is also anticommutative on elements of, for supposing that,hence
More generally, if is a permutation of the integers, and,,..., are elements of, it follows that
where is the signature of the permutation.
In particular, if for some, then the following generalization of the alternating property also holds:
Together with the distributive property of the exterior product, one further generalization is that a necessary and sufficient condition for to be a linearly dependent set of vectors is that
Exterior power
The th exterior power of, denoted, is the vector subspace of spanned by elements of the formIf, then is said to be a -vector. If, furthermore, can be expressed as an exterior product of elements of, then is said to be decomposable. Although decomposable -vectors span, not every element of is decomposable. For example, given with a basis, the following 2-vector is not decomposable:
Basis and dimension
If the dimension of is and is a basis for, then the setis a basis for. The reason is the following: given any exterior product of the form
every vector can be written as a linear combination of the basis vectors ; using the bilinearity of the exterior product, this can be expanded to a linear combination of exterior products of those basis vectors. Any exterior product in which the same basis vector appears more than once is zero; any exterior product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis -vectors can be computed as the minors of the matrix that describes the vectors in terms of the basis.
By counting the basis elements, the dimension of is equal to a binomial coefficient:
where is the dimension of the vectors, and is the number of vectors in the product. The binomial coefficient produces the correct result, even for exceptional cases; in particular, for.
Any element of the exterior algebra can be written as a sum of -vectors. Hence, as a vector space the exterior algebra is a direct sum
, and therefore its dimension is equal to the sum of the binomial coefficients, which is.
Rank of a ''k''-vector
If, then it is possible to express as a linear combination of decomposable -vectors:where each is decomposable, say
The rank of the -vector is the minimal number of decomposable -vectors in such an expansion of. This is similar to the notion of tensor rank.
Rank is particularly important in the study of 2-vectors . The rank of a 2-vector can be identified with half the rank of the matrix of coefficients of in a basis. Thus if is a basis for, then can be expressed uniquely as
where . The rank of the matrix is therefore even, and is twice the rank of the form.
In characteristic 0, the 2-vector has rank if and only if
Graded structure
The exterior product of a -vector with a -vector is a -vector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding sectiongives the exterior algebra the additional structure of a graded algebra, that is
Moreover, if is the base field, we have
The exterior product is graded anticommutative, meaning that if and, then
In addition to studying the graded structure on the exterior algebra, studies additional graded structures on exterior algebras, such as those on the exterior algebra of a graded module.