Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors formula_1 and formula_2, denoted by formula_3, is called a bivector and lives in a space called the "exterior square", a vector space that is distinct from the original space of vectors. The magnitude of formula_3 can be interpreted as the area of the parallelogram with sides formula_1 and formula_2, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that formula_7 for all vectors formula_1 and formula_2, but, unlike the cross product, the exterior product is associative.
When regarded in this manner, the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number "k" of vectors can be defined and is sometimes called a "k"-blade. It lives in a space known as the "k-"th exterior power. The magnitude of the resulting "k"-blade is the volume of the "k"-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors.
The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not only "k"-blades, but sums of "k"-blades; such a sum is called a "k"-vector. The "k"-blades, because they are simple products of vectors, are called the simple elements of the algebra. The "rank" of any "k"-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that formula_10 for any elements formula_11. The "k"-vectors have degree "k", meaning that they are sums of products of "k" vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.
The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms, and the pairing between the exterior algebra and its dual is given by the interior product.
Motivating examples.
Areas in the plane.
The Cartesian plane R2 is a real vector space equipped with a basis consisting of a pair of unit vectors
Suppose that
are a pair of given vectors in R2, 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 alternating, and in particular formula_16. (The fact that the exterior product is alternating also forces formula_17.) 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:
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.
Cross and triple products.
For vectors in a 3-dimensional oriented vector space with a bilinear scalar product, the exterior algebra is closely related to the cross product and triple product. Using a standard basis , the exterior product of a pair of vectors
and
is
where is a basis for the three-dimensional space Λ2(R3). The coefficients above are the same as those in the usual definition of the cross product of vectors in three dimensions with a given orientation, the only differences being that the exterior product is not an ordinary vector, but instead is a 2-vector, and that the exterior product does not depend on the choice of orientation.
Bringing in a third vector
the exterior product of three vectors is
where is the basis vector for the one-dimensional space Λ3(R3). The scalar coefficient is the triple product of the three vectors.
The cross product and triple product in a three dimensional Euclidean vector space each admit both geometric and algebraic interpretations. The cross product can be interpreted as a vector which is perpendicular to both u and v 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 u and v. The triple product of u, v, and w is a signed scalar representing a geometric oriented volume. Algebraically, it is the determinant of the matrix with columns u, v, and w. The exterior product in three dimensions allows for similar interpretations: it, too, can be identified with oriented lines, areas, volumes, etc., that are spanned by one, two or more vectors. The exterior product generalizes these geometric notions to all vector spaces and to any number of dimensions, even in the absence of a scalar product.
Formal definitions and algebraic properties.
The exterior algebra of a vector space over a field is defined as the quotient algebra of the tensor algebra by the two-sided ideal generated by all elements of the form for (i.e. all tensors that can be expressed as the tensor product of a vector in by itself). The ideal "I" contains the ideal "J" generated by elements of the form formula_23 and these ideals coincide if (and only if) formula_24:
We define 
The exterior product of two elements of is the product induced by the tensor product of . That is, if
is the canonical surjection, and and are in , then there are formula_28 and formula_29 in such that formula_30 and formula_31 and
It results from the definition of a quotient algebra that the value of formula_33 does not depend on a particular choice of formula_28 and formula_29.
As , , and formula_36, the inclusions of and in induce injections of and into . These injections are commonly considered as inclusions, and called "natural embeddings", "natural injections" or "natural inclusions". The word "canonical" is also commonly used in place of "natural".
Alternating product.
The exterior product is by construction "alternating" on elements of formula_37, which means that formula_38 for all formula_39, by the above construction. It follows that the product is also anticommutative on elements of formula_37, for supposing that formula_41,
hence
More generally, if "σ" is a permutation of the integers , and "x"1, "x"2, ..., "x""k" are elements of "V", it follows that
where sgn("σ") is the signature of the permutation "σ".
In particular, if "x""i" = "x""j" for some , then the following generalization of the alternating property also holds:
Exterior power.
The "k"th exterior power of "V", denoted Λ"k"("V"), is the vector subspace of Λ("V") spanned by elements of the form
If , then "α" is said to be a "k"-vector. If, furthermore, "α" can be expressed as an exterior product of "k" elements of "V", then "α" is said to be decomposable. Although decomposable "k"-vectors span Λ"k"("V"), not every element of Λ"k"("V") is decomposable. For example, in R4, the following 2-vector is not decomposable:
Basis and dimension.
If the dimension of is and is a basis for , then the set
is 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
(where by convention , the field underlying , and   ), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2.
Rank of a "k"-vector.
If , then it is possible to express "α" as a linear combination of decomposable "k"-vectors:
where each "α"("i") is decomposable, say
The rank of the "k"-vector "α" is the minimal number of decomposable "k"-vectors in such an expansion of "α". 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 "α" can be identified with half the rank of the matrix of coefficients of "α" in a basis. Thus if "e""i" is a basis for "V", then "α" can be expressed uniquely as
where (the matrix of coefficients is skew-symmetric). The rank of the matrix "a""ij" is therefore even, and is twice the rank of the form "α".
In characteristic 0, the 2-vector "α" has rank "p" if and only if
Graded structure.
The exterior product of a "k"-vector with a "p"-vector is a -vector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section
gives 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 (a module that already carries its own gradation).
Universal property.
Let be a vector space over the field . Informally, multiplication in is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity formula_62 for . Formally, is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative -algebra containing with alternating multiplication on must contain a homomorphic image of . In other words, the exterior algebra has the following universal property:
Given any unital associative -algebra and any -linear map such that for every in , then there exists "precisely one" unital algebra homomorphism such that for all in (here is the natural inclusion of in , see above).
To construct the most general algebra that contains and whose multiplication is alternating on , it is natural to start with the most general associative algebra that contains , the tensor algebra , and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided ideal in generated by all elements of the form for in , and define as the quotient
(and use as the symbol for multiplication in . It is then straightforward to show that contains and satisfies the above universal property.
As a consequence of this construction, the operation of assigning to a vector space its exterior algebra is a functor from the category of vector spaces to the category of algebras.
Rather than defining first and then identifying the exterior powers as certain subspaces, one may alternatively define the spaces first and then combine them to form the algebra . This approach is often used in differential geometry and is described in the next section.
Generalizations.
Given a commutative ring "R" and an "R"-module "M", we can define the exterior algebra Λ("M") just as above, as a suitable quotient of the tensor algebra T("M"). It will satisfy the analogous universal property. Many of the properties of Λ("M") also require that "M" be a projective module. Where finite dimensionality is used, the properties further require that "M" be finitely generated and projective. Generalizations to the most common situations can be found in .
Exterior algebras of vector bundles are frequently considered in geometry and topology. There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the Serre–Swan theorem. More general exterior algebras can be defined for sheaves of modules.
Alternating tensor algebra.
If "K" is a field of characteristic 0, then the exterior algebra of a vector space "V" over "K" can be canonically identified with the vector subspace of T("V") consisting of antisymmetric tensors. Recall that the exterior algebra is the quotient of T("V") by the ideal "I" generated by elements of the form .
Let T"r"("V") be the space of homogeneous tensors of degree "r". This is spanned by decomposable tensors
The antisymmetrization (or sometimes the skew-symmetrization) of a decomposable tensor is defined by
where the sum is taken over the symmetric group of permutations on the symbols This extends by linearity and homogeneity to an operation, also denoted by Alt, on the full tensor algebra T("V"). The image Alt(T("V")) is the alternating tensor algebra, denoted A("V"). This is a vector subspace of T("V"), and it inherits the structure of a graded vector space from that on T("V"). It carries an associative graded product formula_66 defined by
Although this product differs from the tensor product, the kernel of "Alt" is precisely the ideal "I" (again, assuming that "K" has characteristic 0), and there is a canonical isomorphism
Index notation.
Suppose that "V" has finite dimension "n", and that a basis of "V" is given. then any alternating tensor can be written in index notation as
where "t""i"1⋅⋅⋅"i""r" is completely antisymmetric in its indices.
The exterior product of two alternating tensors "t" and "s" of ranks "r" and "p" is given by
The components of this tensor are precisely the skew part of the components of the tensor product , denoted by square brackets on the indices:
The interior product may also be described in index notation as follows. Let formula_72 be an antisymmetric tensor of rank "r". Then, for , "i""α""t" is an alternating tensor of rank , given by
where "n" is the dimension of "V".
Duality.
Alternating operators.
Given two vector spaces "V" and "X" and a natural number "k", an alternating operator from "V""k" to "X" is a multilinear map
such that whenever "v"1, ..., "v""k" are linearly dependent vectors in "V", then
The map
which associates to formula_77 vectors from formula_78 their exterior product, i.e. their corresponding formula_77-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on formula_80; given any other alternating operator formula_81, there exists a unique linear map formula_82 with formula_83. This universal property characterizes the space formula_84 and can serve as its definition.
Alternating multilinear forms.
The above discussion specializes to the case when , the base field. In this case an alternating multilinear function
is called an alternating multilinear form. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating. By the universal property of the exterior power, the space of alternating forms of degree "k" on "V" is naturally isomorphic with the dual vector space (Λ"k""V")∗. If "V" is finite-dimensional, then the latter is naturally isomorphic to Λ"k"("V"∗). In particular, if "V" is "n"-dimensional, the dimension of the space of alternating maps from "V""k" to "K" is the binomial coefficient formula_86
Under this identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose and are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables. It is defined as follows:
where, if the characteristic of the base field "K" is 0, the alternation Alt of a multilinear map is defined to be the average of the sign-adjusted values over all the permutations of its variables:
When the field "K" has finite characteristic, an equivalent version of the above expression without any factorials or any constants is well-defined:
where here is the subset of ("k","m") shuffles: permutations "σ" of the set such that , and .
Interior product.
Suppose that "V" is finite-dimensional. If "V"∗ denotes the dual space to the vector space "V", then for each , it is possible to define an antiderivation on the algebra Λ("V"),
This derivation is called the interior product with "α", or sometimes the insertion operator, or contraction by "α".
Suppose that . Then w is a multilinear mapping of "V"∗ to "K", so it is defined by its values on the "k"-fold Cartesian product . If "u"1, "u"2, ..., "u""k"−1 are elements of "V"∗, then define
Additionally, let whenever "f" is a pure scalar (i.e., belonging to Λ0"V").
Axiomatic characterization and properties.
The interior product satisfies the following properties:
These three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.
Further properties of the interior product include:
Hodge duality.
Suppose that "V" has finite dimension "n". Then the interior product induces a canonical isomorphism of vector spaces
by the recursive definition
In the geometrical setting, a non-zero element of the top exterior power Λ"n"("V") (which is a one-dimensional vector space) is sometimes called a volume form (or orientation form, although this term may sometimes lead to ambiguity). The name orientation form comes from the fact that a choice of preferred top element determines an orientation of the whole exterior algebra, since it is tantamount to fixing an ordered basis of the vector space. Relative to the preferred volume form "σ", the isomorphism is given explicitly by
If, in addition to a volume form, the vector space "V" is equipped with an inner product identifying "V" with "V"∗, then the resulting isomorphism is called the Hodge star operator, which maps an element to its Hodge dual:
The composition of formula_100 with itself maps and is always a scalar multiple of the identity map. In most applications, the volume form is compatible with the inner product in the sense that it is an exterior product of an orthonormal basis of "V". In this case,
where id is the identity mapping, and the inner product has metric signature — "p" pluses and "q" minuses.
Inner product.
For "V" a finite-dimensional space, an inner product (or a pseudo-Euclidean inner product) on "V" defines an isomorphism of "V" with "V"∗, and so also an isomorphism of Λ"k""V" with (Λ"k""V")∗. The pairing between these two spaces also takes the form of an inner product. On decomposable "k"-vectors,
the determinant of the matrix of inner products. In the special case , the inner product is the square norm of the "k"-vector, given by the determinant of the Gramian matrix . This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on Λ"k""V". If "e""i", , form an orthonormal basis of "V", then the vectors of the form
constitute an orthonormal basis for Λ"k"("V").
With respect to the inner product, exterior multiplication and the interior product are mutually adjoint. Specifically, for , , and ,
where is the musical isomorphism, the linear functional defined by
for all . This property completely characterizes the inner product on the exterior algebra.
Indeed, more generally for , , and , iteration of the above adjoint properties gives
where now is the dual "l"-vector defined by
for all .
Bialgebra structure.
There is a correspondence between the graded dual of the graded algebra Λ("V") and alternating multilinear forms on "V". The exterior algebra (as well as the symmetric algebra) inherits a bialgebra structure, and, indeed, a Hopf algebra structure, from the tensor algebra. See the article on tensor algebras for a detailed treatment of the topic.
The exterior product of multilinear forms defined above is dual to a coproduct defined on Λ("V"), giving the structure of a coalgebra. The coproduct is a linear function which is given by
on elements "v"∈"V". The symbol 1 stands for the unit element of the field "K". Recall that so that the above really does lie in This definition of the coproduct is lifted to the full space Λ("V") by (linear) homomorphism. 
The correct form of this homomorphism is not what one might naively write, but has to be the one carefully defined in the coalgebra article. In this case, one obtains
Expanding this out in detail, one obtains the following expression on decomposable elements:
where the second summation is taken over all -shuffles. The above is written with a notational trick, to keep track of the field element 1: the trick is to write formula_111, and this is shuffled into various locations during the expansion of the sum over shuffles. The shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements formula_112 is "preserved" in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.
Observe that the coproduct preserves the grading of the algebra. Extending to the full space Λ("V"), one has
The tensor symbol ⊗ used in this section should be understood with some caution: it is "not" the same tensor symbol as the one being used in the definition of the alternating product. Intuitively, it is perhaps easiest to think it as just another, but different, tensor product: it is still (bi-)linear, as tensor products should be, but it is the product that is appropriate for the definition of a bialgebra, that is, for creating the object Any lingering doubt can be shaken by pondering the equalities and , which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols. This distinction is developed in greater detail in the article on tensor algebras. Here, there is much less of a problem, in that the alternating product Λ clearly corresponds to multiplication in the bialgebra, leaving the symbol ⊗ free for use in the definition of the bialgebra. In practice, this presents no particular problem, as long as one avoids the fatal trap of replacing alternating sums of ⊗ by the wedge symbol, with one exception. One can construct an alternating product from ⊗, with the understanding that it works in a different space. Immediately below, an example is given: the alternating product for the "dual space" can be given in terms of the coproduct. The construction of the bialgebra here parallels the construction in the tensor algebra article almost exactly, except for the need to correctly track the alternating signs for the exterior algebra.
In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:
where the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely, , where "ε" is the counit, as defined presently).
The counit is the homomorphism that returns the 0-graded component of its argument. The coproduct and counit, along with the exterior product, define the structure of a bialgebra on the exterior algebra.
With an antipode defined on homogeneous elements by formula_115, the exterior algebra is furthermore a Hopf algebra.
Functoriality.
Suppose that "V" and "W" are a pair of vector spaces and is a linear map. Then, by the universal property, there exists a unique homomorphism of graded algebras
such that