exterior algebra

1 Introductory remarks.

We begin with some informal remarks to motivate the formal definitions found in the next sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Throughout, V is a vector spaceMathworldPlanetmath over a field K. Many of the concepts and constructions discussed below apply verbatim to modules over commutative rings, but we will stick to vector spaces to keep things simple.

The exterior product, commonly denoted by the wedge symbol and also known as the wedge product, is an antisymmetric variant of the tensor productPlanetmathPlanetmathPlanetmath. The former, like the latter is an associative, bilinear operation. Thus, for all u,vV and a,b,c,dK, we have

(au+bv)(cu+dv) =ac(uu)+ad(uv)+bc(vu)+bd(vv) (1)
(au+bv)(cu+dv) =ac(uu)+ad(uv)+bc(vu)+bd(vv). (2)

The essential differencePlanetmathPlanetmath between the two operationsMathworldPlanetmath is that all squares formed using the exterior product vanish, by definition. Thus,

vv=0, (3)

whereas vv0. Hence, the expressions in (1) are equal to (ad-bc)uv, but there is no way to simplify further the right-hand side of (2).

A polarization argumentPlanetmathPlanetmath shows that for u,vV we have

0 =(u+v)(u+v)

Therefore, if the characteristicPlanetmathPlanetmath (http://planetmath.org/characteristic) of the underlying field K is not equal to 2, that is if 1+10, then the key postulateMathworldPlanetmath (3) is logically equivalent to the antisymmetry condition

uv=-vu,u,vV. (4)

However, if the characteristic is 2, that is if K is a field where 1=-1, then (3) does not, necessarily, follow from (4). Therefore, to keep things as general as possible, we must use (3) to formulate the essential identityPlanetmathPlanetmathPlanetmathPlanetmath satisfied by the exterior product.

So far so good, but we have not yet given a meaning to the symbol uv. The geometric interpretationMathworldPlanetmathPlanetmath of uv is that of an oriented area element in the plane spanned by u and v. Without additional structure, there is no way to assign a area measurement to a parallelogram in a vector space. However, parallelograms that lie in the same plane are commensurate. If we adopt the parallelogram spanned by u and v as the standard area, we can say that the oriented area of another parallelogram, say one that is spanned by au+bv and cu+dv, has an area that is ad-bc times the area of the first parallelogram. The exterior product allows us to express this algebraically. To wit,


The analogous interpretation for vectors is that of an oriented length element on a line. For this reason, the object uv is referred to as a bivector.

From a more algebraic point of view, a bivector uv can be considered as a formal antisymmetric productPlanetmathPlanetmathPlanetmath of vectors u and v, in much the same way that uv can be regarded as a formal non-commutative product of two vectors. Such descriptions can hardly serve as rigorous definitions, but an explicit construction is not really the way to go here.

Take the case of the tensor product. Formal sums of formal products uv, where u,vV, form a certain vector space, which we denote as VV. However, rather than saying that VV is such and such a thing, it is better to state a certain universal propertyMathworldPlanetmath that describes VV up to vector space isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. The property in question is that every bilinear map f:V×VW determines a unique linear map from g:VVW such that


Similarly, formal sums of bivectors constitute a vector space Λ2(V), called the second exterior power of V. This vector space is defined, up to isomorphism, by the condition that every antisymmetric, bilinear map f:V×VW determines a unique linear map g:Λ2(V)W with


Thus, in the same way that the tensor product replaces bilinear maps with a certain kind of linear map, the exterior product replaces bilinear, antisymmetric maps with linear maps from Λ2V.

More generally, k-multivectors are k-fold products v1vk, and the kth exterior power, Λk(V), is the vector space of formal sums of k-multivectors. The product of a k-multivector and an -multivector is a (k+)-multivector. So, the direct sumMathworldPlanetmathPlanetmathPlanetmathPlanetmath kΛk(V) forms an associative algebra, which is closed with respect to the wedge product. This algebraPlanetmathPlanetmath, commonly denoted by Λ(V), is called the exterior algebra of V.

Again, the analogyMathworldPlanetmath with the tensor product is useful. The tensor algebra T(V) can be characterized as the associative, non-commutative algebra freely generated by V. If the characteristic of k is not 2, then the wedge product satisfies the supercommutativity relationsMathworldPlanetmathPlanetmathPlanetmath


Thus, Λ(V) can be characterized as the supercommutative algebra which is freely generated by V.

2 Formal definitions.

Supercommutative algebras.

For the purposes of this discussion, we define a supercommutative algebra to be an associative, unital K-algebra A with an 2-grading, A=A+A-, such that for all odd aA- we have


and such that for all even bA+ and all aA, we have


Using a polarization argument we see that the first condition implies that for all odd a,bA- we have


If the characteristic of K is different from 2, then the converseMathworldPlanetmath is true, and we recover the usual definition of supercommutativity, namely that


with the minus sign employed if both a and b or odd, and with + employed otherwise.

Exterior algebra.

Let E be a supercommutative algebra and ι:VE- a linear map. We will say that (E,ι) is a model for the exterior algebra of V, if every linear map f:VA-, where A a supercommutative algebra, to a unique algebra homomorphism g:EA, where “lifts” means that f=gι. Diagrammatically:

The above condition on E is a universal property; this implies that all models are isomorphic as algebras. Thus, when we speak of Λ(V), the exterior algebra of V, we are referring to the isomorphism class of all such models. It is also common to identify V with its image ι(V), and to write v rather than ι(v).

Exterior powers.

For the purposes of the present entry, we define an antisymmetric map to be a k-multilinear map f:V×kW such that f(,v,v,)=0 for all vV. A polarization argument then implies the usual antisymmetry condition, namely that for every permutationMathworldPlanetmath π of {1,2,,k} we have


As usual, if the characteristic of K is different from 2, the two assertions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath. However if 1=-1, then the first assertion is stronger, and that is why we adopt it as the definition of antisymmetry.

We now define a model of the kth exterior power of V to be a vector space Ek and an antisymmetric map :V×kEk such that every antisymmetric map f:V×kW lifts to a unique linear map g:EkW, where “lifts” means that


As above, all models are isomorphic as vector spaces, and we use Λk(V) to denote the isomorphism class of all such.

The standard model.

A model of the exterior algebra Λ(V), and the exterior powers Λk(V) can be easily constructed as the antisymmetrized quotientsPlanetmathPlanetmath of the tensor algebra

T(V)=k=0Vk,Vk=VV (k times).

To that end, let S(V) denote the two sided ideal of T(V) generated by elements of the form vv,vV. Then


and let


denote the indicated quotients, with a:T(V)E(V) and ak:VkEk(V) denoting the corresponding antisymmetrization surjections. It is easy to see that S1(V) is the trivial vector space, and hence that E1(V)V. We leave it as an exercise for the reader to show that Ek(V),k2 is a model of the kth exterior power, while E(V) together with the map VE1(V) is a model of the full exterior algebra.

The canonical grading.

An inspection of the above construction reveals that


Indeed, every model of exterior algebra carries a canonical grading. Let E be a particular model of the exterior algebra of V. For k=1,2,, we will call αE, a k-primitive elementMathworldPlanetmath if α=v1vk, for some viV. We now let EkE denote the vector space spanned by all k-primitive elements, and let E0=K.

Proposition 1.

The subspacePlanetmathPlanetmathPlanetmath Ek is a model for the k𝑡ℎ exterior power of V. Furthermore,


Categorical formulation.

The above definition of exterior product has a very appealing categorical formulation. Let 𝕊 denote the category of supercommutative K-algebras, let 𝕍 denote category of vector spaces over K, and let ()-:𝕊𝕍 denote the forgetful functorMathworldPlanetmathPlanetmath AA-. We may now say that the exterior algebra function Λ:𝕍𝕊 is the left adjoint of ()-. In other words,


with the isomorphism natural in V and A.

It is useful to compare the above definition to the categorical definition of the tensor algebra. Let 𝔸 denote the category of associative, unital K-algebras, and let F:𝔸𝕍 be the forgetful functor that gives the underlying vector space structure of a K-algebra. We can then define the tensor algebra T(V) of a vector space V by saying that T:𝕍𝔸 is the left-adjoint of F:𝔸𝕍. Thus, whereas T(V) as the associative algebra freely generated by V, the exterior algebra Λ(V) is the supercommutative algebra freely generated by V. The antisymmetrization quotient map aV:T(V)Λ(V) is a natural transformation between these two functorsMathworldPlanetmath.

3 Finite dimensional models.

Basis models.

If V is an n-dimensional vector space, there are some down-to-earth constructions of Λ(V) that go a long way to illuminate the nature of the exterior product. Suppose then, that V is n-dimensional, and let e1,,en be a basis of V. For every ascending sequencePlanetmathPlanetmath


let us introduce the symbol eI=ei1ik to represent the primitive k-multivector ei1eik. If I is the empty sequence, we let eI denote the unit element of the field K.

Proposition 2.

The (nk)-dimensional vector space spanned by ei1ik is a model of Λk(V).

Note that Λ1(V) is just the n-dimensional space spanned by the basis symbols e1,,en. As such, Λ1(V) is naturally isomorphic to V. For disjoint sequences I and J, let us define


where [IJ] denotes the ascending sequence composed of the union of I and J, and where sgn(IJ)=±1 denotes the parity of the permutation that takes the sorted list [IJ] to the unsorted concatenation IJ. If I and J have one or more elements in common, we define


Here are some examples:

Proposition 3.

The 2n dimensional vector spanned by the symbols eI, together with the above product and the linear isomorphism from V to Λ1(V) is a model of the exterior algebra Λ(V).

Evidently, any list of numbers between 1 and n with length greater than n will contain duplicates. Thus, an immediate consequence of this construction is that Λk(V)=0 for k>n, and hence that


Alternating forms.

If V is finite-dimensional, we have the natural isomorphism between V and the double-dual V**. We can exploit this natural isomorphism to construct the following model of exterior algebra. Let Ak(V) denote the vector space of k-multilinear mappings from (V*)××(V*) (k times) to K. Such an mapping is known as an alternating k-form. Using the above duality we can prove that Ak(V) is a model for the kth exterior power of V.

Given alternating forms uAk(V) and vA(V), let us define uvAk+(V) according to


where a1,,ak+V*, and where the sum is taken over all permutations π of {1,2,,k+} such that π1<π2<<πk and πk+1<<πk+, and where sgnπ=±1 according to whether π is an even or odd permutationMathworldPlanetmath. With this definition, we can show that


together with the above product, and the linear isomorphism VA1(V)V** is a model for the exterior algebra Λ(V).

4 Historical Notes.

The exterior algebra is also known as the Grassmann algebra after its inventor http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Grassmann.htmlHermann Grassmann who created it to give algebraic treatment of linear geometry. Grassmann was also one of the first people to talk about the geometry of an n-dimensional space with n an arbitrary natural numberMathworldPlanetmath. The axiomatics of the exterior product are needed to define differential forms and therefore play an essential role in the theory of integration on manifolds. Exterior algebra is also an essential prerequisite to understanding de Rham’s theory of differential cohomology.

Title exterior algebra
Canonical name ExteriorAlgebra
Date of creation 2013-03-22 12:34:14
Last modified on 2013-03-22 12:34:14
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 35
Author rmilson (146)
Entry type Definition
Classification msc 15A75
Synonym Grassmann algebra
Related topic AntiSymmetric
Defines exterior product
Defines wedge product
Defines multivector
Defines exterior power