Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of vectors is used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. Also like the cross product, the exterior product is alternating, meaning that u ∧ u = 0 for all vectors u, or equivalently^[1] u ∧ v = -v ∧ u for all vectors u and v. In linear algebra, the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a linear transformation that is basis-independent, and is fundamentally related to ideas of rank and linear independence.

The exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann^[2]) of a given vector space V over a field K is the unital associative algebra Λ(V) generated by the exterior product. It is widely used in contemporary geometry, especially differential geometry and algebraic geometry through the algebra of differential forms, as well as in multilinear algebra and related fields. In terms of category theory, the exterior algebra is a type of functor on vector spaces, given by a universal construction. The universal construction allows the exterior algebra to be defined, not just for vector spaces over a field, but also for modules over a commutative ring, and for other structures of interest. 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 wedge product. This dual algebra is precisely the algebra of alternating multilinear forms on V, and the pairing between the exterior algebra and its dual is given by the interior product.

Contents 1 Motivating examples 1.1 Areas in the plane 1.2 Cross and triple products 2 Formal definitions and algebraic properties 2.1 Anticommutativity of the wedge product 2.2 The exterior power 2.2.1 Basis and dimension 2.2.2 Rank of a multivector 2.3 Graded structure 2.4 Universal property 2.5 Generalizations 3 Duality 3.1 Alternating operators 3.2 Alternating multilinear forms 3.3 Bialgebra structure 3.4 Interior product 3.4.1 Axiomatic characterization and properties 3.5 Hodge duality 3.6 Inner product 4 Functoriality 4.1 Exactness 4.2 Direct sums 5 The alternating tensor algebra 5.1 Index notation 6 Applications 6.1 Linear geometry 6.2 Projective geometry 6.3 Differential geometry 6.4 Representation theory 6.5 Physics 6.6 Lie algebra homology 6.7 Homological algebra 7 History 8 See also 9 Notes 10 References 10.1 Mathematical references 10.2 Historical references 10.3 Other references and further reading

Motivating examples

Areas in the plane

The Cartesian plane R² is a 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 wedge product, and the last uses the fact that the wedge product is alternating, and in particular e₂ ∧ e₁ = -e₁ ∧ e₂. 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.

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 A(v,w) denotes the signed area of the parallelogram determined by the pair of vectors v and w, then A must satisfy the following properties:

1. A(av,bw) = a b A(v,w) for any real numbers a and b, 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).
2. A(v,v) = 0, since the area of the degenerate parallelogram determined by v (i.e., a line segment) is zero.
3. A(w,v) = −A(v,w), since interchanging the roles of v and w reverses the orientation of the parallelogram.
4. A(v + aw,w) = A(v,w), since adding a multiple of w to v affects neither the base nor the height of the parallelogram and consequently preserves its area.
5. A(e₁, e₂) = 1, since the area of the unit square is one.

With the exception of the last property, the wedge product satisfies the same formal properties as the area. In a certain sense, the wedge product generalizes the final property by allowing the area of a parallelogram to be compared to that of any "standard" chosen parallelogram. In other words, the exterior product in two-dimensions is a basis-independent formulation of area.^[3]

Cross and triple products

For vectors in R³, the exterior algebra is closely related to the cross product and triple product. Using the standard basis {e₁, e₂, e₃}, the wedge product of a pair of vectors

and

is

where {e₁ Λ e₂, e₁ Λ e₃, e₂ Λ e₃} is the basis for the three-dimensional space Λ²(R³). This imitates the usual definition of the cross product of vectors in three dimensions.

Bringing in a third vector

the wedge product of three vectors is

where e₁ Λ e₂ Λ e₃ is the basis vector for the one-dimensional space Λ³(R³). This imitates the usual definition of the triple product.

The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product u×v 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 geometrically a (signed) 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. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions.

Formal definitions and algebraic properties

The exterior algebra Λ(V) over a vector space V is defined as the quotient algebra of the tensor algebra by the two-sided ideal I generated by all elements of the form x ⊗ x such that x ∈ V.^[4] Symbolically,

The wedge product ∧ of two elements of Λ(V) is defined by

Anticommutativity of the wedge product

This product is anticommutative on elements of V, for supposing that x, y ∈ V,

whence

More generally, if x₁, x₂, ..., x_k are elements of V, and σ is a permutation of the integers [1,...,k], then

where sgn(σ) is the signature of the permutation σ.^[5]

The exterior power

The kth exterior power of V, denoted Λ^k(V), is the vector subspace of Λ(V) spanned by elements of the form

If α ∈ Λ^k(V), then α is said to be a k-multivector. If, furthermore, α can be expressed as a wedge product of k elements of V, then α is said to be decomposable. Although decomposable multivectors span Λ^k(V), not every element of Λ^k(V) is decomposable. For example, in R⁴, the following 2-multivector is not decomposable:

(This is in fact a symplectic form, since α ∧ α ≠ 0.^[6])

Basis and dimension

If the dimension of V is n and {e₁,...,e_n} is a basis of V, then the set

is a basis for Λ^k(V). The reason is the following: given any wedge product of the form

then every vector v_j can be written as a linear combination of the basis vectors e_i; using the bilinearity of the wedge product, this can be expanded to a linear combination of wedge products of those basis vectors. Any wedge product in which the same basis vector appears more than once is zero; any wedge 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 k-vectors can be computed as the minors of the matrix that describes the vectors v_j in terms of the basis e_i.

By counting the basis elements, the dimension of Λ^k(V) is the binomial coefficient C(n,k). In particular, Λ^k(V) = {0} for k > n.

Any element of the exterior algebra can be written as a sum of multivectors. Hence, as a vector space the exterior algebra is a direct sum

(where by convention Λ⁰(V) = K and Λ¹(V) = V), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2ⁿ.

Rank of a multivector

If α ∈ Λ^k(V), then it is possible to express α as a linear combination of decomposable multivectors:

where each α⁽ⁱ⁾ is decomposable, say

The rank of the multivector α is the minimal number of decomposable multivectors in such an expansion of α. This is similar to the notion of tensor rank.

Rank is particularly important in the study of 2-multivectors (Sternberg 1974, §III.6) (Bryant et al. 1991). The rank of a 2-multivector α can be identified with 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 a_ij = −a_ji (the matrix of coefficients is skew-symmetric). The rank of α agrees with the rank of the matrix a_ij.

In characteristic 0, the 2-multivector α has rank p if and only if

and

Graded structure

The wedge product of a k-multivector with a p-multivector is a (k+p)-multivector, 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. Symbolically,

Moreover, the wedge product is graded anticommutative, meaning that if α ∈ Λ^k(V) and β ∈ Λ^p(V), then

In addition to studying the graded structure on the exterior algebra, Bourbaki (1989) 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 V be a vector space over the field K. Informally, multiplication in Λ(V) is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity v ∧ v = 0 for v ∈ V. Formally, Λ(V) is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative K-algebra containing V with alternating multiplication on V must contain a homomorphic image of Λ(V). In other words, the exterior algebra has the following universal property:^[7]

To construct the most general algebra that contains V and whose multiplication is alternating on V, it is natural to start with the most general algebra that contains V, the tensor algebra T(V), and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided ideal I in T(V) generated by all elements of the form v⊗v for v in V, and define Λ(V) as the quotient

(and use Λ as the symbol for multiplication in Λ(V)). It is then straightforward to show that Λ(V) contains V and satisfies the above universal property.

As a consequence of this construction, the operation of assigning to a vector space V its exterior algebra Λ(V) is a functor from the category of vector spaces to the category of algebras.

Rather than defining Λ(V) first and then identifying the exterior powers Λ^k(V) as certain subspaces, one may alternatively define the spaces Λ^k(V) first and then combine them to form the algebra Λ(V). 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 (Bourbaki 1989).

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.

Duality

Alternating operators

Given two vector spaces V and X, an alternating operator (or anti-symmetric operator) from V^k to X is a multilinear map

f: V^k → X

such that whenever v₁,...,v_k are linearly dependent vectors in V, then

f(v₁,...,v_k) = 0.

A well-known example is the determinant, an alternating operator from (Kⁿ)ⁿ to K.

The map

w: V^k → Λ^k(V)

which associates to k vectors from V their wedge product, i.e. their corresponding k-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on V^k: given any other alternating operator f : V^k → X, there exists a unique linear map φ: Λ^k(V) → X with f = φ o w. This universal property characterizes the space Λ^k(V) and can serve as its definition.

Alternating multilinear forms

The above discussion specializes to the case when X = K, 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^kV)^∗. If V is finite-dimensional, then the latter is naturally isomorphic to Λ^k(V^∗). In particular, the dimension of the space of anti-symmetric maps from V^k to K is the binomial coefficient n choose k.

Under this identification, the wedge product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : V^k → K and η : V^m → K are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their wedge product is the sum of the numbers of their variables. It is defined as follows:

where the alternation Alt of a multilinear map is defined to be the signed average of the values over all the permutations of its variables:

This definition of the wedge product is well-defined even if the field K has finite characteristic, if one considers an equivalent version of the above that does not use factorials or any constants:

where here Sh_k,m ⊂ S_k+m is the subset of (k,m) shuffles: permutations σ of the set {1,2,…,k+m} such that σ(1) < σ(2) < … < σ(k), and σ(k+1) < σ(k+2)< … <σ(k+m).^[8]

Bialgebra structure

In formal terms, there is a correspondence between the graded dual of the graded algebra Λ(V) and alternating multilinear forms on V. The wedge 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 Δ : Λ(V) → Λ(V) ⊗ Λ(V) given on decomposable elements by

For example,

This extends by linearity to an operation defined on the whole exterior algebra. In terms of the coproduct, the wedge 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, α∧β = ε o (α⊗β) o Δ, where ε is the counit, as defined presently).

The counit is the homomorphism ε : Λ(V) → K which returns the 0-graded component of its argument. The coproduct and counit, along with the wedge product, define the structure of a bialgebra on the exterior algebra.

Interior product

Suppose that V is finite-dimensional. If V* denotes the dual space to the vector space V, then for each α ∈ V^*, 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 w ∈ Λ^kV. Then w is a multilinear mapping of V^* to K, so it is defined by its values on the k-fold Cartesian product V^*× V^*× ... × V^*. If u₁, u₂, ..., u_k-1 are k-1 elements of V^*, then define

Additionally, let i_αf = 0 whenever f is a pure scalar (i.e., belonging to Λ⁰V).

Axiomatic characterization and properties

The interior product satisfies the following properties:

1. For each k and each α ∈ V^*,

   

   (By convention, Λ⁻¹ = 0.)

2. If v is an element of V ( = Λ¹V), then i_αv = α(v) is the dual pairing between elements of V and elements of V^*.
3. For each α ∈ V^*, i_α is a graded derivation of degree −1:

   

In fact, 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

In the geometrical setting, a non-zero element of the top exterior power Λⁿ(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.) Relative to a given 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 dual (or more commonly the Hodge star operator)

The composite of * with itself maps Λ^k(V) → Λ^k(V) 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 a wedge product of an orthonormal basis of V. In this case,

where I is the identity, and the inner product has metric signature (p,q) — p plusses and q minuses.

Inner product

For V a finite-dimensional space, an inner product on V defines an isomorphism of V with V^∗, and so also an isomorphism of Λ^kV with (Λ^kV)^∗. The pairing between these two spaces also takes the form of an inner product. On decomposable k-multivectors,

the determinant of the matrix of inner products. In the special case v_i = w_i, the inner product is the square norm of the multivector, 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^kV. If e_i, i=1,2,...,n, 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 v ∈ Λ^k−1(V), w ∈ Λ^k(V), and x ∈ V,

where is the linear functional defined by

for all y ∈ V. This property completely characterizes the inner product on the exterior algebra.

Functoriality

Suppose that V and W are a pair of vector spaces and f : V → W is a linear transformation. Then, by the universal construction, there exists a unique homomorphism of graded algebras

such that

In particular, Λ(f) preserves homogeneous degree. The k-graded components of Λ(f) are given on decomposable elements by

Let

The components of the transformation Λ(k) relative to a basis of V and W is the matrix of k × k minors of f. In particular, if V = W and V is of finite dimension n, then Λⁿ(f) is a mapping of a one-dimensional vector space Λⁿ to itself, and is therefore given by a scalar: the determinant of f.

Exactness

If

is a short exact sequence of vector spaces, then

is an exact sequence of graded vector spaces^[9] as is

^[10]

Direct sums

In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:

This is a graded isomorphism; i.e.,

Slightly more generally, if

is a short exact sequence of vector spaces then Λ^k(V) has a filtration

with quotients :. In particular, if U is 1-dimensional then

is exact, and if W is 1-dimensional then

is exact.^[11]

The alternating tensor algebra

If K is a field of characteristic 0,^[12] then the exterior algebra of a vector space V 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 x ⊗ x.

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 {1,...,r}. 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 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 e₁, ..., e_n of V is given. then any alternating tensor t ∈ A^r(V) ⊂ T^r(V) can be written in index notation as

where t^{i₁ ... i_r} is completely antisymmetric in its indices.

The wedge 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 s ⊗ t, denoted by square brackets on the indices:

The interior product may also be described in index notation as follows. Let be an antisymmetric tensor of rank r. Then, for α ∈ V^*, i_αt is an alternating tensor of rank r-1, given by

where n is the dimension of V.

Applications

Linear geometry

The decomposable k-vectors have geometric interpretations: the bivector represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented parallelogram with sides u and v. Analogously, the 3-vector represents the spanned 3-space weighted by the volume of the oriented parallelepiped with edges u, v, and w.

Projective geometry

Decomposable k-vectors in Λ^kV correspond to weighted k-dimensional subspaces of V. In particular, the Grassmannian of k-dimensional subspaces of V, denoted Gr_k(V), can be naturally identified with an algebraic subvariety of the projective space P(Λ^kV). This is called the Plücker embedding.

Differential geometry

The exterior algebra has notable applications in differential geometry, where it is used to define differential forms. A differential form at a point of a differentiable manifold is an alternating multilinear form on the tangent space at the point. Equivalently, a differential form of degree k is a linear functional on the k-th exterior power of the tangent space. As a consequence, the wedge product of multilinear forms defines a natural wedge product for differential forms. Differential forms play a major role in diverse areas of differential geometry.

In particular, the exterior derivative gives the exterior algebra of differential forms on a manifold the structure of a differential algebra. The exterior derivative commutes with pullback along smooth mappings between manifolds, and it is therefore a natural differential operator. The exterior algebra of differential forms, equipped with the exterior derivative, is a differential complex whose cohomology is called the de Rham cohomology of the underlying manifold and plays a vital role in the algebraic topology of differentiable manifolds.

Representation theory

In representation theory, the exterior algebra is one of the two fundamental Schur functors on the category of vector spaces, the other being the symmetric algebra. Together, these constructions are used to generate the irreducible representations of the general linear group.

Physics

The exterior algebra is an archetypal example of a superalgebra, which plays a fundamental role in physical theories pertaining to fermions and supersymmetry. For a physical discussion, see Grassmann number. For various other applications of related ideas to physics, see superspace and supergroup (physics).

Lie algebra homology

Let L be a Lie algebra over a field k, then it is possible to define the structure of a chain complex on the exterior algebra of L. This is a k-linear mapping

defined on decomposable elements by

The Jacobi identity holds if and only if ∂∂ = 0, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra L to be a Lie algebra. Moreover, in that case ΛL is a chain complex with boundary operator ∂. The homology associated to this complex is the Lie algebra homology.

Homological algebra

The exterior algebra is the main ingredient in the construction of the Koszul complex, a fundamental object in homological algebra.

History

The exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension.^[13] This referred more generally to an algebraic (or axiomatic) theory of extended quantities and was one of the early precursors to the modern notion of a vector space. Saint-Venant also published similar ideas of exterior calculus for which he claimed priority over Grassmann.^[14]

The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors. It was thus a calculus, much like the propositional calculus, except focused exclusively on the task of formal reasoning in geometrical terms.^[15] In particular, this new development allowed for an axiomatic characterization of dimension, a property that had previously only been examined from the coordinate point of view.

The import of this new theory of vectors and multivectors was lost to mid 19th century mathematicians,^[16] until being thoroughly vetted by Giuseppe Peano in 1888. Peano's work also remained somewhat obscure until the turn of the century, when the subject was unified by members of the French geometry school (notably Henri Poincaré, Élie Cartan, and Gaston Darboux) who applied Grassmann's ideas to the calculus of differential forms.

A short while later, Alfred North Whitehead, borrowing from the ideas of Peano and Grassmann, introduced his universal algebra. This then paved the way for the 20th century developments of abstract algebra by placing the axiomatic notion of an algebraic system on a firm logical footing.

See also

• symmetric algebra, the symmetric analog
• Clifford algebra, a quantum deformation of the exterior algebra by a quadratic form
• Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic form
• multilinear algebra
• tensor algebra
• geometric algebra
• Koszul complex

Notes

References

Mathematical references

• Bishop, R.; Goldberg, S.I. (1980), Tensor analysis on manifolds, Dover, ISBN 0-486-64039-6 

Includes a treatment of alternating tensors and alternating forms, as well as a detailed discussion of Hodge duality from the perspective adopted in this article.

• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9 

This is the main mathematical reference for the article. It introduces the exterior algebra of a module over a commutative ring (although this article specializes primarily to the case when the ring is a field), including a discussion of the universal property, functoriality, duality, and the bialgebra structure. See chapters III.7 and III.11.

• Bryant, R.L.; Chern, S.S.; Gardner, R.B.; Goldschmidt, H.L.; Griffiths, P.A. (1991), Exterior differential systems, Springer-Verlag 

This book contains applications of exterior algebras to problems in partial differential equations. Rank and related concepts are developed in the early chapters.

• MacLane, S.; Birkhoff, G. (1999), Algebra, AMS Chelsea, ISBN 0-8218-1646-2 

Chapter XVI sections 6-10 give a more elementary account of the exterior algebra, including duality, determinants and minors, and alternating forms.

• Sternberg, Shlomo (1964), Lectures on Differential Geometry, Prentice Hall 

Contains a classical treatment of the exterior algebra as alternating tensors, and applications to differential geometry.

Historical references

• Bourbaki, Nicolas (1989). "Historical note on chapters II and III". Elements of mathematics, Algebra I. Springer-Verlag. 
• Clifford, W. (1878), "Applications of Grassmann's Extensive Algebra", American Journal of Mathematics 1 (4): 350–358, doi:10.2307/2369379 
• Forder, H. G. (1941), The Calculus of Extension, Cambridge University Press 
• Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik  (The Linear Extension Theory - A new Branch of Mathematics)
• Kannenberg, Llyod (2000), Extension Theory (translation of Grassmann's Ausdehnungslehre), American Mathematical Society, ISBN 0821820311 
• Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva ; Kannenberg, Lloyd (1999), Geometric calculus: according to the Ausdehnungslehre of H. Grassmann, Birkhäuser, ISBN 978-0817641269 .
• Whitehead, Alfred North (1898), A Treatise on Universal Algebra, with Applications, Cambridge 

Other references and further reading

• Browne, J.M. (2007), Grassmann algebra - Exploring applications of Extended Vector Algebra with Mathematica, Published on line 

An introduction to the exterior algebra, and geometric algebra, with a focus on applications. Also includes a history section and bibliography.

• Spivak, Michael (1965), Calculus on manifolds, Addison-Wesley, ISBN 978-0805390216 

Includes applications of the exterior algebra to differential forms, specifically focused on integration and Stokes's theorem. The notation Λ^kV in this text is used to mean the space of alternating k-forms on V; i.e., for Spivak Λ^kV is what this article would call Λ^kV*. Spivak discusses this in Addendum 4.

• Strang, G. (1993), Introduction to linear algebra, Wellesley-Cambridge Press, ISBN 978-0961408855 

Includes an elementary treatment of the axiomatization of determinants as signed areas, volumes, and higher-dimensional volumes.

• Onishchik, A.L. (2001), "Exterior algebra", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/E/e037080.htm 
• Wendell H. Fleming (1965) Functions of Several Variables, Addison-Wesley.

Chapter 6: Exterior algebra and differential calculus, pages 205-38. This textbook in multivariate calculus introduces the exterior algebra of differential forms adroitly into the calculus sequence for colleges.