Exterior algebra

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

and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.

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.

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.

In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions.

In general, the resulting coefficients of the basis k-vectors can be computed as the minors of the matrix that describes the vectors

In other words, the exterior algebra has the following universal property:[8] Given any unital associative K-algebra A and any K-linear map

⁠ (or higher than the dimension of the vector space), one or the other definition of the product could be used, as the two algebras are isomorphic (see V. I. Arnold or Kobayashi-Nomizu).

Then any alternating tensor t ∈ Ar(V) ⊂ Tr(V) can be written in index notation with the Einstein summation convention as where ti1⋅⋅⋅ir is completely antisymmetric in its indices.

Under such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones.

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

As this might look very specific and fine tuned, an equivalent raw version is to sum in the above formula over permutations in left cosets of Sk+m / (Sk × Sm).

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

⁠, the isomorphism is given explicitly by If, in addition to a volume form, the vector space V is equipped with an inner product identifying

In the special case vi = wi, the inner product is the square norm of the k-vector, given by the determinant of the Gramian matrix (⟨vi, vj⟩).

Any lingering doubt can be shaken by pondering the equalities (1 ⊗ v) ∧ (1 ⊗ w) = 1 ⊗ (v ∧ w) and (v ⊗ 1) ∧ (1 ⊗ w) = v ⊗ w, which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols.

In practice, this presents no particular problem, as long as one avoids the fatal trap of replacing alternating sums of

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, α ∧ β = ε ∘ (α ⊗ β) ∘ Δ, where

This is also the intimate connection between exterior algebra and differential forms, as to integrate we need a 'differential' object to measure infinitesimal volume.

Likewise, the k × k minors of a matrix can be defined by looking at the exterior products of column vectors chosen k at a time.

represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented parallelogram with sides

Equivalently, a differential form of degree k is a linear functional on the kth exterior power of the tangent space.

The exterior algebra of differential forms, equipped with the exterior derivative, is a cochain 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.

The exterior algebra over the complex numbers is the archetypal example of a superalgebra, which plays a fundamental role in physical theories pertaining to fermions and supersymmetry.

The exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension.

[18] 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.

[19] The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors.

[20] 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.

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.