16.12.2 Tensor Wedge Product Alternating Construction

The wedge product constructs alternating tensors by antisymmetrizing multilinear maps, essential in differential geometry and algebraic structures.

Tensor Wedge Product Alternating Construction is the formal quotient-based procedure for building the exterior algebra from the full tensor algebra, defining Λ(V) as T(V) modulo the two-sided ideal generated by all elements of the form v⊗v, and thereby constructing an alternating multiplication out of raw materials that individually carry no alternation at all.

The Ingredients Before Construction

Starting from the Full Tensor Algebra

The construction begins with the full tensor algebra:

T (V) = \oplus_{k = 0}^{\infty} V^{\otimes k}

built from ordinary tensor products, with no antisymmetry imposed at any stage; every element of V⊗V is, in general, neither symmetric nor alternating.

The Ideal Generated by Squares

The construction singles out the two-sided ideal I ⊂ T(V) generated by all elements v⊗v for v ∈ V:

I = ⌈ v \otimes v : v \in V ⌉

meaning I consists of every element of T(V) expressible as a sum of terms, each containing a factor v⊗v for some v, multiplied on either side by arbitrary tensors.

The Quotient Construction

Defining the Exterior Algebra

The exterior algebra is defined as the quotient:

Λ (V) = T (V) / I

with the wedge product ∧ defined as the induced multiplication on equivalence classes, inherited directly from the tensor product on T(V).

Why the Quotient Forces Alternation

Because every element v⊗v is identified with zero in the quotient, the image of v⊗v under the quotient map — which is exactly v∧v by definition of the induced product — must equal zero in Λ(V), giving the fundamental vanishing relation directly as a consequence of the construction rather than as an additionally imposed axiom.

Deriving Anticommutativity from the Construction

Expanding (u+v)⊗(u+v)

Since (u+v)⊗(u+v) ∈ I by definition of I, its image in the quotient is zero. Expanding using bilinearity of the tensor product:

(u + v) \otimes (u + v) = u \otimes u + u \otimes v + v \otimes u + v \otimes v

Passing to the quotient, the first and last terms vanish (each is individually in I), leaving u∧v + v∧u = 0 in Λ(V), which is exactly the anticommutation relation, derived purely from the quotient construction rather than assumed as a starting definition.

Confirming the Grading Survives the Quotient

I Respects the Tensor Algebra's Grading

The ideal I is homogeneous — generated entirely by degree-2 elements v⊗v, and closed under multiplication by tensors of any degree — which ensures the quotient T(V)/I inherits a well-defined grading by degree, giving Λ(V) = ⊕ₖ Λᵏ(V) directly from the construction rather than as a separately verified add-on.

Each Graded Piece as a Quotient

Explicitly, each graded piece of the exterior algebra is itself a quotient of the corresponding graded piece of the tensor algebra:

Λ^{k} (V) = V^{\otimes k} / (I \cap V^{\otimes k})

showing the alternating tensors of rank k are literally the equivalence classes of rank-k ordinary tensors modulo the antisymmetry-forcing identification.

Relationship to the Antisymmetrization Construction

Two Equivalent Routes to the Same Algebra

This quotient construction and the earlier antisymmetrization-operator construction (Λᵏ(V) ≅ image of Alt) produce isomorphic results in characteristic zero; the quotient route emphasizes what is being discarded (symmetric-type redundancy), while the projection route emphasizes what is being selected (the alternating part), and both descriptions are standard, interchangeable ways of arriving at the same exterior algebra.