Exterior algebra

The exterior algebra $⋀ • 𝑉$ of a vector space $𝑉$ is the freëst alternating K-monoid containing $𝑉$ , falg as formalized by the Universal property.

The exterior algebra is in a sense generalized by, or rather quantized by, the Clifford algebra. Conceptually similar is the Symmetric algebra.

Universal property

Let $𝑉$ be a vector space over $𝕂$ the associated exterior algebra is a pair consisting of an alternating K-monoid $⋀ ∙ 𝑉$ and a linear map $𝜄 : 𝑉 \to ⋀ ∙ 𝑉$ such that given any unital associative algebra $𝐴$ , a linear map $𝑓 : 𝑉 \to 𝐴$ satisfying the identity $𝑓 (𝑣) 2 = 0$ factorizes uniqely through $𝜄$

such that $¯ 𝑓 : ⋀ ∙ 𝑉 \to 𝐴$ is a unital algebra homomorphism. This admits a unique extension to a functor $⋀ ∙ : 𝖵 𝖾 𝖼 𝗍 𝕂 \to 𝖴 𝖠 𝗌 𝖠 𝗅 𝗀 𝕂$ such that $𝜄 : 1 \Rightarrow ⋀ ∙ : 𝖵 𝖾 𝖼 𝗍 𝕂 \to 𝖵 𝖾 𝖼 𝗍 𝕂$ becomes a natural transformation.

Construction

The exterior algebra may be constructed as a quotient of the tensor algebra

⋀ • 𝑉 = 𝑇 • 𝑉 ⟨ 𝑣 \otimes 𝑣 : 𝑣 \in 𝑉 ⟩ ⊴ 𝑇 ∙ 𝑉

where the divisor is the algebra ideal generated by tensors of the form $𝑣 \otimes 𝑣$ , where the wedge product $𝑣 \land 𝑤$ is the quotient algebra product.

Proof of universal property

proof

Graded structure

Like the tensor algebra, the exterior algebra is $ℕ 0$ -graded into exterior powers

⋀ • 𝑉 = ⋀ 0 𝑉 \oplus ⋀ 1 𝑉 \oplus ⋀ 2 𝑉 \oplus \dots

such that $⋀ 𝑘 𝑉 \land ⋀ 𝑝 𝑉 \subseteq ⋀ 𝑘 + 𝑝 𝑉$ . If ${𝑒 𝑖} 𝑛 𝑖 = 1$ is a basis for $𝑉$ , then

{𝑒 𝑖 1 \land 𝑒 𝑖 2 \land \dots \land 𝑒 𝑖 𝑘 ∣ 1 \leq 𝑖 1 < 𝑖 2 < \dots < 𝑖 𝑘 \leq 𝑛}

is a basis for $⋀ 𝑘 𝑉$ , hence

d i m ⋀ 𝑘 𝑉 = (𝑛 𝑘)

Elements of the form $⋀ 𝑘 𝑖 = 1 𝑣 𝑘$ where $𝑣 𝑘 \in 𝑉$ are called $𝑘$ -blades, whereas $𝑘$ -vectors are in general linear combinations of $𝑘$ -blades. The distinction is the same as that of separable and entangled tensors. In particular, if $d i m 𝑉 = 𝑛$ then

An $(𝑛 - 1)$ -vector is a pseudovector ( $d i m ⋀ 𝑛 - 1 𝑉 = 𝑛$ )
An $𝑛$ -vector is a pseudoscalar ( $d i m ⋀ 𝑛 𝑉 = 1$ )

Geometric interpretation

Geometrically, the magnitude of a $𝑘$ -blade represents the $𝑘$ -hypervolume of the $𝑘$ -hyperparallelotope spanned defined by some vectors. Hence it generalizes the cross product, which can be thought of as resulting from the linear isomorphism from $⋀ 𝑘 𝕂 3$ to $𝕂 3$ , which is natural if $𝕂 3$ is taken as an oriented vector space.

As antisymmetric tensors

Let $𝜋 : 𝑇 ∙ 𝑉 \to ⋀ ∙ 𝑉$ be the ^graded natural projection. If $𝑘!$ is invertible in the ground field, in particular if [[Characteristic| $c h a r 𝕂 = 0$ ]], then $⋀ 𝑘 𝑉$ may be identified as a vector space with the subspace $𝑇 𝑘 - 𝑉$ of [[Tensor algebra| $𝑇 𝑘 𝑉$ ]] consisting of antisymmetric tensors via the linear section

A l t 𝑘 ⋀ 𝑖 = 1 𝑣 𝑘 = 1 𝑘! \sum 𝜎 \in 𝑆 𝑘 s g n (𝜎) 𝑘 ⨂ 𝑗 = 1 𝑣 𝜎 (𝑗)

or more generally for homogenous vectors $𝑣, 𝑤 \in ⋀ ∙ 𝑉$

A l t (𝑣 \land 𝑤) = 𝑣 \otimes 𝑤 - 𝑤 \otimes 𝑣 d e g 𝑣 + d e g 𝑤 = 𝑣 \otimes - 𝑤 d e g 𝑣 + d e g 𝑤

This is just the Antisymmetrization and symmetrization of tensors factored via the Universal property.

Properties

The $𝑘$ -blades satisfy the Plücker relations, enabling the Plücker embedding of the Grassmannian $G r 𝑘 (𝑉)$ into the Projectivization $P (⋀ 𝑘 𝑉)$

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Table of Contents

Exterior algebra

Universal property

Construction

Graded structure

Geometric interpretation

As antisymmetric tensors

Properties

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