Tensor algebra

The tensor algebra $𝑇 • 𝑉$ of a vector space $𝑉$ is the direct sum of all tensor powers of $𝑉$ together with the outer product $(\otimes) : 𝑇 • 𝑉 \times 𝑇 • 𝑉 \to 𝑇 • 𝑉$ , #m/def/ralg i.e. denoting $𝑇 𝑘 𝑉 = 𝑉 \otimes$ ,

𝑇 • 𝑉 = \infty ⨁ 𝑘 = 0 𝑇 𝑘 𝑉

where $𝑇 0 𝑉 = 𝕂$ . The tensor algebra is a very simple $𝕂$ -monoid¹ and $ℕ 0$ -graded algebra.

Universal property

The tensor algebra has a unique extension to a functor $𝑇 • : 𝖵 𝖾 𝖼 𝗍 𝕂 \to 𝖠 𝗌 𝖠 𝗅 𝗀 𝕂$ so that the canonical inclusion becomes a natural transformation $𝜄 : i d 𝖵 𝖾 𝖼 𝗍 𝕂 \to 𝐹 \circ 𝑇 •$ , where $𝐹 : 𝖠 𝗌 𝖠 𝗅 𝗀 𝕂 \to 𝖵 𝖾 𝖼 𝗍 𝕂$ is the forgetful functor (thus creating a Free-forgetful adjunction). This is enabled by characterising $(𝑇 • 𝑉, 𝜄 𝑉)$ with the following universal property:

If $𝐴 \in 𝖠 𝗌 𝖠 𝗅 𝗀 𝕂$ and $𝑓 \in 𝖵 𝖾 𝖼 𝗍 𝕂 (𝑉, 𝐴)$ is a linear map of vector spaces there exists a unique $¯ 𝑓 \in 𝖠 𝗌 𝖠 𝗅 𝗀 𝕂$ so that $¯ 𝑓 𝜄 𝐴 = 𝑓$ , i.e. the following diagram commutes

Proof

#missing/proof

Graded structure

The tensor algebra is $ℕ 0$ -graded, since $𝑇 𝑖 𝑉 \otimes 𝑇 𝑗 𝑉 \subseteq 𝑇 𝑖 + 𝑗 𝑉$ . If $𝑉$ is itself a $𝔄$ -graded vector space for some monoid $𝔄$ , then $𝑇 ∙ 𝑉$ possesses an additional unique gradation extending that of $𝑉$ so that $𝑉 𝛼 \otimes 𝑉 𝛽 \leq (𝑇 ∙ 𝑉) 𝛼 + 𝛽$ .

#state/tidy | #lang/en | #SemBr

Indeed, there is a sense in which it is the most simple, i.e. a Free-forgetful adjunction. ↩

Tensor algebra

Universal property

Graded structure

Related

Footnotes