Symmetric algebra

The symmetric algebra $𝑆 ∙ 𝑉$ of a vector space $𝑉$ is the universal commutative $𝕂$ -monoid containing $𝑉$ , as formalized by the Universal property. Compare this to the exterior algebra, which is has the alternating property.

The symmetric algebra is in a sense generalized by, or rather quantized by, the Weyl algebra. Conceptually similar is the Exterior algebra.

Universal property

The symmetric algebra is a pair consisting of a commutative $𝕂$ -monoid $𝑆 ∙ 𝑉$ and a linear map $𝜄 : 𝑉 ↪ 𝑆 ∙ 𝑉$ such that given any commutative unital associative algebra $𝐴$ and any linear map $𝑓 : 𝑉 \to 𝐴$ , there exists a unique unital algebra homomorphism $¯ 𝑓 : 𝑆 ∙ 𝑉 \to 𝐴$ for which the following diagram commutes: #m/def/ralg

$𝑆 ∙ : 𝖵 𝖾 𝖼 𝗍 𝕂 \to 𝖢 𝗈 𝗆 𝖠 𝗌 𝖠 𝗅 𝗀 𝕂$ has a unique extension to a functor such that $𝜄 : 1 \Rightarrow 𝑆 ∙ : 𝖵 𝖾 𝖼 𝗍 𝕂 \to 𝖵 𝖾 𝖼 𝗍 𝕂$ becomes a natural transformation.

Construction

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

𝑆 ∙ 𝑉 = 𝑇 ∙ 𝑉 ⟨ 𝑣 \otimes 𝑤 - 𝑤 \otimes 𝑣 : 𝑤, 𝑣 \in 𝑉 ⟩ ⊴ 𝑇 ∙ 𝑉

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

Proof of universal property

#missing/proof

Graded structure

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

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