[[K-monoid]]
# Symmetric algebra

The **symmetric algebra** $S^\bullet V$ 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]] $S^\bullet V$ 
and a [[linear map]] $\iota : V \hookrightarrow S^\bullet V$
such that given any commutative unital associative algebra $A$ and any linear map $f : V \to A$,
there exists a unique [[Algebra homomorphism|unital algebra homomorphism]] $\bar{f} : S^\bullet V \to A$ for which the following diagram commutes: #m/def/falg 

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$S^\bullet : \Vect_{\mathbb{K}} \to \cat{ComAsAlg}_{\mathbb{K}}$ has a unique extension to a [[functor]] such that $\iota : 1 \Rightarrow S^\bullet : \Vect_{\mathbb{K}} \to \Vect_{\mathbb{K}}$ becomes a [[natural transformation]].

## Construction

The symmetric algebra may be constructed as a [[quotient algebra|quotient]] of the [[tensor algebra]]
$$
\begin{align*}
S^\bullet V = \frac{T^\bullet V}{\langle v \otimes w - w \otimes v : w,v \in V \rangle }_{\trianglelefteq T^\bullet V}
\end{align*}
$$
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.

> [!missing]- Proof of universal property
> #missing/proof

## Graded structure

The symmetric algebra $\mathbb{N}_{0}$-[[Graded algebra|graded]], since $S^i V \cdot S^j V \leq S^{i + j} V$.
If $V$ is itself a $\mathfrak{A}$-[[graded vector space]] for some abelian [[monoid]] $\mathfrak{A}$,
then $S^\bullet V$ possesses an additional unique gradation extending that of $V$ so that $V_{\alpha} \cdot V_{\beta} \leq (S^\bullet V)_{\alpha+\beta}$.

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