---
tag: YZCBSRCZC7
kind: definition
name: Symmetric algebra
msc: [15A72, 15A69, 16S10]
defines: PBGF9TXMB7
uses: [000AYPA9AJ, P4J5JZDW0B, 0CD87SFM2N, 0076HWHGHQ]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_algebra
---

## Definition

Let $F$ be a field, let $V$ be a [[vector space|000AYPA9AJ]] over $F$, and let $T(V)$ be its [[tensor algebra|P4J5JZDW0B]].
Let $I$ be the [[two-sided ideal|0CD87SFM2N]] of $T(V)$ generated by the elements $v \otimes w - w \otimes v$ with $v, w \in V$.
The *symmetric algebra* of $V$ is the [[quotient ring|0076HWHGHQ]]
$$S(V) = T(V) / I.$$

## Remarks

$S(V)$ is a commutative associative unital $F$-algebra, graded by degree; its degree $k$ part $S^k(V)$ is the $k$-th symmetric power of $V$.
Universal property: every linear map from $V$ to a commutative associative unital $F$-algebra $A$ extends uniquely to a unital algebra homomorphism $S(V) \to A$.
Example: if $V$ has basis $x_1, \ldots, x_n$, then $S(V) \cong F[x_1, \ldots, x_n]$, and $S^k(V)$ has dimension $\binom{n + k - 1}{k}$; for $n = 2$, $k = 2$ the basis is $x_1^2, x_1 x_2, x_2^2$.
Non-example: for $\dim V \ge 2$, the tensor algebra $T(V)$ itself is not commutative, so it is not $S(V)$.
The [[exterior algebra|0MDMZ965VE]] uses instead the ideal generated by the elements $v \otimes v$.