---
tag: PBGF9TXMB7
kind: concept
name: Symmetric algebra
msc: [15A72, 15A69, 16S10]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_algebra
same_as:
  wikidata: Q1052674
  wikipedia: Symmetric algebra
---

## Summary

The symmetric algebra of a vector space $V$ is the [[tensor algebra|P4J5JZDW0B]] of $V$ made commutative.
It is the most general commutative algebra generated by $V$.
If $V$ has a finite basis $x_1, \ldots, x_n$, it is the [[polynomial ring|0SDC934R52]] $F[x_1, \ldots, x_n]$, so it gives a coordinate-free description of polynomials.

## Explanations

### L1 human

In the tensor algebra the order of the factors matters, so $x \otimes y$ and $y \otimes x$ are different.
The symmetric algebra forgets the order: products of vectors behave like products of numbers.
With a basis $x, y$ of $V$, the elements of degree $2$ are the combinations of $x^2$, $xy$ and $y^2$.
So the symmetric algebra is the ring of polynomials in the basis vectors.