---
tag: P4J5JZDW0B
kind: concept
name: Tensor algebra
msc: [15A72, 15A69, 16S10]
sources:
  - url: https://en.wikipedia.org/wiki/Tensor_algebra
same_as:
  wikidata: Q2296021
  wikipedia: Tensor algebra
---

## Summary

The tensor algebra of a vector space $V$ is the direct sum of all [[tensor powers|0BF2DE2SYK]] of $V$, with multiplication given by the tensor product.
It is the most general associative algebra generated by $V$.
Every linear map from $V$ to an associative unital algebra extends uniquely to a unital algebra homomorphism.
The [[symmetric algebra|PBGF9TXMB7]], the [[exterior algebra|0MDMZ965VE]] and the [[Clifford algebra|A38B1QFK20]] are quotients of it.

## Explanations

### L1 human

Take the elements of $V$ as letters and form words by writing letters side by side with the tensor product.
The tensor algebra contains all linear combinations of such words of every length, including the empty word $1$.
Multiplication puts one word after another, and the letters never commute.
If $V$ has a basis $x, y$, then $x \otimes y$ and $y \otimes x$ are different elements.
If $V$ has dimension $1$ with basis $x$, the tensor algebra is just the polynomial ring $F[x]$.