{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "5X7NA4TTK8"
    },
    {
      "field": "uses",
      "target": "YZCBSRCZC7"
    },
    {
      "field": "defines",
      "target": "VTFQYCP1WA"
    }
  ],
  "body": "\n## Summary\n\nThe 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.\nIt is the most general associative algebra generated by $V$.\nEvery linear map from $V$ to an associative unital algebra extends uniquely to a unital algebra homomorphism.\nThe [[symmetric algebra|PBGF9TXMB7]], the [[exterior algebra|0MDMZ965VE]] and the [[Clifford algebra|A38B1QFK20]] are quotients of it.\n\n## Explanations\n\n### L1 human\n\nTake the elements of $V$ as letters and form words by writing letters side by side with the tensor product.\nThe tensor algebra contains all linear combinations of such words of every length, including the empty word $1$.\nMultiplication puts one word after another, and the letters never commute.\nIf $V$ has a basis $x, y$, then $x \\otimes y$ and $y \\otimes x$ are different elements.\nIf $V$ has dimension $1$ with basis $x$, the tensor algebra is just the polynomial ring $F[x]$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "15A72",
      "15A69",
      "16S10"
    ],
    "name": "Tensor algebra",
    "same_as": {
      "wikidata": "Q2296021",
      "wikipedia": "Tensor algebra"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tensor_algebra"
      }
    ],
    "tag": "P4J5JZDW0B"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/P4J5JZDW0B/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/P4J5JZDW0B.md",
  "msc": [
    "15A72",
    "15A69",
    "16S10"
  ],
  "name": "Tensor algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P4J/P4J5JZDW0B.md",
  "status": null,
  "tag": "P4J5JZDW0B",
  "uri": "m2://P4J5JZDW0B",
  "url": "https://math2.tamnd.com/wiki/tensor-algebra/",
  "writing": {
    "tag": "P4J5JZDW0B",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}