DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Tensor algebra of a vector space

Definition

For a vector space $V$ over $k$ , set $V^{\otimes 0} = k$ and define the tensor algebra

$T (V) = ⨁_{n \geq 0} V^{\otimes n} .$

After fixing the canonical tensor-product associators, multiplication on homogeneous pure tensors is concatenation:

$(v_{1} \otimes \dots \otimes v_{m}) (w_{1} \otimes \dots \otimes w_{n}) = v_{1} \otimes \dots \otimes v_{m} \otimes w_{1} \otimes \dots \otimes w_{n} .$

Extend this bilinearly. Each element has finite degree support, so products are finite sums. Tensor associativity makes concatenation associative, and $1 \in k = V^{\otimes 0}$ is its unit. Thus $T (V)$ is a graded unital associative $k$ -algebra. The degree-one inclusion is denoted $j : V \to T (V)$ .

Depends on

Used by

Symmetric algebra of a vector space Definition
Universal enveloping algebra Definition
Universal property of the tensor algebra Theorem

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Etingof, MIT 18.745 notes, §12.1, printed pp. 69–70 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.1, printed pp. 71–72 (standard reference, not scraped)