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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $j : V \to T (V)$ .

Tensor algebra of a vector space

Definition

Depends on

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