TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Universal property of the tensor algebra

Statement

If $A$ is a unital associative $k$ -algebra, every linear map $f : V \to A$ extends uniquely to a unital $k$ -algebra homomorphism $\hat{f} : T (V) \to A$ .

Facts & Assumptions

Given: A vector space $V$ , a unital associative $k$ -algebra $A$ , and a linear map $f : V \to A$ .

[L1]

Multilinear maps on $V^{n}$ factor uniquely through $V^{\otimes n}$ (Finite iterated tensor products represent multilinear maps independently of parenthesization).

[L2]

Maps from a direct sum are determined uniquely by their restrictions to the summands (Universal property of a direct sum of modules).

[L3]

The grading and concatenation multiplication are those of Tensor algebra of a vector space.

Proof

technique · direct

1.1L1construct

For $n \geq 1$ , the map $(v_{1}, \dots, v_{n}) \mapsto f (v_{1}) \dots f (v_{n})$ is multilinear, so [L1] gives a linear map $f_{n} : V^{\otimes n} \to A$ with $f_{n} (v_{1} \otimes \dots \otimes v_{n}) = f (v_{1}) \dots f (v_{n})$ . Put $f_{0} (a) = a 1_{A}$ .

2.1step 1.1L2L3algebra

By [L2], the maps $f_{n}$ combine uniquely to a linear map $\hat{f} : T (V) \to A$ . On pure homogeneous tensors, concatenation gives $\hat{f} (u v) = \hat{f} (u) \hat{f} (v)$ , and bilinearity extends this to all finite sums; also $\hat{f} (1) = 1_{A}$ and $\hat{f} \circ j = f$ .

3.1step 2.1L2L3algebra

If $F : T (V) \to A$ is any unital algebra homomorphism with $F j = f$ , then $F (v_{1} \otimes \dots \otimes v_{n}) = F (j v_{1}) \dots F (j v_{n}) = f (v_{1}) \dots f (v_{n})$ and $F (1) = 1_{A}$ . Pure tensors span every homogeneous summand, so [L2] gives $F = \hat{f}$ .

4.1step 2.1step 3.1∎

The map constructed in step 2.1 is therefore the unique unital algebra extension of $f$ , including the boundary case $V = 0$ , where $T (V) = k$ .

Universal property of the tensor algebra

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources