Tensor algebra - Knowlpedia

Definition

Let $M$ be a module over a commutative ring $R$ . Its tensor algebra is the graded $R$ -module

T_R(M)=\bigoplus_{n\geq0}M^{\otimes_R n}, \qquad M^{\otimes_R0}=R,

with multiplication given by concatenation of tensors. It is a unital associative $R$ -algebra.

Universal property

For every unital associative $R$ -algebra $A$ and every $R$ -linear map $f:M\to A$ , there is a unique unital $R$ -algebra homomorphism

\widetilde f:T_R(M)\longrightarrow A

whose restriction to the degree-one summand $M$ is $f$ . Thus $T_R(M)$ is the free unital associative $R$ -algebra generated by $M$ .

Quotients

Many familiar algebras impose relations on this free algebra. The symmetric algebra kills commutators, the exterior algebra kills squares of generators, and a Clifford algebra imposes quadratic Clifford relations.

References

Nicolas Bourbaki, Algebra I: Chapters 1–3, Springer, 1989. DOI record. Relevant: Chapter III, tensor algebras.