PlanetMath: tensor algebra

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tensor algebra

(Definition)

Let be a commutative ring, and an -module. The tensor algebra

$\displaystyle \mathcal{T}(M) = \bigoplus_{n=0}^\infty \mathcal{T}_n(M)$

is the graded

-algebra with $n^{th}$ graded component simply the $n^{th}$ tensor power:

$\displaystyle \mathcal{T}_n(M) = M^{\otimes n} =\overbrace{M\otimes \cdots \otimes M}^{n\text{ times}},\quad n=1,2,\ldots,$

and $\mathcal{T}_0(M)=R$ . The multiplication $m:\mathcal{T}(M)\times \mathcal{T}(M)\to\mathcal{T}(M)$ is given by the usual tensor product:

$\displaystyle m(a,b)=a\otimes b,\quad a\in M^{\otimes n},\; b\in M^{\otimes m}.$

Remark 1.

One can generalize the above definition to cover the case where the ground ring

is non-commutative by requiring that the module

is a bimodule with

acting on both the left and the right.

Remark 2.

From the point of view of category theory, one can describe the tensor algebra construction as a functor $\mathcal{T}$ from the category of

-module to the category of

-algebras that is left-adjoint to the forgetful functor $\mathcal{F}$ from algebras to modules. Thus, for

-module and

-algebra, every module homomorphism $M\to \mathcal{F}(S)$ extends to a unique algebra homomorphism $\mathcal{T}(M)\to S$ .

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"tensor algebra" is owned by rmilson. [ full author list (3) | owner history (1) ]

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