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tensor algebra
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(Definition)
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Let $R$ be a commutative ring, and $M$ an $R$ -module. The tensor algebra $$ \mc{T}(M) = \bigoplus_{n=0}^\infty \mc{T}_n(M $$ is the graded $R$ -algebra with $n^{th}$ graded component simply the $n^{th}$ tensor power: $$ \mc{T}_n(M) = M^{\otimes n} =\overbrace{M\otimes \cdots \otimes M}^{n\text{ times}},\quad n=1,2,\ldots $$ and $\mc{T}_0(M)=R$ . The multiplication $m:\mc{T}(M)\times \mc{T}(M)\to\mc{T}(M)$
is given by the usual tensor product: $$ m(a,b)=a\otimes b,\quad a\in M^{\otimes n},\; b\in M^{\otimes m} $$
One can generalize the above definition to cover the case where the ground ring $R$ is non-commutative by requiring that the module $M$ is a bimodule with $R$ acting on both the left and the right.
From the point of view of category theory, one can describe the tensor algebra construction as a functor $\mc{T}$ from the category of $R$ -module to the category of $R$ -algebras that is left-adjoint to the forgetful functor $\mc{F}$ from algebras to modules. Thus, for $M$ an $R$ -module and $S$ an $R$ -algebra, every module homomorphism $M\to \mc{F}(S)$ extends to a unique algebra homomorphism $\mc{T}(M)\to S$ .
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"tensor algebra" is owned by rmilson. [ full author list (3) | owner history (1) ]
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Cross-references: homomorphism, algebras, forgetful functor, category, functor, category theory, point, right, bimodule, module, non-commutative, ground ring, cover, tensor product, multiplication, component, commutative ring
There are 13 references to this entry.
This is version 10 of tensor algebra, born on 2002-12-18, modified 2007-03-22.
Object id is 3776, canonical name is TensorAlgebra.
Accessed 13203 times total.
Classification:
| AMS MSC: | 15A69 (Linear and multilinear algebra; matrix theory :: Multilinear algebra, tensor products) |
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Pending Errata and Addenda
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