tensor algebra

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


is the graded R-algebraPlanetmathPlanetmathPlanetmath with nt⁒h graded componentPlanetmathPlanetmath simply the nt⁒h tensor power:

𝒯n⁒(M)=MβŠ—n=MβŠ—β‹―βŠ—M⏞n⁒ times,n=1,2,…,

and 𝒯0⁒(M)=R. The multiplication m:𝒯⁒(M)×𝒯⁒(M)→𝒯⁒(M) is given by the usual tensor productPlanetmathPlanetmathPlanetmath:


Remark 1.

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.

Remark 2.

From the point of view of category theoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath, one can describe the tensor algebra construction as a functorMathworldPlanetmath 𝒯 from the categoryMathworldPlanetmath of R-module to the category of R-algebras that is left-adjoint to the forgetful functorMathworldPlanetmathPlanetmath β„± from algebras to modules. Thus, for M an R-module and S an R-algebra, every module homomorphismMathworldPlanetmath M→ℱ⁒(S) extends to a unique algebra homomorphism 𝒯⁒(M)β†’S.

Title tensor algebra
Canonical name TensorAlgebra
Date of creation 2013-03-22 13:17:21
Last modified on 2013-03-22 13:17:21
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 13
Author rmilson (146)
Entry type Definition
Classification msc 15A69
Related topic FreeAssociativeAlgebra
Defines tensor power