tensor algebra
Let be a commutative ring, and an -module. The tensor algebra
is the graded -algebra with
graded component
simply the tensor power:
and .
The multiplication is given
by the usual tensor product:
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
from the category
of -module to the category of -algebras that
is left-adjoint to the forgetful functor
from algebras to
modules. Thus, for an -module and an -algebra, every
module homomorphism
extends to a unique algebra
homomorphism .
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 |