Tensor Algebra - Construction

Construction

Let V be a vector space over a field K. For any nonnegative integer k, we define the kth tensor power of V to be the tensor product of V with itself k times:

That is, TkV consists of all tensors on V of rank k. By convention T0V is the ground field K (as a one-dimensional vector space over itself).

We then construct T(V) as the direct sum of TkV for k = 0,1,2,…

The multiplication in T(V) is determined by the canonical isomorphism

given by the tensor product, which is then extended by linearity to all of T(V). This multiplication rule implies that the tensor algebra T(V) is naturally a graded algebra with TkV serving as the grade-k subspace. This grading can be extended to a Z grading by appending subspaces for negative integers k.

The construction generalizes in straightforward manner to the tensor algebra of any module M over a commutative ring. If R is a non-commutative ring, one can still perform the construction for any R-R bimodule M. (It does not work for ordinary R-modules because the iterated tensor products cannot be formed.)