Tensor algebra

In mathematics, the tensor algebra of a vector space V, denoted T(V) or T ^•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below).

The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, and a more complicated one, which yields a bialgebra, and can be extended with an antipode to a Hopf algebra structure.

Note: In this article, all algebras are assumed to be unital and associative.

Construction

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

That is, T^kV 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).