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 order) 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 is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.

The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bi-algebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure.

Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.

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:

T

k

V

=

V

⊗

k

=

V

⊗

V

⊗

⋯

⊗

V

.

{\displaystyle T^{k}V=V^{\otimes k}=V\otimes V\otimes \cdots \otimes V.}

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