# 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.

### Other articles related to "tensor algebra, algebra, tensors, algebras":

Symmetric Algebra - Construction
... It is possible to use the tensor algebra T(V) to describe the symmetric algebra S(V) ... In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative if elements of V commute, then tensors in them must, so that we construct the symmetric algebra ...
Universal Property - Examples - Tensor Algebras
... vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K (assumed to be unital and associative) ... Let U K-Alg → K-Vect be the forgetful functor which assigns to each algebra its underlying vector space ... Given any vector space V over K we can construct the tensor algebra T(V) of V ...
Tensor Algebra - Coalgebra Structures - Bialgebra and Hopf Algebra Structure
... Finally, the tensor algebra becomes a Hopf algebra with antipode given by extended linearly to all of TV ... This is just the standard Hopf algebra structure on a free algebra, where one defines the comultiplication on by and then extends to via Similarly one defines the antipode on ...