Editing Tensor algebra (section)

{{Short description|Universal construction in multilinear algebra}}
In [[mathematics]], the '''tensor algebra''' of a [[vector space]] ''V'', denoted ''T''(''V'') or ''T''{{i sup|•}}(''V''), is the [[algebra over a field|algebra]] of [[tensor]]s 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 [[#Adjunction and universal property|below]]).

The tensor algebra is important because many other algebras arise as [[quotient associative algebra|quotient algebra]]s of ''T''(''V'').  These include the [[exterior algebra]], the [[symmetric algebra]], [[Clifford algebra]]s, the [[Weyl algebra]] and [[universal enveloping algebra]]s.

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 algebra|unital]] and [[associative algebra|associative]]. The unit is explicitly required to define the [[coproduct]].