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.

Read more about Tensor Algebra: Construction, Adjunction and Universal Property, Non-commutative Polynomials, Quotients, Coalgebra Structures

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