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Tensor AlgebraIn mathematics, the tensor algebra is an abstract algebra construction of a unital associative algebra T(V) from a vector space V. In a sense, T(V) is the "most general" algebra containing V. If we take basis vectors for V, those become non-commuting variables in T(V), subject to no constraints (beyond associativity, the distributive law and K-linearity, where V is defined over the field K). Therefore, T(V) looked at in terms that aren't intrinsic, can be seen as the algebra of polynomials in n non-commuting variables over K, if V has dimension n. Because of the generality of the tensor algebra, many other algebras of interest are constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain quotients of T(V). Examples of this are the exterior algebra, Clifford algebras and universal enveloping algebras. The formal construction of T(V) is as a direct sum of graded parts Tk for k = 0,1,2, ... ; where Tk is the tensor product of V with itself k times, and T0 is K as one-dimensional vector space. -
The multiplication map on Ti and Tj to Ti+j is the natural juxtaposition on pure tensors, extended by bilinearity. That is, the tensor algebra contains all covariant tensors on V, of any rank. One can also refer to T(V) as the free algebra on the vector space V. In fact, T is a functor from the category of K-vector spaces to the category of unital associative K-algebras, and it is left adjoint to the functor taking any unital associative K-algebra to its underlying vector space. Spelled out, this translates into the following universal property of T(V): any K-linear map from V to some unital associative K-algebra A can be uniquely extended to a unital algebra homomorphism from T(V) to A. The construction generalises straightforwardly to the tensor algebra of any module M over a commutative ring. If R is a non-commutative ring, one can still perform the construction for any R-R bimodule M. (It doesn't work for ordinary R-modules because the iterated tensor products cannot be formed.)
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