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Tensor algebra

In mathematics, the tensor algebra is an abstract algebra construction of an algebra T(V) from a vector space 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. Other algebras of interest such as the exterior algebra appear as quotients of T(V), as relations are imposed on generators.

The 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 T'\'0 is K as one-dimensional vector space. The multiplication map on Ti and Tj is the mapping to Ti+j is the natural juxtaposition on pure tensors, extended by bilinearity. That is, the tensor algebra is representative of algebra with tensors that are formed from V and covariant, of any rank. To have the complete algebra of tensors, contravariant as well as covariant, one should take T(W) where W is the direct sum of V and its dual space - this will consist of all tensors T''IJ with upper indices J and lower indices I, in classical notation.

One can also refer to T(V) as the free algebra on the vector space V. In fact the functor taking a K-algebra A to its underlying K-vector space is in a pair of adjoint functors with T, which is its left adjoint. The free algebra point of view is useful for constructions like those of Clifford algebras and universal enveloping algebras, where the existence question can be settled by starting with T(V) and then imposing the required relations.

The construction generalises straightforwardly to the tensor algebra of any module M over a commutative ring.