Tensor algebra: Difference between revisions

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In [[mathematics]], the '''tensor algebra''' is an [[abstract algebra]] construction of an [[unital]] [[associative algebra]] T(V) from a [[vector space]] V. If we take [[basis vector]]s 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 (mathematics)|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 T^''k'' for ''k'' = 0,1,2, ... ; where T^''k'' is the [[tensor product]] of V with itself ''k'' times, and T^''0'' is K as one-dimensional vector space. The multiplication map on T^''i'' and T^''j'' is the mapping to T^''i''+''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[[tensor]]s ''T''_I^J 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 algebra]]s and [[universal enveloping algebra]]s, 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 (mathematics)|module]] M over a '''''commutative''''' ring.