Tensor algebra

In mathematics, the Tensorial algebra is (within the abstract algebra) a construction of an associative algebra (T(V),+, ){displaystyle scriptstyle ({text{T}}(V),+,otimes)} from a vector space (V,K,+){displaystyle scriptstyle (V,mathbb {K}+)} (on the body K{displaystyle scriptstyle mathbb {K} }). Tensory algebras can be seen as a generalization of the tensorial calculation.

Construction

A tensorial algebra is built from a base of the same: starting from a vector base of the base vector space V{displaystyle scriptstyle V}, defines a non-commutative tensorial product of these elements and not subject to any restriction (beyond associative, distributive law and the K{displaystyle scriptstyle mathbb {K} }-linearities, where T(V){displaystyle scriptstyle {text{T}(V)}} is defined on the body K{displaystyle scriptstyle mathbb {K} }). Once these products are defined, the whole T(V){displaystyle scriptstyle {text{T}(V)}} consists of combinations of these same products. So, T(V){displaystyle scriptstyle {text{T}(V)}}, look in terms that are not intrinsic, can be seen as the polynomial algebra in n variables that do not switch about K, if V has dimension n. Other interesting algebras such as the outer algebra appear as quotients T(V){displaystyle scriptstyle {text{T}(V)}}like relationships imposed by generators.

Construction T(V){displaystyle scriptstyle {text{T}(V)}} is a direct sum of T graduates^k for k = 0, 1, 2,...

T(V)= kTk(V)=T0(V) T1(V) T2(V) ...... {displaystyle {text{T}}(V)=bigoplus _{k}{text{T}}}{k(V)={text{T}}}{oplus {text{T}}}{text{1}{1}{1(V)oplus {text{T}{2}(V)oplus dots}}{

Where T^k is the tensor product of V with itself k times:

Tk(V)=V V k{displaystyle {text{T}}^{k}(V)=overbrace {Votimes dots otimes V} ^{k}}}

and T0(V)=K{displaystyle scriptstyle {text{T}}{0}(V)=mathbb {K} }} is a one-dimensional vector space. The T multiplication functionⁱ and T^j map T ^{i + j} and is the natural juxtaposition of pure tensors, enlarged by bilineality. I mean, the Tensorial algebra is representative of algebras with tensors covariants that form V and of any rank. To have the full algebra of tensors, countervariants as well as covariants, you should take T(W){displaystyle scriptstyle {text{T}(W)}} where W{displaystyle scriptstyle W} is the direct sum of V{displaystyle scriptstyle V} and its dual space - this will consist of all tensors T_I^J with the upper J and lower I rates, in the classical notation.

One can also refer to T(V){displaystyle scriptstyle {text{T}(V)}} like Free algebra about the vector space V{displaystyle scriptstyle V}. In fact, the funtor who carries a K-algebra A to his underlying K-vectorial space is in a pair of T attachments, which is his Deputy left. The point of view of free algebra is useful for constructions such as a Clifford algebra or a universal enveloping algebra, where the question about existence can be resolved starting with T(V) and then imposing the required relationships.

The construction is easily generalized to the tensor algebra of any module M over a commutative ring.

Tensor algebra of a manifold

Given a differentiable manifold, a tangent space can be defined locally from which a tensor algebra can be defined. Since a tangent space can be defined at each point, and given two points their respective tangent spaces are isomorphic, a tensor algebra associated with the entire differentiable manifold can be constructed.