- Tensor algebra
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 algebraon "V", in the sense of being left adjointto 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 a coalgebra structure.
"Note": In this article, all algebras are assumed to be
Let "V" be a
vector spaceover a field "K". For any nonnegative integer"k", we define the "k"th tensor power of "V" to be the tensor productof "V" with itself "k" times::That is, "T""k""V" consists of all tensors on "V" of rank "k". By convention "T"0"V" is the ground field "K" (as a one-dimensional vector space over itself).
We then construct "T"("V") as the
direct sumof "T""k""V" for "k" = 0,1,2,…:The multiplication in "T"("V") is determined by the canonical isomorphism:given by the tensor product, which is then extended by linearity to all of "T"("V"). This multiplication rule implies that the tensor algebra "T"("V") is naturally a graded algebrawith "T""k""V" serving as the grade-"k" subspace.
The construction generalizes in straightforward manner 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 does not work for ordinary "R"-modules because the iterated tensor products cannot be formed.)
Adjunction and universal property
The tensor algebra "T"("V") is also called the
free algebraon the vector space "V", and is functorial. As with other free constructions, the functor "T" is left adjoint to some forgetful functor, here the functor which sends each "K"-algebra to its underlying vector space.
Explicitly, the tensor algebra satisfies the following
universal property, which formally expresses the statement that it is the most general algebra containing "V":: Any linear transformation"f" : "V" → "A" from "V" to an algebra "A" over "K" can be uniquely extended to an algebra homomorphismfrom "T"("V") to "A" as indicated by the following commutative diagram:
Here "i" is the canonical inclusion of "V" into "T"("V") (the unit of the adjunction). One can, in fact, define the tensor algebra "T"("V") as the unique algebra satisfying this property (specifically, it is unique
up toa unique isomorphism), but one must still prove that an object satisfying this property exists.
The above universal property shows that the construction of the tensor algebra is "functorial" in nature. That is, "T" is a
functorfrom the "K"-Vect, category of vector spacesover "K", to "K"-Alg, the category of "K"-algebras. The functoriality of "T" means that any linear map from "V" to "W" extends uniquely to an algebra homomorphism from "T"("V") to "T"("W").
If "V" has finite dimension "n", another way of looking at the tensor algebra is as the "algebra of polynomials over "K" in "n" non-commuting variables". If we take
basis vectors for "V", those become non-commuting variables (or "indeterminants") in "T"("V"), subject to no constraints (beyond associativity, the distributive lawand "K"-linearity).
Note that the algebra of polynomials on "V" is not , but rather : a (homogeneous) linear function on "V" is an element of .
Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain
quotient algebras of "T"("V"). Examples of this are the exterior algebra, the symmetric algebra, Clifford algebras and universal enveloping algebras.
coalgebrastructure on the tensor algebra is given as follows. The coproduct Δ is defined by:extended by linearity to all of "TV". The counit is given by ε("v") = 0-graded component of "v". Note that Δ : "TV" → "TV" ⊗ "TV" respects the grading:and ε is also compatible with the grading.
The tensor algebra is "not" a
bialgebrawith this coproduct. However, the following more complicated coproduct does yield a bialgebra::where the summation is taken over all (p,m-p)-shuffles. Finally, the tensor algebra becomes a Hopf algebrawith antipode given by:extended linearly to all of "TV".
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