Tensor algebra

In 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 algebra on "V", in the sense of being left adjoint to 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 is important because many other algebras arise as quotient algebras of "T"("V"). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras.
The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure.
"Note": In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.
Construction.
Let "V" be a vector space over a field "K". For any nonnegative integer "k", we define the "k"th tensor power of "V" to be the tensor product of "V" with itself "k" times:
That is, "T""k""V" consists of all tensors on "V" of order "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 sum of "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 algebra with "T""k""V" serving as the grade-"k" subspace. This grading can be extended to a Z grading by appending subspaces formula_4 for negative integers "k".
The construction generalizes in a 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 algebra on the vector space "V", and is functorial. As with other free constructions, the functor "T" is left adjoint to some forgetful functor. In this case, it's the functor that 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":
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 to a 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 functor from K"-Vect, the category of vector spaces over "K", to K"-Alg, the category of "K"-algebras. The functoriality of "T" means that any linear map between "K"-vector spaces "U" and "W" extends uniquely to a "K"-algebra homomorphism from "T"("U") to "T"("W").
Non-commutative polynomials.
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 "indeterminates") in "T"("V"), subject to no constraints beyond associativity, the distributive law and "K"-linearity.
Note that the algebra of polynomials on "V" is not formula_5, but rather formula_6: a (homogeneous) linear function on "V" is an element of formula_7 for example coordinates formula_8 on a vector space are covectors, as they take in a vector and give out a scalar (the given coordinate of the vector).
Quotients.
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, the Weyl algebra and universal enveloping algebras.
Coalgebra.
The tensor algebra has two different coalgebra structures. One is compatible with the tensor product, and thus can be extended to a bialgebra, and can be further be extended with an antipode to a Hopf algebra structure. The other structure, although simpler, cannot be extended to a bialgebra. The first structure is developed immediately below; the second structure is given in the section on the cofree coalgebra, further down.
The development provided below can be equally well applied to the exterior algebra, using the wedge symbol formula_9 in place of the tensor symbol formula_10; a sign must also be kept track of, when permuting elements of the exterior algebra. This correspondence also lasts through the definition of the bialgebra, and on to the definition of a Hopf algebra. That is, the exterior algebra can also be given a Hopf algebra structure.
Similarly, the symmetric algebra can also be given the structure of a Hopf algebra, in exactly the same fashion, by replacing everywhere the tensor product formula_10 by the symmetrized tensor product formula_12, i.e. that product where formula_13
In each case, this is possible because the alternating product formula_9 and the symmetric product formula_12 obey the required consistency conditions for the definition of a bialgebra and Hopf algebra; this can be explicitly checked in the manner below. Whenever one has a product obeying these consistency conditions, the construction goes thorough; insofar as such a product gave rise to a quotient space, the quotient space inherits the Hopf algebra structure.
In the language of category theory, one says that there is a functor from the category of -vector spaces to the category of -associate algebras. But there is also a functor taking vector spaces to the category of exterior algebras, and a functor taking vector spaces to symmetric algebras. There is a natural map from to each of these. Verifying that quotienting preserves the Hopf algebra structure is the same as verifying that the maps are indeed natural.
Coproduct.
The coalgebra is obtained by defining a coproduct or diagonal operator
Here, formula_17 is used as a short-hand for formula_5 to avoid an explosion of parentheses. The formula_19 symbol is used to denote the "external" tensor product, needed for the definition of a coalgebra. It is being used to distinguish it from the "internal" tensor product formula_10, which is already "taken" and being used to denote multiplication in the tensor algebra (see the section "Multiplication", below, for further clarification on this issue). In order to avoid confusion between these two symbols, most texts will replace formula_10 by a plain dot, or even drop it altogether, with the understanding that it is implied from context. This then allows the formula_10 symbol to be used in place of the formula_19 symbol. This is not done below, and the two symbols are used independently and explicitly, so as to show the proper location of each. The result is a bit more verbose, but should be easier to comprehend.
The definition of the operator formula_24 is most easily built up in stages, first by defining it for elements formula_25 and then by homomorphically extending it to the whole algebra. A suitable choice for the coproduct is then
and
where formula_28 is the unit of the field formula_29. By linearity, one obviously has
for all formula_31 It is straight-forward to verify that this definition satisfies the axioms of a coalgebra: that is, that
where formula_33 is the identity map on formula_17. Indeed, one gets
and likewise for the other side. At this point, one could invoke a lemma, and say that formula_24 extends trivially, by linearity, to all of formula_17, because formula_17 is a free object and formula_39 is a generator of the free algebra, and formula_24 is a homomorphism. However, it is insightful to provide explicit expressions. So, for formula_41, one has (by definition) the homomorphism
Expanding, one has
In the above expansion, there is no need to ever write formula_44 as this is just plain-old scalar multiplication in the algebra; that is, one trivially has that formula_45
The extension above preserves the algebra grading. That is,
Continuing in this fashion, one can obtain an explicit expression for the coproduct acting on a homogenous element of order "m":
where the formula_48 symbol, which should appear as ш, the sha, denotes the shuffle product. This is expressed in the second summation, which is taken over all (p,m-p+1)-shuffles. The above is written with a notational trick, to keep track of the field element 1: the trick is to write formula_49, and this is shuffled into various locations during the expansion of the sum over shuffles. The shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements formula_50 is "preserved" in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right. Any one given shuffle obeys
As before, the algebra grading is preserved:
Counit.
The counit formula_53 is given by the projection of the field component out from the algebra. This can be written as formula_54 for formula_55 and formula_56 for formula_57. By homomorphism under the tensor product formula_10, this extends to 
for all formula_60
It is a straight-forward matter to verify that this counit satisfies the needed axiom for the coalgebra:
Working this explicitly, one has
where, for the last step, one has made use of the isomorphism formula_63, as is appropriate for the defining axiom of the counit.
Bialgebra.
A bialgebra defines both multiplication, and comultiplication, and requires them to be compatible.
Multiplication.
Multiplication is given by an operator
which, in this case, was already given as the "internal" tensor product. That is,
That is, formula_66 The above should make it clear why the formula_19 symbol needs to be used: the formula_10 was actually one and the same thing as formula_69; and notational sloppiness here would lead to utter chaos. To strengthen this: the tensor product formula_10 of the tensor algebra corresponds to the multiplication formula_69 used in the definition of an algebra, whereas the tensor product formula_19 is the one required in the definition of comultiplication in a coalgebra. These two tensor products are "not" the same thing!
Unit.
The unit for the algebra
is just the embedding, so that
That the unit is compatible with the tensor product formula_10 is "trivial": it is just part of the standard definition of the tensor product of vector spaces. That is, formula_76 for field element "k" and any formula_77 More verbosely, the axioms for an associative algebra require the two homomorphisms (or commuting diagrams):
on formula_79, and that symmetrically, on formula_80, that
where the right-hand side of these equations should be understood as the scalar product.
Compatibility.
The unit and counit, and multiplication and comultiplication, all have to satisfy compatibility conditions. It is straightforward to see that 
Similarly, the unit is compatible with comultiplication:
The above requires the use of the isomorphism formula_84 in order to work; without this, one loses linearity. Component-wise,
with the right-hand side making use of the isomorphism.
Multiplication and the counit are compatible:
whenever "x" or "y" are not elements of formula_29, and otherwise, one has scalar multiplication on the field: formula_88 The most difficult to verify is the compatibility of multiplication and comultiplication:
where formula_90 exchanges elements. The compatibility condition only needs to be verified on formula_91; the full compatibility follows as a homomorphic extension to all of formula_92 The verification is verbose but straight-forward; it is not given here, except for the final result:
For formula_94 an explicit expression for this was given in the coalgebra section, above.
Hopf algebra.
The Hopf algebra adds an antipode to the bialgebra axioms. The antipode formula_95 on formula_57 is given by
This is sometimes called the "anti-identity". The antipode on formula_98 is given by
and on formula_100 by
This extends homomorphically to
Compatibility.
Compatibility of the antipode with multiplication and comultiplication requires that
This is straight-forward to verify componentwise on formula_104:
Similarly, on formula_55:
Recall that
and that 
for any formula_110 that is "not" in formula_111
One may proceed in a similar manner, by homomorphism, verifying that the antipode inserts the appropriate cancellative signs in the shuffle, starting with the compatibility condition on formula_112 and proceeding by induction.
Cofree cocomplete coalgebra.
One may define a different coproduct on the tensor algebra, simpler than the one given above. It is given by
Here, as before, one uses the notational trick formula_114 (recalling that formula_115 trivially).
This coproduct gives rise to a coalgebra. It describes a coalgebra that is dual to the algebra structure on "T"("V"∗), where "V"∗ denotes the dual vector space of linear maps "V" → F. In the same way that the tensor algebra is a free algebra, the corresponding coalgebra is termed cocomplete co-free. With the usual product this is not a bialgebra. It "can" be turned into a bialgebra with the product formula_116 where "(i,j)" denotes the binomial coefficient for formula_117. This bialgebra is known as the divided power Hopf algebra.
The difference between this, and the other coalgebra is most easily seen in the formula_112 term. Here, one has that
for formula_120, which is clearly missing a shuffled term, as compared to before.