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.
Let V be a vector space over a field K. For any nonnegative integer k, we define the kth tensor power of V to be the tensor product of V with itself k times:
TkV=V ⊗ =V ⊗ V ⊗ … ⊗ V.
We then construct T(V) as the direct sum of TkV for k = 0,1,2,…
T(V)=
infty | |
oplus | |
k=0 |
TkV=K ⊕ V ⊕ (V ⊗ V) ⊕ (V ⊗ V ⊗ V) ⊕ … .
TkV ⊗ T\ellV\toTkV
TkV=\{0\}
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.)
The tensor algebra is also called the free algebra on the vector space, and is functorial; this means that the map
V\mapstoT(V)
Explicitly, the tensor algebra satisfies the following universal property, which formally expresses the statement that it is the most general algebra containing V:
f:V\toA
Here is the canonical inclusion of into . As for other universal properties, the tensor algebra can be defined as the unique algebra satisfying this property (specifically, it is unique up to a unique isomorphism), but this definition requires to prove that an object satisfying this property exists.
The above universal property implies that is a functor from the category of vector spaces over, to the category of -algebras. This means that any linear map between -vector spaces and extends uniquely to a -algebra homomorphism from to .
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
T(V)
T(V*)
V*,
x1,...,xn
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.
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
\wedge
⊗
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
⊗
⊗ Sym
v ⊗ Symw=w ⊗ Symv.
In each case, this is possible because the alternating product
\wedge
⊗ Sym
In the language of category theory, one says that there is a functor from the category of -vector spaces to the category of -associative 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.
The coalgebra is obtained by defining a coproduct or diagonal operator
\Delta:TV\toTV\boxtimesTV
Here,
TV
T(V)
\boxtimes
⊗
⊗
⊗
\boxtimes
The definition of the operator
\Delta
v\inV\subsetTV
\Delta:v\mapstov\boxtimes1+1\boxtimesv
\Delta:1\mapsto1\boxtimes1
where
1\inK=T0V\subsetTV
K
\Delta(k)=k(1\boxtimes1)=k\boxtimes1=1\boxtimesk
for all
k\inK.
(idTV\boxtimes\Delta)\circ\Delta=(\Delta\boxtimesidTV)\circ\Delta
where
idTV:x\mapstox
TV
((idTV\boxtimes\Delta)\circ\Delta)(v)= v\boxtimes1\boxtimes1+1\boxtimesv\boxtimes1+1\boxtimes1\boxtimesv
\Delta
TV
TV
V
\Delta
v ⊗ w\inT2V
\Delta:v ⊗ w\mapsto\Delta(v) ⊗ \Delta(w)
\begin{align}\Delta(v ⊗ w)&=(v\boxtimes1+1\boxtimesv) ⊗ (w\boxtimes1+1\boxtimesw)\\ &=(v ⊗ w)\boxtimes1+v\boxtimesw+w\boxtimesv+1\boxtimes(v ⊗ w)\end{align}
In the above expansion, there is no need to ever write
1 ⊗ v
1 ⊗ v=1 ⋅ v=v.
The extension above preserves the algebra grading. That is,
\Delta:T2V\to
2 | |
oplus | |
k=0 |
TkV\boxtimesT2-kV
Continuing in this fashion, one can obtain an explicit expression for the coproduct acting on a homogenous element of order m:
\begin{align} \Delta(v1 ⊗ … ⊗ vm)&= \Delta(v1) ⊗ … ⊗ \Delta(vm)\\ &=
m | |
\sum | |
p=0 |
\left(v1 ⊗ … ⊗ vp\right) \omega \left(vp+1 ⊗ … ⊗ vm\right)\ &=
m | |
\sum | |
p=0 |
\sum\sigma\inSh(p,m-p) \left(v\sigma(1) ⊗ ... ⊗ v\sigma(p)\right)\boxtimes\left(v\sigma(p+1) ⊗ ... ⊗ v\sigma(m)\right) \end{align}
\omega
\begin{aligned} \operatorname{Sh}(p,q) =\{\sigma:\{1,...,p+q\}\to\{1,...,p+q\} \mid &\sigmaisbijective, \sigma(1)<\sigma(2)< … <\sigma(p),\\ &and \sigma(p+1)<\sigma(p+2)< … <\sigma(m)\}. \end{aligned}
By convention, one takes that Sh(m,0) and Sh(0,m) equals