Editing Tensor algebra (section)

==Cofree cocomplete coalgebra==
{{main article|Cofree coalgebra}}
One may define a different coproduct on the tensor algebra, simpler than the one given above.  It is given by
:<math>\Delta(v_1 \otimes \dots \otimes v_k) := \sum_{j=0}^{k} (v_0 \otimes \dots \otimes v_j) \boxtimes (v_{j+1} \otimes \dots \otimes v_{k+1})</math>

Here, as before, one uses the notational trick <math>v_0=v_{k+1}=1\in K</math> (recalling that <math>v\otimes 1=v</math> trivially).

This coproduct gives rise to a coalgebra. It describes a coalgebra that is [[duality (linear algebra)|dual]] to the algebra structure on ''T''(''V''<sup>&lowast;</sup>), where ''V''<sup>&lowast;</sup> 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 <math>v_i\cdot v_j=(i,j)v_{i+j}</math> where ''(i,j)'' denotes the binomial coefficient for <math>\tbinom{i+j}{i}</math>. This bialgebra is known as the [[divided power structure|divided power Hopf algebra]].

The difference between this, and the other coalgebra is most easily seen in the <math>T^2V</math> term.  Here, one has that
:<math>\Delta(v\otimes w) = 1\boxtimes (v\otimes w) + v \boxtimes w + (v\otimes w) \boxtimes 1</math>
for <math>v,w\in V</math>, which is clearly missing a shuffled term, as compared to before.