A quantum Grothendieck group? Question 1: Given a co-commutative bialgebra, does there exist a sort of Grothendieck group type construction? Presumably this should take the form of a functor from co-commutative bialgebras to hopf algebras?
My motivation: the finite particle vectors in the symmetric Fock space $\mathbb{C}\Omega\oplus \bigoplus_{n=1}^\infty H^{\vee n}$ have the natural structure of a graded bialgebra. Just set


*

*$m(f^{\otimes n},g^{\otimes m})=Sym(f^{\otimes n}\otimes g^{\otimes m})$,

*$\eta(\lambda)=\lambda\Omega$,

*$\Delta(f^{\otimes n})=\sum_{k=0}^n{ {n\choose k} f^{\otimes k}\otimes f^{\otimes n-k}}$, where $f^{\otimes 0}:=\Omega$, and

*$\epsilon(\cdot)=\langle \Omega,\cdot\rangle$.


but it doesn't seem to have an antipode.
Question 2: Can we make a quantum group containing this bialgebra?
Actually it occurs to me that it has probably been looked at, since a finite dimensional $H$ gives us an algebra isomorphic to $\mathbb{C}[x_1,\ldots,x_{\mathrm{dim} H}]$? I should say also that I'm neither an algebraist, nor a quantum groupie, so I'd appreciate any references/constructions readable by a non-expert!
 A: Connected graded bialgebras have an antipode (which is unique):
The following book gives two formulae:
MR2724388
Aguiar, Marcelo; Mahajan, Swapneel
Monoidal functors, species and Hopf algebras.
CRM Monograph Series, 29. American Mathematical Society, Providence, RI, 2010. lii+784 pp. ISBN: 978-0-8218-4776-3 
These formulae are for the antipode of a connected graded bialgebra and are given in 2.3.3.
These are the Takeuchi formula and the Milnor and Moore formula.
The original references are
MR0292876 (45 #1958)
Takeuchi, Mitsuhiro
Free Hopf algebras generated by coalgebras.
J. Math. Soc. Japan 23 (1971), 561–582. 
MR0174052 (30 #4259)
Milnor, John W.; Moore, John C.
On the structure of Hopf algebras.
Ann. of Math. (2) 81 1965 211–264. 
A: The forgetful functor from the category of Hopf algebras to the category of bialgebras has a left adjoint.  This means that given a bialgebra $B$, there is a Hopf algebra $H(B)$ with a bialgebra morphism $\iota : B \to H(B)$ such that any bialgebra morphism from $B$ to a Hopf algebra $H$ factors through $\iota$ via a morphism of Hopf algebras.  
I do not know whether $H(B)$ is cocommutative if $B$ is, and I also do not know whether the morphism $\iota$ is always injective.  For the latter question I sort of suspect the answer to be negative, just as a semigroup does not always inject into its Grothendieck group.
For discussion of this and related issues, see http://arxiv.org/abs/0905.2613 (very short) and references therein, especially to the lecture notes of Bodo Pareigis available here: http://www.mathematik.uni-muenchen.de/~pareigis/pa_schft.html
