**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


1. $m(f^{\otimes n},g^{\otimes m})=Sym(f^{\otimes n}\otimes g^{\otimes m})$,
2. $\eta(\lambda)=\lambda\Omega$,
3. $\Delta(f^{\otimes n})=\sum\_{k=0}^n{ {n\choose k} f^{\otimes k}\otimes f^{\otimes n-k}}$, where $f^{\otimes 0}:=\Omega$, and
4. $\epsilon(\cdot)=\langle \Omega,\cdot\rangle$.


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 quantum Grothendieck group?
Connected graded bialgebras have an antipode (which is unique):


The following book gives two formulae:


[MR2724388](https://mathscinet.ams.org/mathscinet-getitem?mr=2724388)
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](https://mathscinet.ams.org/mathscinet-getitem?mr=292876) (45 #1958)
Takeuchi, Mitsuhiro
[Free Hopf algebras generated by coalgebras](https://doi.org/10.2969/jmsj/02340561).
J. Math. Soc. Japan 23 (1971), 561–582. 


[MR0174052](https://mathscinet.ams.org/mathscinet-getitem?mr=174052) (30 #4259)
Milnor, John W.; Moore, John C.
[On the structure of Hopf algebras](https://doi.org/10.2307/1970615).
Ann. of Math. (2) 81 1965 211–264. 


