A quantum Grothendieck group?

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.

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