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Question 1: Given a co-commutative bialgebra, does there exist a sort of Grothendick 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!

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up vote 4 down vote accepted

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.

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That certainly answers my second question. Any idea about the non-graded case? – Ollie Margetts Apr 3 '12 at 20:01
You mean a filtered case? – Yannic May 18 at 2:35

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 (very short) and references therein, especially to the lecture notes of Bodo Pareigis available here:

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