nLab free Hopf algebra

There is no left adjoint to the forgetful functor from Hopf algebras to vector spaces. However, Takeuchi in

• M. Takeuchi, Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), No.4, pp. 561–582.

constructed a left adjoint to the forgetful functor from Hopf algebras to coalgebras (his purpose was to construct the historically first example of a Hopf algebra with a noninvertible antipode map).

Let $C$ be a coalgebra over a commutative unital ring $k$. Let $C_i = C$ for $i$ an even nonnegative integer, and $C_i = C_i^{{\mathrm {cop}}}$ (the cooposite coalgebra of $C$) for $i$ an odd positive integer. Then define $V$ to be the external direct sum (coproduct) of coalgebras $C_i$,

$$V = \coprod C_i.$$

The tensor algebra $T(V)$ of $V$, as the tensor algebra of any coalgebra, has a unique bialgebra structure such that the natural inclusion $i_V : V \rightarrow T(V)$ is a morphism of coalgebras. Then $T(V^{\mathrm{cop}}) \cong T(V)^{\mathrm{cop}}$. Define a $k$-linear map $S_V : V \rightarrow V^{\mathrm{cop}}$ by

$$S_V(v_0, v_1, v_2,\ldots) = (0, v_0, v_1, v_2, \ldots).$$

There is a unique bialgebra map $S : T(V) \rightarrow T(V)^{\mathrm{cop}}$ extending $S_V$. Let $I_S$ be the 2-sided ideal in $T(V)$ generated by all elements of the form $\sum c_{(1)} S(c_{(2)}) - \epsilon(c) 1$ and $\sum S(c_{(1)}) c_{(2)} - \epsilon(c) 1$, $c \in C_i$, $i = 1,2, \ldots$. This 2-sided ideal is a biideal and $S(I_S) \subset I_S$, hence it induces a bialgebra map

$$S : T(V)/I_S \rightarrow (T(V)/I_S)^{\mathrm{cop}}.$$

It follows that $H(C) = T(V)/I_S$ is a Hopf algebra with antipode $S$, the free Hopf algebra on $C$. For any Hopf algebra $H'$ and a coalgebra map $\phi : C \rightarrow H'$ there is a unique Hopf algebra map $\phi' : H(C) \rightarrow H'$ such that $\phi' \circ i = \phi$ where $i : C \rightarrow H(C)$ is the composition of inclusion into $T(V)$ and projection $T(V) \rightarrow H(C)$. Takeuchi’s free Hopf algebra construction is functorial.

A comparison with Manin’s closely related construction of a Hopf envelope of a bialgebra can be found in section 13.2 of

• Z. Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003.