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free Hopf algebra

There is no left adjoint to the forgetful functor from Hopf algebras to vector spaces. However, Takuechi 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 $C_i$,

$$V=\coprod C_i.$$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 $i_V:V\to T(V)$ is a morphism of coalgebras. Then $T(V^{\mathrm{cop}})\cong T(V{)}^{\mathrm{cop}}$. Define a $k$-linear map $S_V:V\to V^{\mathrm{cop}}$ by

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

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

$$S:T(V)/I_S\to (T(V)/I_S{)}^{\mathrm{cop}}.$$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 $H\prime$ and a coalgebra map $\varphi :C\to H\prime$ there is a unique Hopf algebra map $\varphi \prime :H(C)\to H\prime$ such that $\varphi \prime \circ i=\varphi$ where $i:C\to H(C)$ is the composition of inclusion into $T(V)$ and projection $T(V)\to H(C)$. 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.