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

V=C i.V = \coprod C_i.

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

S V(v 0,v 1,v 2,)=(0,v 0,v 1,v 2,).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)T(V) copS : T(V) \rightarrow T(V)^{\mathrm{cop}} extending S VS_V. Let I SI_S be the 2-sided ideal in T(V)T(V) generated by all elements of the form c (1)S(c (2))ϵ(c)1\sum c_{(1)} S(c_{(2)}) - \epsilon(c) 1 and S(c (1))c (2)ϵ(c)1\sum S(c_{(1)}) c_{(2)} - \epsilon(c) 1, cC ic \in C_i, i=1,2,i = 1,2, \ldots . This 2-sided ideal is a biideal and S(I S)I SS(I_S) \subset I_S, hence it induces a bialgebra map

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

It follows that H(C)=T(V)/I SH(C) = T(V)/I_S is a Hopf algebra with antipode SS, the free Hopf algebra on CC. For any Hopf algebra HH' and a coalgebra map ϕ:CH\phi : C \rightarrow H' there is a unique Hopf algebra map ϕ:H(C)H\phi' : H(C) \rightarrow H' such that ϕi=ϕ\phi' \circ i = \phi where i:CH(C)i : C \rightarrow H(C) is the composition of inclusion into T(V)T(V) and projection T(V)H(C)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.
Revised on June 24, 2009 18:33:23 by Toby Bartels (71.104.230.172)