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 be a coalgebra over a commutative unital ring . Let for an even nonnegative integer, and (the cooposite coalgebra of ) for an odd positive integer. Then define to be the external direct sum (coproduct) of coalgebras ,
The tensor algebra of , as the tensor algebra of any coalgebra, has a unique bialgebra structure such that the natural inclusion is a morphism of coalgebras. Then . Define a -linear map by
There is a unique bialgebra map extending . Let be the 2-sided ideal in generated by all elements of the form and , , . This 2-sided ideal is a biideal and , hence it induces a bialgebra map
It follows that is a Hopf algebra with antipode , the free Hopf algebra on . For any Hopf algebra and a coalgebra map there is a unique Hopf algebra map such that where is the composition of inclusion into and projection . 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