TITLE: Does the forgetful functor {Hopf Algebras}→{Algebras} have a right adjoint?
QUESTION [24 upvotes]: (Alternate title: Find the Adjoint: Hopf Algebra edition)
I was chatting with Jonah about his question Hopf algebra structure on $\prod_n A^{\otimes n}$ for an algebra $A$. It's very closely related to the following question:

For a $k$-algebra $A$, is there a Hopf Algebra $H(A)$ such that for any Hopf algebra $B$, we have
  $\mathop{Hom}_{k\text{-alg}}(B,A)\cong \mathop{Hom}_{k\text{-Hopf-alg}}(B,H(A))$?

In other words, does the forgetful functor from $k$-Hopf-algebras to $k$-algebras have a right adjoint? There are a few related questions, any of which I'd be interested in knowing the answer to:

Does the forgetful functor from Hopf algebras to augmented algebras (sending the counit to the augmentation) have a right adjoint?
Does the forgetful functor from Hopf algebras to algebras with distinguished anti-automorphism (sending the antipode to the anti-automorphism) have a right adjoint?
Does the forgetful functor from Hopf algebras to algebras with augmentation and distinguished anti-automorphism have a right adjoint?

Unfortunately, I don't feel like I can motivate this question very well. My motivation is that the better I know which forgetful functors have adjoints, the better I sleep at night.

REPLY [13 votes]: The problem is famous and you will find in the book Sweedler. Ana Agore (one of my student in master) prove the result in june 2009 -- and her paper will apper in Comm. Algebra. Is this one http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.2613v3.pdf
However, related to the problem is also another paper of her more recently: http://front.math.ucdavis.edu/1003.0318 
On the other hand, two months latter (july 2009) a colleque of her (Alex Chirvasitu) give the explicit construction of it in this paper http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.2881v2.pdf that allready appered in J. Algebra at the begining of 2010. 
Cheers! and a nice site.
Gigel Militaru