# Normal Hopf ideals and Hopf subalgebras, quotient group schemes

There is a following theorem:

$H$ is a commutative Hopf algebra over a field $k$. Then there exists a bijective correspondence between $$\{ \textrm{Hopf subalgebras }K\subset H \} \quad \leftrightarrow\quad\{\textrm{ normal Hopf ideals } I\subset H\}$$

Where Hopf subalgebra is a subalgebra $K$ satisfying $\Delta K\subset K\otimes K$, and a normal Hopf ideal is a Hopf ideal $I$ which satisfies $ad(I)\subset I\otimes A$ where $ad$ is defined by $\mathrm{ad}:A\to A\otimes A$, $a\mapsto a_2\otimes a_1Sa_3$. The correspondence is defined by $$K\mapsto HK^+,\quad I\mapsto H^I:= k\square_{H/I}H=\{h\in H|\Delta h\equiv h\otimes 1\pmod{H\otimes I}\}$$.

This is proven in M.Takeuchi, A CORRESPONDENCE BETWEEN HOPF IDEALS AND SUB-HOPF ALGEBRAS (1972).

The following is my personal impression on the proof. The easy part is to show that $K\mapsto HK^+\mapsto H^{HK^+}$ is $K$, which depends on rather complicated proof of the faithful flatness of $H$ over $K$. For the other way around, the paper uses a variant of argument Chevalley's theorem on algebraic groups, which is not truely "algebraic".

I found some papers which simplifies the proof that $H$ is faithfully flat over $K$. Some of those are: Schneider, Principal Homogeneous Spaces for Arbitrary Hopf Algebras(1990) Masuoka and Wigner, Faithful Flatness of Hopf Algebras (1992)

However, I couldn't find any simplification of the proof of $H(H^I)^+=I$.

I would like to know about more recent result related to this theorem. Is there any simplification on the part that I mentioned? Is there any generalization for, like quantum groups?

I come to this question while I was learning basic theory of algebraic groups or group schemes. Most references construct a quotient by realizing it as an orbit space of certain action on a projective plane. Then I thought there could be a way of constructing it in terms of coordinate rings. I heard that there is a chapter in SGA which deals with a construction of quotient, but I don't have access to french literatures yet... I would appreciate if someone instead explain their approach a bit, or give some references written in english.

Thank you.

• Is this the theorem that a normal subgroup scheme of an affine group scheme over a field admits a quotient? I think Waterhouse proves this in his book. See also the notes on Milne's home page.
– anon
May 26, 2015 at 23:15
• @anon Yes, this theorem gives a coordinate ring of quotient. The proof in Waterhouse's book is similar to that of Takeuchi's paper. What I am looking for is a proof without using Chevalley's theorem, if such things exist. May 27, 2015 at 3:46
• Chevalley's theorem is pretty basic. I don't see much point in trying to avoid it.
– anon
May 28, 2015 at 12:13

I am not sure I have an answer to your question but it seems to me that the statement you refer to needs a property which is really of somewhat geometric nature: i.e. that every normal algebraic subgroup of an affine algebraic group is observable. Observability is precisely the property that selects subgroups behaving nicely when treated algebraically, in a way.

Let me say a little more about extensions of this result to non commutative Hopf algebras which are usually applied to quantum groups. In Parshall-Wang Memoirs of AMS 439 (1991), there are definitions of left and righ normal quantum subgroups.

However in this general setting faithful flatness is not granted in general. Thus no nice correspondence without imposing additional faithful flatness hypothesis (at times under disguise, e.g. whith some semisemplicity or cosemisemplicity assumptions, see arXiv:1110.6701) which in the commutative case you can (or have to, it depends on your point of view...) prove. The paper H-J. Schneider, Some remarks on exact sequences of quantum groups, Comm. Alg. 21(9), 3337-3357 (1993), contains much of the most relevant informations on the subject. The paper by Schauenburg "Faithful flatness over Hopf subalgebras: counterexamples" (cannot trace at moment where it was published) gives you what the title says.

So even what you consider the easy part needs some fixing in the quantum case.

Of course the first, other obvious generalization of suche result is, already in the commutative case, removing the normality requirement, which means passing from normal subgroup-- quotient subgroup correspondence to subgroup--homogeneous space correspondence and much of what you say generalizes under the observability requirement. When you then move to quantum case this again survives into a correspondence between left ideals which are 2-sided coideals on one hand and left coideal subalgebras (with unity) on the other hand. Correspondence which is 1:1 only under suitable faithful flatness-faithful coflatness requirements...

Masuoka Takeuchi Schneider are again the authors where you'll find more material.

For a general review of many of the things we mentiones: Susan Montgomery "Hopf algebras and their action on rings" AMS 1993.

Hope this helps.

• Thank you so much for introducing various related results! Yes, my objective was to understand algebraic nature of something geometric. I have had some time parsing arguments in the papers you've indicated, and it seems (not 100% sure) that their results are enough to prove the part what I have mentioned in the question. Jun 2, 2015 at 12:26