Noah Snyder gave a great answer to [this](https://mathoverflow.net/questions/10036/) question about different versions of a quantum group $U\_q(\mathfrak g)$ when $q$ is a root of unity. I want to ask about forms of the deformed coordinate ring $\mathcal O\_q(G)$ when $q$ is a root of unity.


Let's focus on $SL(2)$. Recall that the "small quantum group" (see Noah's answer) is obtained by dividing $U\_q(\mathfrak s\mathfrak l\_2)$ by the ideal generated by $E^e,F^e,K^e-1$ (which is central since $q^e=1$). The quotient is a finite-dimensional Hopf algebra $\overline{U\_q}(\mathfrak s\mathfrak l\_2)$. Since $\mathcal O\_q(SL(2))$ and $U\_q(\mathfrak s\mathfrak l\_2)$ are in duality, one expects that there is a corresponding finite-dimensional subalgebra of $\mathcal O\_q(SL(2))$ (the set of elements annihilated by $E^e,F^e,K^e-1$). What is it, and are there references about it? Is there a good reason why it is hard to work with compared to the Hopf algebra perspective?



Quantum coordinate ring at root of unity
For what concerns De Concini-like integer form the Sl\_2 case (and more) is treated in quite some detail in
"Quantum function algebra at roots of 1" De Concini-Lyubashenko, Adv. Math. 108, 205-262 (1994).
The powers of usual $a,b,c,d$ generators form a commutative Hopf subalgebra and the duality relation is explained in detail (Proposition 1.4 of ref. cit.)


A number of general algebraic properties are contained in two papers by Benjamin Enriquez


"Le centre des algèbres de coordonnèes des group quantiques aux racines $p^\alpha$-ièmes de l'unité" Bull. Soc. Math. Fr. 122, 443-485 (1994)


"Integrity, Integral closedness and finiteness over their centers of the coordinate algebras of quantum groups at $p^\nu$-th roots of unity" Ann. Sci. Math. Quebec 19, 21-47 (1995).


The multiparameter cas was considered by 
Costantini-Varagnolo
"Multiparameter quantum function algebras at roots of 1" Math. Ann. 306, 759-780 (1996).


More recently, also,


Costantini "On the quantum function algebra at roots of 1" Comm. Algebra 32, 2377-2383 (2004).


