Noah Snyder gave a great answer to this 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?

Noah Snyder gave a great answer to this 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?