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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?

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I think that if you take in $Q_q(G)$ elements of the form $X^n$ they form an ideal. Small check - if I take elements from the same column of $Q_q(GL)$ they are q-commuting variables xy=qyx (well-known), so x^n , y^n are central in column-subalgebra. However "duality O(G) and U(g)" is somewhat subtle - it requires completions - since U(g) - is "delta function at identity for O(G)"... These completions may not well respect q^n=1... So whether it is true that "(the set of elements annihilated by ..." - I am not sure – Alexander Chervov Jan 24 '12 at 5:26
By the way, what are finite-dim irreps of $Q_q(GL)$ for q^n=1 ? If we take elements from the same column - they are q-commuting xy=qyx so as a reperesentation we can take 'shift' and 'FT(shift)'. Can we say something interesting about FT (Fourier transform) from quantum groups point of view ? – Alexander Chervov Jan 24 '12 at 5:30

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).

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