Coideal subalgebras of quantum $SL_2$ at roots of unity

TL;DR

This work provides a complete classification of right coideal subalgebras for the small quantum group $ar{U}_q(rak{sl}_2)$ and its dual $ar{O}_q(SL_2)$ at odd roots of unity, giving explicit generators and defining relations for all families. By developing a lifting framework for finite quantum linear spaces and employing a partial-derivative technique, the authors describe coideals in terms of intersections with grouplike elements and skew-primitive generators, identifying Taft-like and other structured algebras. The duality between $U$ and $U^*$ (via Masuoka–Skryabin) yields parallel classifications on $ar{O}_q(SL_2)$, with several families explicitly computed and semisimplicity criteria established through minimal polynomials. Additional remarks address normal coideal subalgebras, Hopf-automorphism orbits, and implications for semisimplicity in coideal settings and for simple objects in representation categories. Overall, the paper advances understanding of quantum homogeneous spaces at roots of unity by providing concrete, generator-relations descriptions and highlighting structural parallels across dual Hopf algebras.

Abstract

We classify right coideal subalgebras of the finite-dimensional quotient of the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$ and that of the quantized coordinate algebra $\mathcal{O}_q(SL_2)$ at a root of unity $q$ of odd order. All those coideal subalgebras are described by generators and relations.

Figures (1)

• Figure 1: Lattice of coideal subalgebras of $\overline{U}_q(\mathfrak{sl}_2)$

Theorems & Definitions (73)

• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Example 3.4
• Example 3.5