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Quantum subgroups of $SL_q(2)$ at roots of unity of arbitrary order

Gaston Andres Garcia, Josefina Vallejos

TL;DR

This work completes the classification of quantum subgroups of $SL_q(2)$ at roots of unity of arbitrary order by describing Hopf algebra quotients of $\mathcal{O}_q(SL_2(\mathbb{C}))$ through two parallel parametrizations. For odd order roots, quotients are in bijection with odd subgroup data, and for even order roots $\ell=2m$ with $m\neq1$, quotients correspond to even subgroup data, each realized via explicit pushouts and exact sequences involving $\mathcal{O}(\Gamma)$ and small quantum groups. Special cases $q=-1$ yield dihedral quotients and $PSL_2$-subgroup constructions; the framework unifies these with the general even/odd-data approach and provides generators/relations for the resulting subgroups. The results extend prior classifications (Müller, AG, Bichon, Chelsea) and deliver a complete, explicit taxonomy of quantum subgroups of $\mathcal{O}_q(SL_2(\mathbb{C}))$, enabling systematic analysis and potential applications to representation theory and noncommutative geometry at roots of unity.

Abstract

We complete the classification of quantum subgroups of $SL_q(2)$ with $q$ a root of unity of arbitrary order, that is, Hopf algebra quotients of the quantum function algebras $\mathcal{O}_{q} (SL_2(\mathbb{C}))$.

Quantum subgroups of $SL_q(2)$ at roots of unity of arbitrary order

TL;DR

This work completes the classification of quantum subgroups of at roots of unity of arbitrary order by describing Hopf algebra quotients of through two parallel parametrizations. For odd order roots, quotients are in bijection with odd subgroup data, and for even order roots with , quotients correspond to even subgroup data, each realized via explicit pushouts and exact sequences involving and small quantum groups. Special cases yield dihedral quotients and -subgroup constructions; the framework unifies these with the general even/odd-data approach and provides generators/relations for the resulting subgroups. The results extend prior classifications (Müller, AG, Bichon, Chelsea) and deliver a complete, explicit taxonomy of quantum subgroups of , enabling systematic analysis and potential applications to representation theory and noncommutative geometry at roots of unity.

Abstract

We complete the classification of quantum subgroups of with a root of unity of arbitrary order, that is, Hopf algebra quotients of the quantum function algebras .
Paper Structure (16 sections, 14 theorems, 39 equations)

This paper contains 16 sections, 14 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hopf algebras
  4. $SL_{2}(\mathbb{C})$, $\mathfrak{sl}_{2}(\mathbb{C})$, $U(\mathfrak{sl}_{2})$ and $\mathcal{O}(SL_2(\mathbb{C}))$
  5. On $PSL_2(\mathbb{C})$ and polyhedral groups
  6. Quantum enveloping algebras $\mathcal{U}_q(\mathfrak{sl}_n)$, for $q$ indeterminate
  7. Quantum groups $\mathcal{O}_q(SL_n(\mathbb{C}))$, for $q$ indeterminate
  8. Small quantum groups
  9. Quantum subgroups of $\mathcal{O}_q(SL_2(\mathbb{C}))$
  10. Quantum subgroups of $\mathcal{O}_{q}(SL_{2}(\mathbb{C}))$, $\text{ord}(q)=\ell$ odd
  11. Construction of quantum subgroups.
  12. Determination of quantum subgroups.
  13. Quantum subgroups of $\mathcal{O}_{-1}(SL_{2}(\mathbb{C}))$
  14. Quantum subgroups of $\mathcal{O}_{q}(SL_{2}(\mathbb{C}))$, ord($q$) even
  15. Construction of quantum subgroups.
  16. ...and 1 more sections

Key Result

Proposition 2.1.2

Let $B \hookrightarrow A$ be an injective morphism of Hopf algebras.

Theorems & Definitions (45)

  • Definition 1.0.1
  • Definition 1.0.2
  • Remark 2.1.1
  • Proposition 2.1.2
  • Proposition 2.1.3
  • Proposition 2.1.4
  • Lemma 2.3.1
  • proof
  • Definition 2.4.1
  • Definition 2.5.1
  • ...and 35 more