Quantum subgroups of GL_{\alpha,\beta}(n)

TL;DR

This paper classifies all Hopf algebra quotients of the quantized coordinate algebra of GL(n) under specific conditions, revealing new non-semisimple Hopf algebras with non-pointed duals.

Contribution

It provides a complete classification of Hopf algebra quotients of O_{a,b}(GL_{n}) for certain parameters, introducing new examples of non-semisimple, non-pointed Hopf algebras.

Findings

Classified all Hopf algebra quotients when a^{-1}b is a primitive l-th root of unity.

Characterized all finite-dimensional quotients when a^{-1}b is not a root of unity.

Discovered new families of non-semisimple, non-pointed Hopf algebras with non-pointed duals.

Abstract

Let a, b be non-zero complex numbers and l an odd natural number bigger that 2. We determine all Hopf algebra quotients of the quantized coordinate algebra O_{a,b}(GL_{n}) when a^{-1}b is a primitive l-th root of unity and a, b satisfy certain mild conditions, and we caracterize all finite-dimensional quotients when a^{-1}b is not a root of unity. As a byproduct we give a new family of non-semisimple and non-pointed Hopf algebras with non-pointed duals which are quotients of O_{a, b}(GL_{n}).