The nodal cubic and quantum groups at roots of unity

TL;DR

This paper explores the structure of quantum groups at roots of unity through the coordinate ring of the nodal cubic, revealing its cleft coalgebra Galois extension and minimal quotient extensions.

Contribution

It introduces a new connection between the coordinate ring of the nodal cubic and quantum groups at roots of unity, detailing the Galois extension structure.

Findings

The coalgebra Galois extension is shown to be cleft.

Minimal quotient extensions are explicitly determined.

The coordinate ring of the nodal cubic is characterized as a quantum homogeneous space.

Abstract

In a recent article, the coordinate ring of the nodal cubic was given the structure of a quantum homogeneous space. Here the corresponding coalgebra Galois extension is expressed in terms of quantum groups at roots of unity, and is shown to be cleft. Furthermore, the minimal quotient extensions are determined.