The Finite Duals of Affine Prime Regular Hopf Algebras of GK-Dimension One

TL;DR

This paper investigates the finite duals of affine prime regular Hopf algebras of GK-dimension one, aiming to construct specific Hopf pairings and understand their dual structures through explicit generators and relations.

Contribution

It provides a detailed computation of finite duals for these algebras and establishes a method to determine Hopf pairings via subalgebras, advancing the understanding of duality in this context.

Findings

Finite duals are explicitly computed with generators and relations.

Hopf pairings are constructed through subalgebras of the duals.

The approach applies to all affine prime regular Hopf algebras of GK-dimension one.

Abstract

This paper is an attempt to construct a special kind of Hopf pairing $\langle-,-\rangle:H^\bullet\otimes H\rightarrow\Bbbk$. Specifically, $H^\bullet$ and $H$ should be both affine, noetherian and of the same GK-dimension. In addition, some properties of them would be dual to each other. We test the ideas in two steps for all the affine prime regular Hopf algebras $H$ of GK-dimension one: 1) We compute the finite duals $H^\circ$ of them, which are given by generators and relations; 2) the Hopf pairings desired are determined by choosing certain Hopf subalgebras $H^\bullet$ of $H^\circ$, where $\langle-,-\rangle$ becomes the evaluation.