The finite dual of commutative-by-finite Hopf algebras

TL;DR

This paper investigates the structure of the finite dual of affine commutative-by-finite Hopf algebras, revealing conditions for its decomposition and analyzing its components using classical theorems.

Contribution

It provides a natural decomposition framework for the finite dual of such Hopf algebras, extending classical theorems to new algebraic structures.

Findings

Decomposition of the finite dual as a crossed or smash product.

Application of Cartier-Gabriel-Kostant theorem to analyze components.

Detailed analysis of examples illustrating the theoretical results.

Abstract

The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra $H$ is studied. Such a Hopf algebra $H$ is an extension of an affine commutative Hopf algebra $A$ by a finite dimensional Hopf algebra $F$. The main theorem gives natural conditions under which $H^{\circ}$ decomposes as a crossed or smash product of $F^{\ast}$ by the finite dual of $A$. This decomposition is then further analysed using the Cartier- Gabriel-Kostant theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the classical components of $A^{\circ}$. The detailed consequences for a number of families of examples are then studied.