Coradically graded Hopf algebras with the dual Chevalley property of tame corepresentation type

TL;DR

This paper characterizes when certain finite-dimensional Hopf algebras with the dual Chevalley property have tame corepresentation type, linking their structure to specific algebraic forms and ideals.

Contribution

It provides a classification of coradically graded Hopf algebras with the dual Chevalley property of tame corepresentation type, identifying their structure as a product involving specific ideals.

Findings

Tame corepresentation type occurs iff the algebra is isomorphic to a specific product form.

Identification of ideals that lead to tame corepresentation type.

Use of link quiver and bosonization methods to analyze Hopf algebra structures.

Abstract

Let $\Bbbk$ be an algebraically closed field of characteristic 0 and $H$ a finite-dimensional Hopf algebra over $\Bbbk$ with the dual Chevalley property. In this paper, we show that $\operatorname{gr}^c(H)$ is of tame corepresentation type if and only if $\operatorname{gr}^c(H)\cong (\Bbbk\langle x,y\rangle/I)^* \times H^\prime$ for some finite-dimensional semisimple Hopf algebra $H^\prime$ and some special ideals $I$. Then, by the method of link quiver and bosonization, we discuss which of the above ideals will occur when $(\Bbbk\langle x,y\rangle/I)^* \times H_0$ is a Hopf algebra of tame corepresentation type under some assumptions.