Hopf algebras with the dual Chevalley property of finite corepresentation type

TL;DR

This work extends the classification of finite-dimensional Hopf algebras with the dual Chevalley property by analyzing finite corepresentation type through the link quiver. It develops two constructions of complete families of non-trivial primitive matrices to capture all arrows in the link quiver and relates arrow counts to dimensions of intersections (C∧D)/(C+D), bridging bicomodule theory with quiver representations. The authors prove that finite corepresentation type is equivalent to coNakayama, and delineate the characteristic-dependent structure: in characteristic zero, finite corepresentation type occurs when H is cosemisimple or when H_(1) is of type A(n,d,μ,q); in positive characteristic, when H is cosemisimple or H_(1) ≅ C_d(n). They show that the link-indecomposable component H_(1) is pointed with a basic-cycle link quiver, and provide explicit examples illustrating both finite and infinite corepresentation types and the non-normality of H_(1) in certain cases. Collectively, the results generalize known classifications for pointed and elementary Hopf algebras and highlight the central role of the dual Chevalley property and comonomial structures in determining corepresentation type.

Abstract

Let $H$ be a finite-dimensional Hopf algebra over an algebraically closed field $\Bbbk$ with the dual Chevalley property. We prove that $H$ is of finite corepresentation type if and only if it is coNakayama, if and only if the link quiver $\mathrm{Q}(H)$ of $H$ is a disjoint union of basic cycles, if and only if the link-indecomposable component $H_{(1)}$ containing $\Bbbk1$ is a pointed Hopf algebra and the link quiver of $H_{(1)}$ is a basic cycle.

Theorems & Definitions (103)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof