Prescribed duality dynamics in comodule categories

TL;DR

This paper constructs specific Hopf algebras with non-bijective antipodes that cannot be derived from bijective antipode Hopf algebras, revealing new structural insights into comodule categories.

Contribution

It demonstrates the existence of Hopf algebras with surjective, non-bijective antipodes that are not quotients of Hopf algebras with bijective antipodes, and introduces a method to realize arbitrary subspace lattice structures as comodule lattices.

Findings

Existence of Hopf algebras with surjective, non-bijective antipodes and no non-trivial morphisms from bijective antipode Hopf algebras.

Construction of Hopf algebras realizing prescribed subspace lattice structures as comodule lattices.

Contrasts with the dual case where injective antipodes embed into bijective antipode Hopf algebras.

Abstract

We prove that there exist Hopf algebras with surjective, non-bijective antipode which admit no non-trivial morphisms from Hopf algebras with bijective antipode; in particular, they are not quotients of such. This answers a question left open in prior work, and contrasts with the dual setup whereby a Hopf algebra has injective antipode precisely when it embeds into one with bijective antipode. The examples rely on the broader phenomenon of realizing pre-specified subspace lattices as comodule lattices: for a finite-dimensional vector space $V$ and a sequence $(\mathcal{L}_r)_r$ of successively finer lattices of subspaces thereof, assuming the minimal subquotients of the supremum $\bigvee_r \mathcal{L}_r$ are all at least 2-dimensional, there is a Hopf algebra equipping $V$ with a comodule structure in such a fashion that the lattice of comodules of the $r^{th}$ dual comodule $V^{r*}$ is precisely the given $\mathcal{L}_r$.