Prescribed duality dynamics in comodule categories

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 $V^{r\ast }$ is precisely the given ${\mathcal{L}}_r$.

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