1. Prescribed duality dynamics in comodule categories
- Author
-
Chirvasitu, Alexandru
- Subjects
Mathematics - Rings and Algebras ,Mathematics - Combinatorics ,Mathematics - Category Theory ,Mathematics - Quantum Algebra ,16S10, 16T05, 16T15, 16T30, 18A40, 18M05 - 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 $\mathcal{L}_r$., Comment: 13 pages + references
- Published
- 2024