Prescribed duality dynamics in comodule categories
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911989870100480 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | 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 $\mathcal{L}_r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08167 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Prescribed duality dynamics in comodule categories Chirvasitu, Alexandru Rings and Algebras Combinatorics Category Theory Quantum Algebra 16S10, 16T05, 16T15, 16T30, 18A40, 18M05 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 $\mathcal{L}_r$. |
| title | Prescribed duality dynamics in comodule categories |
| topic | Rings and Algebras Combinatorics Category Theory Quantum Algebra 16S10, 16T05, 16T15, 16T30, 18A40, 18M05 |
| url | https://arxiv.org/abs/2408.08167 |