A Categorification of Subword Complexes and Its Hall Algebra
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912995587653632 |
|---|---|
| author | Gorsky, Mikhail Li, Zijun |
| author_facet | Gorsky, Mikhail Li, Zijun |
| contents | Bergeron and Ceballos defined a Hopf algebra structure on equivalence classes of subword complexes. We introduce a category of subword complexes, endow it with a proto-exact-like structure, and show that the corresponding dual Hall Hopf algebra is isomorphic to the algebra of Bergeron-Ceballos. We prove that the full subcategory of root-independent objects is proto-abelian in the sense of Dyckerhoff. We give a categorical lift of flips in subword complexes.
We consider a version of a category of formal direct sums of subobjects for a root-independent subword complex and interpret it in terms of quivers. If the corresponding quiver is a tree, the category is endowed with a proto-exact structure. We show that its Hall algebra is isomorphic to the Hall algebra of the category of representations of this quiver over $\mathbb{F}_1$. Under certain conditions, a flip corresponds to changing a proto-exact structure while keeping the category the same up to isomorphism, which corresponds to a non-trivial automorphism of the Hall algebra. In type $A$, this leads to a realization of the nilpotent part of the universal enveloping algebra and its automorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00879 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Categorification of Subword Complexes and Its Hall Algebra Gorsky, Mikhail Li, Zijun Representation Theory Combinatorics Category Theory Bergeron and Ceballos defined a Hopf algebra structure on equivalence classes of subword complexes. We introduce a category of subword complexes, endow it with a proto-exact-like structure, and show that the corresponding dual Hall Hopf algebra is isomorphic to the algebra of Bergeron-Ceballos. We prove that the full subcategory of root-independent objects is proto-abelian in the sense of Dyckerhoff. We give a categorical lift of flips in subword complexes. We consider a version of a category of formal direct sums of subobjects for a root-independent subword complex and interpret it in terms of quivers. If the corresponding quiver is a tree, the category is endowed with a proto-exact structure. We show that its Hall algebra is isomorphic to the Hall algebra of the category of representations of this quiver over $\mathbb{F}_1$. Under certain conditions, a flip corresponds to changing a proto-exact structure while keeping the category the same up to isomorphism, which corresponds to a non-trivial automorphism of the Hall algebra. In type $A$, this leads to a realization of the nilpotent part of the universal enveloping algebra and its automorphisms. |
| title | A Categorification of Subword Complexes and Its Hall Algebra |
| topic | Representation Theory Combinatorics Category Theory |
| url | https://arxiv.org/abs/2604.00879 |