Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.06890 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913230957314048 |
|---|---|
| author | Zhou, Fan |
| author_facet | Zhou, Fan |
| contents | In this paper we homologically construct a (functorial) BGG resolution of the finite-dimensional simple module of the nilBrauer algebra by using infinity-categorical methods following the reconstruction-from-stratification philosophy, e.g. appearing in Ayala-Mazel-Gee-Rozenblyum. To do so, we prove a fact of independent interest, that half of the nilBrauer algebra is Koszul. This BGG resolution categorifies a character formula of Brundan-Wang-Webster. More generally, we have a (functorial) ``BGG spectral sequence'' which converges to any desired module; this spectral sequence is secretly a resolution when the desired module is finite-dimensional. This spectral sequence also categorifies the character formulae of Brundan-Wang-Webster for any (possibly infinite-dimensional) simple module. We expect the methods used here for producing BGG resolutions to be applicable to other (graded) triangular-based algebras also, especially diagrammatic ones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06890 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | BGG Resolutions, Koszulity, and Stratifications, Part I: the nilBrauer Algebra Zhou, Fan Representation Theory Quantum Algebra In this paper we homologically construct a (functorial) BGG resolution of the finite-dimensional simple module of the nilBrauer algebra by using infinity-categorical methods following the reconstruction-from-stratification philosophy, e.g. appearing in Ayala-Mazel-Gee-Rozenblyum. To do so, we prove a fact of independent interest, that half of the nilBrauer algebra is Koszul. This BGG resolution categorifies a character formula of Brundan-Wang-Webster. More generally, we have a (functorial) ``BGG spectral sequence'' which converges to any desired module; this spectral sequence is secretly a resolution when the desired module is finite-dimensional. This spectral sequence also categorifies the character formulae of Brundan-Wang-Webster for any (possibly infinite-dimensional) simple module. We expect the methods used here for producing BGG resolutions to be applicable to other (graded) triangular-based algebras also, especially diagrammatic ones. |
| title | BGG Resolutions, Koszulity, and Stratifications, Part I: the nilBrauer Algebra |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2402.06890 |