Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908491140038656 |
|---|---|
| author | Chen, Chih-Whi Cheng, Shun-Jen Mazorchuk, Volodymyr |
| author_facet | Chen, Chih-Whi Cheng, Shun-Jen Mazorchuk, Volodymyr |
| contents | We study various categories of Whittaker modules over a type I Lie superalgebra realized as cokernel categories that fit into the framework of properly stratified categories. These categories are the target of the Backelin functor $Γ_ζ$. We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category $\mathcal O$ and of certain singular categories of Harish-Chandra $(\mathfrak g,\mathfrak g_{\bar 0})$-bimodules. We also show that $Γ_ζ$ is a realization of the Serre quotient functor. We further investigate a $q$-symmetrized Fock space over a quantum group of type A and prove that, for general linear Lie superalgebras our Whittaker categories, the functor $Γ_ζ$ and various realizations of Serre quotients and Serre quotient functors categorify this $q$-symmetrized Fock space and its $q$-symmetrizer. In this picture, the canonical and dual canonical bases in this $q$-symmetrized Fock space correspond to tilting and simple objects in these Whittaker categories, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_00541 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras Chen, Chih-Whi Cheng, Shun-Jen Mazorchuk, Volodymyr Representation Theory 17B10, 17B55 We study various categories of Whittaker modules over a type I Lie superalgebra realized as cokernel categories that fit into the framework of properly stratified categories. These categories are the target of the Backelin functor $Γ_ζ$. We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category $\mathcal O$ and of certain singular categories of Harish-Chandra $(\mathfrak g,\mathfrak g_{\bar 0})$-bimodules. We also show that $Γ_ζ$ is a realization of the Serre quotient functor. We further investigate a $q$-symmetrized Fock space over a quantum group of type A and prove that, for general linear Lie superalgebras our Whittaker categories, the functor $Γ_ζ$ and various realizations of Serre quotients and Serre quotient functors categorify this $q$-symmetrized Fock space and its $q$-symmetrizer. In this picture, the canonical and dual canonical bases in this $q$-symmetrized Fock space correspond to tilting and simple objects in these Whittaker categories, respectively. |
| title | Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras |
| topic | Representation Theory 17B10, 17B55 |
| url | https://arxiv.org/abs/2203.00541 |