RECORD
Lie algebra actions on module categories for truncated shifted Yangians
Representation Theory · Submitted 23 Mar 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Thus, we regard our action as a categorification of the geometric Satake correspondence.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's induction and restriction functors for Cherednik algebras, but their definition uses different tools. After this general definition, we focus on quiver gauge theories attached to a quiver $\Gamma$. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak{g}_{\Gamma}$ on category $ \mathcal O $ for these Coulomb branch algebras. When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence. To establish this categorical action, we define a new class of "flavoured" KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. | contribution 1.00 | 0.94 | 0.12 | asserted 1.00 |
| s1 | Our functors generalize Bezrukavnikov-Etingof's induction and restriction functors for Cherednik algebras, but their definition uses different tools. | background 0.64 | 0.48 | 0.04 | asserted 1.00 |
| s2 | After this general definition, we focus on quiver gauge theories attached to a quiver $\Gamma$. | method 0.39 | 0.64 | 0.02 | asserted 0.93 |
| s3 | The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak{g}_{\Gamma}$ on category $ \mathcal O $ for these Coulomb branch algebras. | contribution 0.52 | 0.91 | 0.43 | asserted 0.99 |
| s4 | When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. | background 0.46 | 0.21 | 0.08 | asserted 1.00 |
| s5 | Thus, we regard our action as a categorification of the geometric Satake correspondence. | interpretation 1.00 | 0.69 | 0.41 | asserted 0.89 |
| s6 | To establish this categorical action, we define a new class of "flavoured" KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. | contribution 0.51 | 0.85 | 0.06 | asserted 0.99 |
| s7 | We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. | result 0.93 | 0.94 | 0.21 | asserted 1.00 |
| s8 | This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra. | interpretation 0.70 | 0.88 | 0.16 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's induction and restriction functors for Cherednik algebras, but their definition uses different tools. After this general definition, we focus on quiver gauge theories attached to a quiver $\Gamma$. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak{g}_{\Gamma}$ on category $ \mathcal O $ for these Coulomb branch algebras. When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence. To establish this categorical action, we define a new class of "flavoured" KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.
Human review
How the literature uses it
2 arXiv papers cite this in 3 sentences
Papers citing it · 2 sentences without a confident label
Builds on it 1 paper
-
In [ref] , we defined restriction and induction functors between category $$ for Coulomb branches, generalizing the Bezrukavnikov-Etingof [ref] restriction and induction functors between Cherednik algebras.
Symplectic resolutions, symplectic duality, and Coulomb branches · Categorical $_Q$ actions coming from Coulomb branches · 0.92
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.