---
_id: '12303'
abstract:
- lang: eng
text: We construct for each choice of a quiver Q, a cohomology theory A, and a poset
P a “loop Grassmannian” GP(Q,A). This generalizes loop Grassmannians of semisimple
groups and the loop Grassmannians of based quadratic forms. The addition of a
“dilation” torus D⊆G2m gives a quantization GPD(Q,A). This construction is motivated
by the program of introducing an inner cohomology theory in algebraic geometry
adequate for the Geometric Langlands program (Mirković, Some extensions of the
notion of loop Grassmannians. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardešić
issue. No. 532, 53–74, 2017) and on the construction of affine quantum groups
from generalized cohomology theories (Yang and Zhao, Quiver varieties and elliptic
quantum groups, preprint. arxiv1708.01418).
acknowledgement: I.M. thanks Zhijie Dong for long-term discussions on the material
that entered this work. We thank Misha Finkelberg for pointing out errors in earlier
versions. His advice and his insistence have led to a much better paper. A part
of the writing was done at the conference at IST (Vienna) attended by all coauthors.
We therefore thank the organizers of the conference and the support of ERC Advanced
Grant Arithmetic and Physics of Higgs moduli spaces No. 320593. The work of I.M.
was partially supported by NSF grants. The work of Y.Y. was partially supported
by the Australian Research Council (ARC) via the award DE190101231. The work of
G.Z. was partially supported by ARC via the award DE190101222.
alternative_title:
- Trends in Mathematics
article_processing_charge: No
arxiv: 1
author:
- first_name: Ivan
full_name: Mirković, Ivan
last_name: Mirković
- first_name: Yaping
full_name: Yang, Yaping
last_name: Yang
- first_name: Gufang
full_name: Zhao, Gufang
id: 2BC2AC5E-F248-11E8-B48F-1D18A9856A87
last_name: Zhao
citation:
ama: 'Mirković I, Yang Y, Zhao G. Loop Grassmannians of Quivers and Affine Quantum
Groups. In: Baranovskky V, Guay N, Schedler T, eds. Representation Theory and
Algebraic Geometry. 1st ed. TM. Cham: Springer Nature; Birkhäuser; 2022:347-392.
doi:10.1007/978-3-030-82007-7_8'
apa: 'Mirković, I., Yang, Y., & Zhao, G. (2022). Loop Grassmannians of Quivers
and Affine Quantum Groups. In V. Baranovskky, N. Guay, & T. Schedler (Eds.),
Representation Theory and Algebraic Geometry (1st ed., pp. 347–392). Cham:
Springer Nature; Birkhäuser. https://doi.org/10.1007/978-3-030-82007-7_8'
chicago: 'Mirković, Ivan, Yaping Yang, and Gufang Zhao. “Loop Grassmannians of Quivers
and Affine Quantum Groups.” In Representation Theory and Algebraic Geometry,
edited by Vladimir Baranovskky, Nicolas Guay, and Travis Schedler, 1st ed., 347–92.
TM. Cham: Springer Nature; Birkhäuser, 2022. https://doi.org/10.1007/978-3-030-82007-7_8.'
ieee: 'I. Mirković, Y. Yang, and G. Zhao, “Loop Grassmannians of Quivers and Affine
Quantum Groups,” in Representation Theory and Algebraic Geometry, 1st ed.,
V. Baranovskky, N. Guay, and T. Schedler, Eds. Cham: Springer Nature; Birkhäuser,
2022, pp. 347–392.'
ista: 'Mirković I, Yang Y, Zhao G. 2022.Loop Grassmannians of Quivers and Affine
Quantum Groups. In: Representation Theory and Algebraic Geometry. Trends in Mathematics,
, 347–392.'
mla: Mirković, Ivan, et al. “Loop Grassmannians of Quivers and Affine Quantum Groups.”
Representation Theory and Algebraic Geometry, edited by Vladimir Baranovskky
et al., 1st ed., Springer Nature; Birkhäuser, 2022, pp. 347–92, doi:10.1007/978-3-030-82007-7_8.
short: I. Mirković, Y. Yang, G. Zhao, in:, V. Baranovskky, N. Guay, T. Schedler
(Eds.), Representation Theory and Algebraic Geometry, 1st ed., Springer Nature;
Birkhäuser, Cham, 2022, pp. 347–392.
date_created: 2023-01-16T10:06:41Z
date_published: 2022-06-16T00:00:00Z
date_updated: 2023-01-27T07:07:31Z
day: '16'
department:
- _id: TaHa
doi: 10.1007/978-3-030-82007-7_8
ec_funded: 1
edition: '1'
editor:
- first_name: Vladimir
full_name: Baranovskky, Vladimir
last_name: Baranovskky
- first_name: Nicolas
full_name: Guay, Nicolas
last_name: Guay
- first_name: Travis
full_name: Schedler, Travis
last_name: Schedler
external_id:
arxiv:
- '1810.10095'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.1810.10095
month: '06'
oa: 1
oa_version: Preprint
page: 347-392
place: Cham
project:
- _id: 25E549F4-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '320593'
name: Arithmetic and physics of Higgs moduli spaces
publication: Representation Theory and Algebraic Geometry
publication_identifier:
eisbn:
- '9783030820077'
eissn:
- 2297-024X
isbn:
- '9783030820060'
issn:
- 2297-0215
publication_status: published
publisher: Springer Nature; Birkhäuser
quality_controlled: '1'
scopus_import: '1'
series_title: TM
status: public
title: Loop Grassmannians of Quivers and Affine Quantum Groups
type: book_chapter
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2022'
...
---
_id: '7940'
abstract:
- lang: eng
text: We prove that the Yangian associated to an untwisted symmetric affine Kac–Moody
Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter
is constructed in [YZ14] as an algebraic formalism of cohomological Hall algebras.
As a consequence, we obtain the Poincare–Birkhoff–Witt (PBW) theorem for this
class of affine Yangians. Another independent proof of the PBW theorem is given
recently by Guay, Regelskis, and Wendlandt [GRW18].
acknowledgement: Gufang Zhao is affiliated to IST Austria, Hausel group until July
of 2018. Supported by the Advanced Grant Arithmetic and Physics of Higgs moduli
spaces No. 320593 of the European Research Council.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Yaping
full_name: Yang, Yaping
id: 360D8648-F248-11E8-B48F-1D18A9856A87
last_name: Yang
- first_name: Gufang
full_name: Zhao, Gufang
id: 2BC2AC5E-F248-11E8-B48F-1D18A9856A87
last_name: Zhao
citation:
ama: Yang Y, Zhao G. The PBW theorem for affine Yangians. Transformation Groups.
2020;25:1371-1385. doi:10.1007/s00031-020-09572-6
apa: Yang, Y., & Zhao, G. (2020). The PBW theorem for affine Yangians. Transformation
Groups. Springer Nature. https://doi.org/10.1007/s00031-020-09572-6
chicago: Yang, Yaping, and Gufang Zhao. “The PBW Theorem for Affine Yangians.” Transformation
Groups. Springer Nature, 2020. https://doi.org/10.1007/s00031-020-09572-6.
ieee: Y. Yang and G. Zhao, “The PBW theorem for affine Yangians,” Transformation
Groups, vol. 25. Springer Nature, pp. 1371–1385, 2020.
ista: Yang Y, Zhao G. 2020. The PBW theorem for affine Yangians. Transformation
Groups. 25, 1371–1385.
mla: Yang, Yaping, and Gufang Zhao. “The PBW Theorem for Affine Yangians.” Transformation
Groups, vol. 25, Springer Nature, 2020, pp. 1371–85, doi:10.1007/s00031-020-09572-6.
short: Y. Yang, G. Zhao, Transformation Groups 25 (2020) 1371–1385.
date_created: 2020-06-07T22:00:55Z
date_published: 2020-12-01T00:00:00Z
date_updated: 2023-08-21T07:06:21Z
day: '01'
department:
- _id: TaHa
doi: 10.1007/s00031-020-09572-6
ec_funded: 1
external_id:
arxiv:
- '1804.04375'
isi:
- '000534874300003'
intvolume: ' 25'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1804.04375
month: '12'
oa: 1
oa_version: Preprint
page: 1371-1385
project:
- _id: 25E549F4-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '320593'
name: Arithmetic and physics of Higgs moduli spaces
publication: Transformation Groups
publication_identifier:
eissn:
- 1531586X
issn:
- '10834362'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: The PBW theorem for affine Yangians
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 25
year: '2020'
...
---
_id: '8539'
abstract:
- lang: eng
text: Cohomological and K-theoretic stable bases originated from the study of quantum
cohomology and quantum K-theory. Restriction formula for cohomological stable
bases played an important role in computing the quantum connection of cotangent
bundle of partial flag varieties. In this paper we study the K-theoretic stable
bases of cotangent bundles of flag varieties. We describe these bases in terms
of the action of the affine Hecke algebra and the twisted group algebra of KostantKumar.
Using this algebraic description and the method of root polynomials, we give a
restriction formula of the stable bases. We apply it to obtain the restriction
formula for partial flag varieties. We also build a relation between the stable
basis and the Casselman basis in the principal series representations of the Langlands
dual group. As an application, we give a closed formula for the transition matrix
between Casselman basis and the characteristic functions.
- lang: fre
text: "Les bases stables cohomologiques et K-théoriques proviennent de l’étude de
la cohomologie quantique et de la K-théorie quantique. La formule de restriction
pour les bases stables cohomologiques a joué un rôle important dans le calcul
de la connexion quantique du fibré cotangent de variétés de drapeaux partielles.
Dans cet article, nous étudions les bases stables K-théoriques de fibré cotangents
des variétés de drapeaux. Nous décrivons ces bases en fonction de l’action de
l’algèbre de Hecke affine et de l’algèbre de Kostant-Kumar. En utilisant cette
description algébrique et la méthode des polynômes de racine, nous donnons une
formule de restriction des bases stables. Nous l’appliquons\r\npour obtenir la
formule de restriction pour les variétés de drapeaux partielles. Nous construisons
également une relation entre la base stable et la base de Casselman dans les représentations
de la série principale du groupe dual de Langlands p-adique. Comme une application,
nous donnons une formule close pour la matrice de transition entre la base de
Casselman et les fonctions caractéristiques. "
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: C.
full_name: Su, C.
last_name: Su
- first_name: Gufang
full_name: Zhao, Gufang
id: 2BC2AC5E-F248-11E8-B48F-1D18A9856A87
last_name: Zhao
- first_name: C.
full_name: Zhong, C.
last_name: Zhong
citation:
ama: Su C, Zhao G, Zhong C. On the K-theory stable bases of the springer resolution.
Annales Scientifiques de l’Ecole Normale Superieure. 2020;53(3):663-671.
doi:10.24033/asens.2431
apa: Su, C., Zhao, G., & Zhong, C. (2020). On the K-theory stable bases of the
springer resolution. Annales Scientifiques de l’Ecole Normale Superieure.
Société Mathématique de France. https://doi.org/10.24033/asens.2431
chicago: Su, C., Gufang Zhao, and C. Zhong. “On the K-Theory Stable Bases of the
Springer Resolution.” Annales Scientifiques de l’Ecole Normale Superieure.
Société Mathématique de France, 2020. https://doi.org/10.24033/asens.2431.
ieee: C. Su, G. Zhao, and C. Zhong, “On the K-theory stable bases of the springer
resolution,” Annales Scientifiques de l’Ecole Normale Superieure, vol.
53, no. 3. Société Mathématique de France, pp. 663–671, 2020.
ista: Su C, Zhao G, Zhong C. 2020. On the K-theory stable bases of the springer
resolution. Annales Scientifiques de l’Ecole Normale Superieure. 53(3), 663–671.
mla: Su, C., et al. “On the K-Theory Stable Bases of the Springer Resolution.” Annales
Scientifiques de l’Ecole Normale Superieure, vol. 53, no. 3, Société Mathématique
de France, 2020, pp. 663–71, doi:10.24033/asens.2431.
short: C. Su, G. Zhao, C. Zhong, Annales Scientifiques de l’Ecole Normale Superieure
53 (2020) 663–671.
date_created: 2020-09-20T22:01:38Z
date_published: 2020-06-01T00:00:00Z
date_updated: 2023-08-22T09:27:57Z
day: '01'
department:
- _id: TaHa
doi: 10.24033/asens.2431
external_id:
arxiv:
- '1708.08013'
isi:
- '000592182600004'
intvolume: ' 53'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1708.08013
month: '06'
oa: 1
oa_version: Preprint
page: 663-671
publication: Annales Scientifiques de l'Ecole Normale Superieure
publication_identifier:
issn:
- 0012-9593
publication_status: published
publisher: Société Mathématique de France
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the K-theory stable bases of the springer resolution
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 53
year: '2020'
...
---
_id: '7004'
abstract:
- lang: eng
text: We define an action of the (double of) Cohomological Hall algebra of Kontsevich
and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov.
We identify this action with the one of the affine Yangian of gl(1). Based on
that we derive the vertex algebra at the corner Wr1,r2,r3 of Gaiotto and Rapčák.
We conjecture that our approach works for a big class of Calabi–Yau categories,
including those associated with toric Calabi–Yau 3-folds.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Miroslav
full_name: Rapcak, Miroslav
last_name: Rapcak
- first_name: Yan
full_name: Soibelman, Yan
last_name: Soibelman
- first_name: Yaping
full_name: Yang, Yaping
last_name: Yang
- first_name: Gufang
full_name: Zhao, Gufang
id: 2BC2AC5E-F248-11E8-B48F-1D18A9856A87
last_name: Zhao
citation:
ama: Rapcak M, Soibelman Y, Yang Y, Zhao G. Cohomological Hall algebras, vertex
algebras and instantons. Communications in Mathematical Physics. 2020;376:1803-1873.
doi:10.1007/s00220-019-03575-5
apa: Rapcak, M., Soibelman, Y., Yang, Y., & Zhao, G. (2020). Cohomological Hall
algebras, vertex algebras and instantons. Communications in Mathematical Physics.
Springer Nature. https://doi.org/10.1007/s00220-019-03575-5
chicago: Rapcak, Miroslav, Yan Soibelman, Yaping Yang, and Gufang Zhao. “Cohomological
Hall Algebras, Vertex Algebras and Instantons.” Communications in Mathematical
Physics. Springer Nature, 2020. https://doi.org/10.1007/s00220-019-03575-5.
ieee: M. Rapcak, Y. Soibelman, Y. Yang, and G. Zhao, “Cohomological Hall algebras,
vertex algebras and instantons,” Communications in Mathematical Physics,
vol. 376. Springer Nature, pp. 1803–1873, 2020.
ista: Rapcak M, Soibelman Y, Yang Y, Zhao G. 2020. Cohomological Hall algebras,
vertex algebras and instantons. Communications in Mathematical Physics. 376, 1803–1873.
mla: Rapcak, Miroslav, et al. “Cohomological Hall Algebras, Vertex Algebras and
Instantons.” Communications in Mathematical Physics, vol. 376, Springer
Nature, 2020, pp. 1803–73, doi:10.1007/s00220-019-03575-5.
short: M. Rapcak, Y. Soibelman, Y. Yang, G. Zhao, Communications in Mathematical
Physics 376 (2020) 1803–1873.
date_created: 2019-11-12T14:01:27Z
date_published: 2020-06-01T00:00:00Z
date_updated: 2023-08-17T14:02:59Z
day: '01'
department:
- _id: TaHa
doi: 10.1007/s00220-019-03575-5
ec_funded: 1
external_id:
arxiv:
- '1810.10402'
isi:
- '000536255500004'
intvolume: ' 376'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1810.10402
month: '06'
oa: 1
oa_version: Preprint
page: 1803-1873
project:
- _id: 25E549F4-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '320593'
name: Arithmetic and physics of Higgs moduli spaces
publication: Communications in Mathematical Physics
publication_identifier:
eissn:
- 1432-0916
issn:
- 0010-3616
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Cohomological Hall algebras, vertex algebras and instantons
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 376
year: '2020'
...
---
_id: '5999'
abstract:
- lang: eng
text: "We introduce for each quiver Q and each algebraic oriented cohomology theory
A, the cohomological Hall algebra (CoHA) of Q, as the A-homology of the moduli
of representations of the preprojective algebra of Q. This generalizes the K-theoretic
Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When A is
the Morava K-theory, we show evidence that this algebra is a candidate for Lusztig's
reformulated conjecture on modular representations of algebraic groups.\r\nWe
construct an action of the preprojective CoHA on the A-homology of Nakajima quiver
varieties. We compare this with the action of the Borel subalgebra of Yangian
when A is the intersection theory. We also give a shuffle algebra description
of this CoHA in terms of the underlying formal group law of A. As applications,
we obtain a shuffle description of the Yangian. "
article_processing_charge: No
arxiv: 1
author:
- first_name: Yaping
full_name: Yang, Yaping
last_name: Yang
- first_name: Gufang
full_name: Zhao, Gufang
id: 2BC2AC5E-F248-11E8-B48F-1D18A9856A87
last_name: Zhao
citation:
ama: Yang Y, Zhao G. The cohomological Hall algebra of a preprojective algebra.
Proceedings of the London Mathematical Society. 2018;116(5):1029-1074.
doi:10.1112/plms.12111
apa: Yang, Y., & Zhao, G. (2018). The cohomological Hall algebra of a preprojective
algebra. Proceedings of the London Mathematical Society. Oxford University
Press. https://doi.org/10.1112/plms.12111
chicago: Yang, Yaping, and Gufang Zhao. “The Cohomological Hall Algebra of a Preprojective
Algebra.” Proceedings of the London Mathematical Society. Oxford University
Press, 2018. https://doi.org/10.1112/plms.12111.
ieee: Y. Yang and G. Zhao, “The cohomological Hall algebra of a preprojective algebra,”
Proceedings of the London Mathematical Society, vol. 116, no. 5. Oxford
University Press, pp. 1029–1074, 2018.
ista: Yang Y, Zhao G. 2018. The cohomological Hall algebra of a preprojective algebra.
Proceedings of the London Mathematical Society. 116(5), 1029–1074.
mla: Yang, Yaping, and Gufang Zhao. “The Cohomological Hall Algebra of a Preprojective
Algebra.” Proceedings of the London Mathematical Society, vol. 116, no.
5, Oxford University Press, 2018, pp. 1029–74, doi:10.1112/plms.12111.
short: Y. Yang, G. Zhao, Proceedings of the London Mathematical Society 116 (2018)
1029–1074.
date_created: 2019-02-14T13:14:22Z
date_published: 2018-05-01T00:00:00Z
date_updated: 2023-09-19T14:37:19Z
day: '01'
department:
- _id: TaHa
doi: 10.1112/plms.12111
external_id:
arxiv:
- '1407.7994'
isi:
- '000431506400001'
intvolume: ' 116'
isi: 1
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1407.7994
month: '05'
oa: 1
oa_version: Preprint
page: 1029-1074
publication: Proceedings of the London Mathematical Society
publication_identifier:
issn:
- 0024-6115
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: The cohomological Hall algebra of a preprojective algebra
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 116
year: '2018'
...