---
_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. <i>Representation Theory and
    Algebraic Geometry</i>. 1st ed. TM. Cham: Springer Nature; Birkhäuser; 2022:347-392.
    doi:<a href="https://doi.org/10.1007/978-3-030-82007-7_8">10.1007/978-3-030-82007-7_8</a>'
  apa: 'Mirković, I., Yang, Y., &#38; Zhao, G. (2022). Loop Grassmannians of Quivers
    and Affine Quantum Groups. In V. Baranovskky, N. Guay, &#38; T. Schedler (Eds.),
    <i>Representation Theory and Algebraic Geometry</i> (1st ed., pp. 347–392). Cham:
    Springer Nature; Birkhäuser. <a href="https://doi.org/10.1007/978-3-030-82007-7_8">https://doi.org/10.1007/978-3-030-82007-7_8</a>'
  chicago: 'Mirković, Ivan, Yaping Yang, and Gufang Zhao. “Loop Grassmannians of Quivers
    and Affine Quantum Groups.” In <i>Representation Theory and Algebraic Geometry</i>,
    edited by Vladimir Baranovskky, Nicolas Guay, and Travis Schedler, 1st ed., 347–92.
    TM. Cham: Springer Nature; Birkhäuser, 2022. <a href="https://doi.org/10.1007/978-3-030-82007-7_8">https://doi.org/10.1007/978-3-030-82007-7_8</a>.'
  ieee: 'I. Mirković, Y. Yang, and G. Zhao, “Loop Grassmannians of Quivers and Affine
    Quantum Groups,” in <i>Representation Theory and Algebraic Geometry</i>, 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.”
    <i>Representation Theory and Algebraic Geometry</i>, edited by Vladimir Baranovskky
    et al., 1st ed., Springer Nature; Birkhäuser, 2022, pp. 347–92, doi:<a href="https://doi.org/10.1007/978-3-030-82007-7_8">10.1007/978-3-030-82007-7_8</a>.
  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'
...