---
_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. <i>Transformation Groups</i>.
    2020;25:1371-1385. doi:<a href="https://doi.org/10.1007/s00031-020-09572-6">10.1007/s00031-020-09572-6</a>
  apa: Yang, Y., &#38; Zhao, G. (2020). The PBW theorem for affine Yangians. <i>Transformation
    Groups</i>. Springer Nature. <a href="https://doi.org/10.1007/s00031-020-09572-6">https://doi.org/10.1007/s00031-020-09572-6</a>
  chicago: Yang, Yaping, and Gufang Zhao. “The PBW Theorem for Affine Yangians.” <i>Transformation
    Groups</i>. Springer Nature, 2020. <a href="https://doi.org/10.1007/s00031-020-09572-6">https://doi.org/10.1007/s00031-020-09572-6</a>.
  ieee: Y. Yang and G. Zhao, “The PBW theorem for affine Yangians,” <i>Transformation
    Groups</i>, 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.” <i>Transformation
    Groups</i>, vol. 25, Springer Nature, 2020, pp. 1371–85, doi:<a href="https://doi.org/10.1007/s00031-020-09572-6">10.1007/s00031-020-09572-6</a>.
  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: 2025-07-10T11:54:50Z
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:
  - 1531-586X
  issn:
  - 1083-4362
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2020'
...
---
OA_place: repository
OA_type: green
_id: '19987'
abstract:
- lang: eng
  text: 'These lecture notes are based on Yang’s talk at the MATRIX program Geometric
    R-Matrices: from Geometry to Probability, at the University of Melbourne, Dec.
    18–22, 2017, and Zhao’s talk at Perimeter Institute for Theoretical Physics in
    January 2018. We give an introductory survey of the results in Yang and Zhao (Quiver
    varieties and elliptic quantum groups, 2017. arxiv1708.01418). We discuss a sheafified
    elliptic quantum group associated to any symmetric Kac-Moody Lie algebra. The
    sheafification is obtained by applying the equivariant elliptic cohomological
    theory to the moduli space of representations of a preprojective algebra. By construction,
    the elliptic quantum group naturally acts on the equivariant elliptic cohomology
    of Nakajima quiver varieties. As an application, we obtain a relation between
    the sheafified elliptic quantum group and the global affine Grassmannian over
    an elliptic curve.'
acknowledgement: 'Y.Y. would like to thank the organizers of the MATRIX program Geometric
  R-Matrices: from Geometry to Probability for their kind invitation, and many participants
  of the program for useful discussions, including Vassily Gorbounov, Andrei Okounkov,
  Allen Knutson, Hitoshi Konno, Paul Zinn-Justin. Proposition 1 and Sect. 3.3 are
  new, for which we thank Hitoshi Konno for interesting discussions and communications.
  These notes were written when both authors were visiting the Perimeter Institute
  for Theoretical Physics (PI). We are grateful to PI for the hospitality.'
alternative_title:
- MATRIX Book Series
article_processing_charge: No
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. How to Sheafify an Elliptic Quantum Group. In: <i>2017 MATRIX
    Annals</i>. Vol 2. MXBS. Cham: Springer International Publishing; 2019:675-691.
    doi:<a href="https://doi.org/10.1007/978-3-030-04161-8_54">10.1007/978-3-030-04161-8_54</a>'
  apa: 'Yang, Y., &#38; Zhao, G. (2019). How to Sheafify an Elliptic Quantum Group.
    In <i>2017 MATRIX Annals</i> (Vol. 2, pp. 675–691). Cham: Springer International
    Publishing. <a href="https://doi.org/10.1007/978-3-030-04161-8_54">https://doi.org/10.1007/978-3-030-04161-8_54</a>'
  chicago: 'Yang, Yaping, and Gufang Zhao. “How to Sheafify an Elliptic Quantum Group.”
    In <i>2017 MATRIX Annals</i>, 2:675–91. MXBS. Cham: Springer International Publishing,
    2019. <a href="https://doi.org/10.1007/978-3-030-04161-8_54">https://doi.org/10.1007/978-3-030-04161-8_54</a>.'
  ieee: 'Y. Yang and G. Zhao, “How to Sheafify an Elliptic Quantum Group,” in <i>2017
    MATRIX Annals</i>, vol. 2, Cham: Springer International Publishing, 2019, pp.
    675–691.'
  ista: 'Yang Y, Zhao G. 2019.How to Sheafify an Elliptic Quantum Group. In: 2017
    MATRIX Annals. MATRIX Book Series, vol. 2, 675–691.'
  mla: Yang, Yaping, and Gufang Zhao. “How to Sheafify an Elliptic Quantum Group.”
    <i>2017 MATRIX Annals</i>, vol. 2, Springer International Publishing, 2019, pp.
    675–91, doi:<a href="https://doi.org/10.1007/978-3-030-04161-8_54">10.1007/978-3-030-04161-8_54</a>.
  short: Y. Yang, G. Zhao, in:, 2017 MATRIX Annals, Springer International Publishing,
    Cham, 2019, pp. 675–691.
date_created: 2025-07-10T13:31:38Z
date_published: 2019-03-25T00:00:00Z
date_updated: 2025-09-23T11:59:52Z
day: '25'
department:
- _id: TaHa
doi: 10.1007/978-3-030-04161-8_54
external_id:
  arxiv:
  - '1803.06627'
intvolume: '         2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1803.06627
month: '03'
oa: 1
oa_version: Preprint
page: 675-691
place: Cham
publication: 2017 MATRIX Annals
publication_identifier:
  eisbn:
  - '9783030041618'
  eissn:
  - 2523-305X
  isbn:
  - '9783030041601'
  issn:
  - 2523-3041
publication_status: published
publisher: Springer International Publishing
quality_controlled: '1'
series_title: MXBS
status: public
title: How to Sheafify an Elliptic Quantum Group
type: book_chapter
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2
year: '2019'
...