---
_id: '5'
abstract:
- lang: eng
  text: In this paper, we introduce a quantum version of the wonderful compactification
    of a group as a certain noncommutative projective scheme. Our approach stems from
    the fact that the wonderful compactification encodes the asymptotics of matrix
    coefficients, and from its realization as a GIT quotient of the Vinberg semigroup.
    In order to define the wonderful compactification for a quantum group, we adopt
    a generalized formalism of Proj categories in the spirit of Artin and Zhang. Key
    to our construction is a quantum version of the Vinberg semigroup, which we define
    as a q-deformation of a certain Rees algebra, compatible with a standard Poisson
    structure. Furthermore, we discuss quantum analogues of the stratification of
    the wonderful compactification by orbits for a certain group action, and provide
    explicit computations in the case of SL2.
article_processing_charge: Yes (via OA deal)
author:
- first_name: Iordan V
  full_name: Ganev, Iordan V
  id: 447491B8-F248-11E8-B48F-1D18A9856A87
  last_name: Ganev
citation:
  ama: Ganev IV. The wonderful compactification for quantum groups. <i>Journal of
    the London Mathematical Society</i>. 2019;99(3):778-806. doi:<a href="https://doi.org/10.1112/jlms.12193">10.1112/jlms.12193</a>
  apa: Ganev, I. V. (2019). The wonderful compactification for quantum groups. <i>Journal
    of the London Mathematical Society</i>. Wiley. <a href="https://doi.org/10.1112/jlms.12193">https://doi.org/10.1112/jlms.12193</a>
  chicago: Ganev, Iordan V. “The Wonderful Compactification for Quantum Groups.” <i>Journal
    of the London Mathematical Society</i>. Wiley, 2019. <a href="https://doi.org/10.1112/jlms.12193">https://doi.org/10.1112/jlms.12193</a>.
  ieee: I. V. Ganev, “The wonderful compactification for quantum groups,” <i>Journal
    of the London Mathematical Society</i>, vol. 99, no. 3. Wiley, pp. 778–806, 2019.
  ista: Ganev IV. 2019. The wonderful compactification for quantum groups. Journal
    of the London Mathematical Society. 99(3), 778–806.
  mla: Ganev, Iordan V. “The Wonderful Compactification for Quantum Groups.” <i>Journal
    of the London Mathematical Society</i>, vol. 99, no. 3, Wiley, 2019, pp. 778–806,
    doi:<a href="https://doi.org/10.1112/jlms.12193">10.1112/jlms.12193</a>.
  short: I.V. Ganev, Journal of the London Mathematical Society 99 (2019) 778–806.
date_created: 2018-12-11T11:44:06Z
date_published: 2019-06-01T00:00:00Z
date_updated: 2023-09-19T10:13:08Z
day: '01'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.1112/jlms.12193
external_id:
  isi:
  - '000470025900008'
file:
- access_level: open_access
  checksum: 1be56239b2cd740a0e9a084f773c22f6
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  creator: kschuh
  date_created: 2020-01-07T13:31:53Z
  date_updated: 2020-07-14T12:46:35Z
  file_id: '7238'
  file_name: 2019_Wiley_Ganev.pdf
  file_size: 431754
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file_date_updated: 2020-07-14T12:46:35Z
has_accepted_license: '1'
intvolume: '        99'
isi: 1
issue: '3'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 778-806
publication: Journal of the London Mathematical Society
publication_status: published
publisher: Wiley
publist_id: '8052'
quality_controlled: '1'
scopus_import: '1'
status: public
title: The wonderful compactification for quantum groups
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 99
year: '2019'
...
---
_id: '322'
abstract:
- lang: eng
  text: We construct quantizations of multiplicative hypertoric varieties using an
    algebra of q-difference operators on affine space, where q is a root of unity
    in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the
    multiplicative hypertoric variety and admits an explicit finite étale splitting.
    The global sections of this Azumaya algebra is a hypertoric quantum group, and
    we prove a localization theorem. We introduce a general framework of Frobenius
    quantum moment maps and their Hamiltonian reductions; our results shed light on
    an instance of this framework.
acknowledgement: "National Science Foundation: Graduate Research Fellowship and grant
  No.0932078000; ERC Advanced Grant “Arithmetic and Physics of Higgs moduli spaces”
  No. 320593 \r\nThe author is grateful to David Jordan for suggesting this project
  and providing guidance throughout, particularly for the formulation of Frobenius
  quantum moment maps and key ideas in the proofs of Theorems 3.12 and 4.8. Special
  thanks to David Ben-Zvi (the author's PhD advisor) for numerous discussions and
  constant encouragement, and for suggesting the term ‘hypertoric quantum group.’
  Many results appearing in the current paper were proven independently by Nicholas
  Cooney; the author is grateful to Nicholas for sharing his insight on various topics,
  including Proposition 3.8. The author also thanks Nicholas Proudfoot for relating
  the definition of multiplicative hypertoric varieties, as well as the content of
  Remark 2.14. The author also benefited immensely from the close reading and detailed
  comments of an anonymous referee, and from conversations with Justin Hilburn, Kobi
  Kremnitzer, Michael McBreen, Tom Nevins, Travis Schedler, and Ben Webster. \r\n\r\n\r\n\r\n"
article_processing_charge: No
arxiv: 1
author:
- first_name: Iordan V
  full_name: Ganev, Iordan V
  id: 447491B8-F248-11E8-B48F-1D18A9856A87
  last_name: Ganev
citation:
  ama: Ganev IV. Quantizations of multiplicative hypertoric varieties at a root of
    unity. <i>Journal of Algebra</i>. 2018;506:92-128. doi:<a href="https://doi.org/10.1016/j.jalgebra.2018.03.015">10.1016/j.jalgebra.2018.03.015</a>
  apa: Ganev, I. V. (2018). Quantizations of multiplicative hypertoric varieties at
    a root of unity. <i>Journal of Algebra</i>. World Scientific Publishing. <a href="https://doi.org/10.1016/j.jalgebra.2018.03.015">https://doi.org/10.1016/j.jalgebra.2018.03.015</a>
  chicago: Ganev, Iordan V. “Quantizations of Multiplicative Hypertoric Varieties
    at a Root of Unity.” <i>Journal of Algebra</i>. World Scientific Publishing, 2018.
    <a href="https://doi.org/10.1016/j.jalgebra.2018.03.015">https://doi.org/10.1016/j.jalgebra.2018.03.015</a>.
  ieee: I. V. Ganev, “Quantizations of multiplicative hypertoric varieties at a root
    of unity,” <i>Journal of Algebra</i>, vol. 506. World Scientific Publishing, pp.
    92–128, 2018.
  ista: Ganev IV. 2018. Quantizations of multiplicative hypertoric varieties at a
    root of unity. Journal of Algebra. 506, 92–128.
  mla: Ganev, Iordan V. “Quantizations of Multiplicative Hypertoric Varieties at a
    Root of Unity.” <i>Journal of Algebra</i>, vol. 506, World Scientific Publishing,
    2018, pp. 92–128, doi:<a href="https://doi.org/10.1016/j.jalgebra.2018.03.015">10.1016/j.jalgebra.2018.03.015</a>.
  short: I.V. Ganev, Journal of Algebra 506 (2018) 92–128.
corr_author: '1'
date_created: 2018-12-11T11:45:49Z
date_published: 2018-07-15T00:00:00Z
date_updated: 2025-04-14T09:12:46Z
day: '15'
department:
- _id: TaHa
doi: 10.1016/j.jalgebra.2018.03.015
ec_funded: 1
external_id:
  arxiv:
  - '1412.7211'
  isi:
  - '000433270600005'
intvolume: '       506'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1412.7211
month: '07'
oa: 1
oa_version: Preprint
page: 92 - 128
project:
- _id: 25E549F4-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '320593'
  name: Arithmetic and physics of Higgs moduli spaces
publication: Journal of Algebra
publication_status: published
publisher: World Scientific Publishing
publist_id: '7543'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Quantizations of multiplicative hypertoric varieties at a root of unity
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 506
year: '2018'
...