Uspekhi Matematicheskikh Nauk
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor
Subscription
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Uspekhi Mat. Nauk, 2020, Volume 75, Issue 5(455), Pages 3–58 (Mi umn9959)  

This article is cited in 1 scientific paper (total in 1 paper)

Yang–Baxter algebras, convolution algebras, and Grassmannians

V. G. Gorbunovabc, C. Korffd, C. Stroppele

a Institute of Mathematics, University of Aberdeen, UK
b National Research University Higher School of Economics
c Moscow Institute of Physics and Technology (National Research University), Laboratory of Algebraic Geometry and Homological Algebra
d School of Mathematics and Statistics, Glasgow University, UK
e Hausdorff Center of Mathematics, University of Bonn, Germany

Abstract: This paper surveys a new actively developing direction in contemporary mathematics which connects quantum integrable models with the Schubert calculus for quiver varieties: there is a purely geometric construction of solutions to the Yang–Baxter equation and their associated Yang–Baxter algebras which play a central role in quantum integrable systems and exactly solvable (integrable) lattice models in statistical physics. A simple but explicit example is given using the classical geometry of Grassmannians in order to explain some of the main ideas. The degenerate five-vertex limit of the asymmetric six-vertex model is considered, and its associated Yang–Baxter algebra is identified with a convolution algebra arising from the equivariant Schubert calculus of Grassmannians. It is also shown how our methods can be used to construct quotients of the universal enveloping algebra of the current algebra $\mathfrak{gl}_2[t]$ (so-called Schur-type algebras) acting on the tensor product of copies of its evaluation representation $\mathbb{C}^2[t]$. Finally, our construction is connected with the cohomological Hall algebra for the $A_1$-quiver.
Bibliography: 125 titles.

Keywords: quantum integrable systems, quiver varieties, quantum cohomologies.

Funding Agency Grant Number
Russian Science Foundation 20-61-46005
Ministry of Education and Science of the Russian Federation 5-100
The research of the first author was supported by the Russian Science Foundation under grant no. 20-61-46005 and the Russian Academic Excellence Project ‘5-100’).


DOI: https://doi.org/10.4213/rm9959

Full text: PDF file (1116 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2020, 75:5, 791–842

Bibliographic databases:

UDC: 515.16+512.734
MSC: Primary 16T25; Secondary 14M15, 16G20, 81R12
Received: 04.06.2020

Citation: V. G. Gorbunov, C. Korff, C. Stroppel, “Yang–Baxter algebras, convolution algebras, and Grassmannians”, Uspekhi Mat. Nauk, 75:5(455) (2020), 3–58; Russian Math. Surveys, 75:5 (2020), 791–842

Citation in format AMSBIB
\Bibitem{GorKorStr20}
\by V.~G.~Gorbunov, C.~Korff, C.~Stroppel
\paper Yang--Baxter algebras, convolution algebras, and Grassmannians
\jour Uspekhi Mat. Nauk
\yr 2020
\vol 75
\issue 5(455)
\pages 3--58
\mathnet{http://mi.mathnet.ru/umn9959}
\crossref{https://doi.org/10.4213/rm9959}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4154847}
\elib{https://elibrary.ru/item.asp?id=44982250}
\transl
\jour Russian Math. Surveys
\yr 2020
\vol 75
\issue 5
\pages 791--842
\crossref{https://doi.org/10.1070/RM9959}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000613189200001}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85099945300}


Linking options:
  • http://mi.mathnet.ru/eng/umn9959
  • https://doi.org/10.4213/rm9959
  • http://mi.mathnet.ru/eng/umn/v75/i5/p3

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. D. V. Talalaev, “Tetrahedron equation: algebra, topology, and integrability”, Russian Math. Surveys, 76:4 (2021), 685–721  mathnet  crossref  crossref  isi
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:244
    References:22
    First page:37

     
    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021