RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



SIGMA:
Year:
Volume:
Issue:
Page:
Find






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


SIGMA, 2008, Volume 4, 070, 21 pp. (Mi sigma323)  

This article is cited in 7 scientific papers (total in 7 papers)

The PBW Filtration, Demazure Modules and Toroidal Current Algebras

Evgeny Feiginab

a I. E. Tamm Department of Theoretical Physics, Lebedev Physics Institute, Leninski Prospect 53, Moscow, 119991, Russia
b Mathematical Institute, University of Cologne, Weyertal 86-90, D-50931, Cologne, Germany

Abstract: Let $L$ be the basic (level one vacuum) representation of the affine Kac–Moody Lie algebra $\widehat{\mathfrak g}$. The $m$-th space $F_m$ of the PBW filtration on $L$ is a linear span of vectors of the form $x_1\cdots x_lv_0$, where $l\le m$, $x_i\in\widehat{\mathfrak g}$ and $v_0$ is a highest weight vector of $L$. In this paper we give two descriptions of the associated graded space $L^{\mathrm{gr}}$ with respect to the PBW filtration. The “top-down” description deals with a structure of $L^{\mathrm{gr}}$ as a representation of the abelianized algebra of generating operators. We prove that the ideal of relations is generated by the coefficients of the squared field $e_\theta(z)^2$, which corresponds to the longest root $\theta$. The “bottom-up” description deals with the structure of $L^{\mathrm{gr}}$ as a representation of the current algebra $\mathfrak g\otimes\mathbb C[t]$. We prove that each quotient $F_m/F_{m-1}$ can be filtered by graded deformations of the tensor products of $m$ copies of $\mathfrak g$.

Keywords: affine Kac–Moody algebras; integrable representations; Demazure modules

DOI: https://doi.org/10.3842/SIGMA.2008.070

Full text: PDF file (332 kB)
Full text: http://emis.mi.ras.ru/.../070
References: PDF file   HTML file

Bibliographic databases:

ArXiv: 0806.4851
MSC: 17B67
Received: July 4, 2008; in final form October 6, 2008; Published online October 14, 2008
Language:

Citation: Evgeny Feigin, “The PBW Filtration, Demazure Modules and Toroidal Current Algebras”, SIGMA, 4 (2008), 070, 21 pp.

Citation in format AMSBIB
\Bibitem{Fei08}
\by Evgeny Feigin
\paper The PBW Filtration, Demazure Modules and Toroidal Current Algebras
\jour SIGMA
\yr 2008
\vol 4
\papernumber 070
\totalpages 21
\mathnet{http://mi.mathnet.ru/sigma323}
\crossref{https://doi.org/10.3842/SIGMA.2008.070}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2470526}
\zmath{https://zbmath.org/?q=an:05555842}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267267800070}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84896060542}


Linking options:
  • http://mi.mathnet.ru/eng/sigma323
  • http://mi.mathnet.ru/eng/sigma/v4/p70

    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. Feigin E., Fourier G., Littelmann P., “PBW filtration and bases for irreducible modules in type A(n)”, Transform Groups, 16:1 (2011), 71–89  crossref  mathscinet  zmath  isi  elib  scopus
    2. Feigin E., Fourier G., Littelmann P., “PBW Filtration and Bases for Symplectic Lie Algebras”, Int Math Res Not, 2011, no. 24, 5760–5784  crossref  mathscinet  zmath  isi  elib  scopus
    3. Feigin E., “Degenerate Flag Varieties and the Median Genocchi Numbers”, Math. Res. Lett., 18:6 (2011), 1163–1178  crossref  mathscinet  zmath  isi  elib  scopus
    4. Feigin E., “G(a)(M) Degeneration of Flag Varieties”, Sel. Math.-New Ser., 18:3 (2012), 513–537  crossref  mathscinet  zmath  isi  elib  scopus
    5. Bhimarthi Ravinder, “Demazure Modules, Chari–Venkatesh Modules and Fusion Products”, SIGMA, 10 (2014), 110, 10 pp.  mathnet  crossref
    6. Kus D., Littelmann P., “Fusion Products and Toroidal Algebras”, Pac. J. Math., 278:2 (2015), 427–445  crossref  mathscinet  zmath  isi  elib  scopus
    7. Feigin E., Makedonskyi I., “Nonsymmetric Macdonald Polynomials and Pbw Filtration: Towards the Proof of the Cherednik-Orr Conjecture”, J. Comb. Theory Ser. A, 135 (2015), 60–84  crossref  mathscinet  zmath  isi  elib  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
    Number of views:
    This page:226
    Full text:52
    References:19

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