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Mosc. Math. J., 2006, Volume 6, Number 1, Pages 119–134 (Mi mmj239)  

This article is cited in 11 scientific papers (total in 11 papers)

Category of $\mathfrak{sp}(2n)$-modules with bounded weight multiplicities

D. Grantcharova, V. V. Serganovab

a Department of Computer Science San Jose State University
b University of California, Berkeley

Abstract: Let $\mathfrak g$ be a finite dimensional simple Lie algebra. Denote by $\mathcal B$ the category of all bounded weight $\mathfrak g$-modules, i.e. those which are direct sum of their weight spaces and have uniformly bounded weight multiplicities. A result of Fernando shows that infinite-dimensional bounded weight modules exist only for $\mathfrak g=\mathfrak{sl}(n)$ and $\mathfrak g=\mathfrak{sp}(2n)$. If $\mathfrak g =\mathfrak{sp}(2n)$ we show that $\mathcal B$ has enough projectives if and only if $n>1$. In addition, the indecomposable projective modules can be parameterized and described explicitly. All indecomposable objects are described in terms of indecomposable representations of a certain quiver with relations. This quiver is wild for $n>2$. For $n=2$ we describe all indecomposables by relating the blocks of $\mathcal B$ to the representations of the affine quiver $A_3^{(1)}$.

Key words and phrases: Lie algebra, indecomposable representations, quiver, weight modules.

DOI: https://doi.org/10.17323/1609-4514-2006-6-1-119-134

Full text: http://www.ams.org/.../abst6-1-2006.html
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MSC: 17B10
Received: December 1, 2005
Language:

Citation: D. Grantcharov, V. V. Serganova, “Category of $\mathfrak{sp}(2n)$-modules with bounded weight multiplicities”, Mosc. Math. J., 6:1 (2006), 119–134

Citation in format AMSBIB
\Bibitem{GraSer06}
\by D.~Grantcharov, V.~V.~Serganova
\paper Category of $\mathfrak{sp}(2n)$-modules with bounded weight multiplicities
\jour Mosc. Math.~J.
\yr 2006
\vol 6
\issue 1
\pages 119--134
\mathnet{http://mi.mathnet.ru/mmj239}
\crossref{https://doi.org/10.17323/1609-4514-2006-6-1-119-134}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2265951}
\zmath{https://zbmath.org/?q=an:1127.17006}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208595700008}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Futorny V., Jardim M., Moura A.A., “On moduli spaces for abelian categories”, Comm. Algebra, 36:6 (2008), 2171–2185  crossref  mathscinet  zmath  isi  elib  scopus
    2. Grantcharov D., Serganova V., “Cuspidal representations of $\mathfrak{sl}(n+1)$”, Adv. Math., 224:4 (2010), 1517–1547  crossref  mathscinet  zmath  isi  scopus
    3. Tomasini G., “Integrability of Weight Modules of Degree 1”, J. Lie Theory, 22:2 (2012), 523–539  mathscinet  zmath  isi
    4. Tomasini G., “Restriction to Levi Subalgebras and Generalization of the Category O”, Ann. Inst. Fourier, 63:1 (2013), 37–88  crossref  mathscinet  zmath  isi  elib  scopus
    5. Futorny V., Grantcharov D., Mazorchuk V., “Weight Modules Over Infinite Dimensional Weyl Algebras”, Proc. Amer. Math. Soc., 142:9 (2014), PII S0002-9939(2014)12071-5, 3049–3057  crossref  mathscinet  zmath  isi  elib
    6. Bai Zh.Q., “Gelfand-Kirillov Dimensions of the Z(2)-Graded Oscillator Representations of Sl(N)”, Acta. Math. Sin.-English Ser., 31:6 (2015), 921–937  crossref  mathscinet  zmath  isi  scopus
    7. Nilsson J., “U(H)-Free Modules and Coherent Families”, J. Pure Appl. Algebr., 220:4 (2016), 1475–1488  crossref  mathscinet  zmath  isi  scopus
    8. Grantcharov D., Serganova V., “on Weight Modules of Algebras of Twisted Differential Operators on the Projective Space”, Transform. Groups, 21:1 (2016), 87–114  crossref  mathscinet  zmath  isi  elib  scopus
    9. Penkov I., Petukhov A., “on Ideals in U(Sl(Infinity), U(O(Infinity)), U(Sp(Infinity))”, Representation Theory - Current Trends and Perspectives, EMS Ser. Congr. Rep., eds. Krause H., Littelmann P., Malle G., Neeb KH., Schweigert C., Eur. Math. Soc., 2017, 565–602  mathscinet  zmath  isi
    10. Cavaness A., Grantcharov D., “Bounded Weight Modules of the Lie Algebra of Vector Fields on C-2”, J. Algebra. Appl., 16:12 (2017), 1750236  crossref  zmath  isi  scopus
    11. Penkov I., Serganova V., Zuckerman G., “On Categories of Admissible (G, Sl (2))-Modules”, Transform. Groups, 23:2 (2018), 463–489  crossref  mathscinet  zmath  isi  scopus
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