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Uspekhi Mat. Nauk, 1974, Volume 29, Issue 6(180), Pages 3–58 (Mi umn4447)  

This article is cited in 10 scientific papers (total in 11 papers)

Free modular lattices and their representations

I. M. Gel'fand, V. A. Ponomarev


Abstract: Let be $L$ a modular lattice, and $V$ a finite-dimensional vector space over a field $k$. A representation of $L$ in $V$ is a morphism from $L$ into the lattice $\mathscr L(V)$ of all subspaces of $V$. In this paper we study representations of finitely generated free modular lattices $D^r$. An element $a$ of a lattice $L$ is called perfect if for every indecomposable representation $\rho\colon L\to\mathscr L(k^n)$ the subspace $\rho(a)$ of $V=k^n$ is such that $\rho(a)=0$ or $\rho(a)=V$. We construct and study certain important sublattices of $D^r$, called “cubicles”. All elements of the cubicles are perfect. There are indecomposable representations connected with the cubicles. It will be shown that almost all these representations, except the elementary ones, have the important property of complete irreducibility; here a representation $\rho$ of $L$ is called completely irreducible if the sublattice $\rho(L)\subset\mathscr L(k^n)$ is isomorphic to the lattice $\mathbf P(\mathbf Q, n-1)$ of linear submanifolds of projective space over the field $\mathbf Q$ of rational numbers.

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English version:
Russian Mathematical Surveys, 1974, 29:6, 1–56

Bibliographic databases:

UDC: 519.4
MSC: 06C05, 13C10, 13B10
Received: 10.06.1974

Citation: I. M. Gel'fand, V. A. Ponomarev, “Free modular lattices and their representations”, Uspekhi Mat. Nauk, 29:6(180) (1974), 3–58; Russian Math. Surveys, 29:6 (1974), 1–56

Citation in format AMSBIB
\Bibitem{GelPon74}
\by I.~M.~Gel'fand, V.~A.~Ponomarev
\paper Free modular lattices and their representations
\jour Uspekhi Mat. Nauk
\yr 1974
\vol 29
\issue 6(180)
\pages 3--58
\mathnet{http://mi.mathnet.ru/umn4447}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=401566}
\zmath{https://zbmath.org/?q=an:0314.15003}
\transl
\jour Russian Math. Surveys
\yr 1974
\vol 29
\issue 6
\pages 1--56
\crossref{https://doi.org/10.1070/RM1974v029n06ABEH001301}


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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. I. M. Gel'fand, V. A. Ponomarev, “Lattices, representations, and algebras connected with them. I”, Russian Math. Surveys, 31:5 (1976), 67–85  mathnet  crossref  mathscinet  zmath
    2. I. M. Gel'fand, V. A. Ponomarev, “Lattices, representations, and algebras connected with them. II”, Russian Math. Surveys, 32:1 (1977), 91–114  mathnet  crossref  mathscinet  zmath
    3. I. M. Gel'fand, V. A. Ponomarev, “Representations of graphs. Perfect subrepresentations”, Funct. Anal. Appl., 14:3 (1980), 177–190  mathnet  crossref  mathscinet  zmath  isi
    4. Christian Herrmann, “Rahmen und erzeugende quadrupel in modularen verbänden”, Algebra univers, 14:1 (1982), 357  crossref  mathscinet  zmath  isi
    5. A. A. Tsyl'ke, “Perfect elements of free modular lattices”, Funct. Anal. Appl., 16:1 (1982), 73–74  mathnet  crossref  mathscinet  zmath  isi
    6. N. N. Bogolyubov, S. G. Gindikin, A. A. Kirillov, A. N. Kolmogorov, S. P. Novikov, L. D. Faddeev, “Izrail' Moiseevich Gel'fand (on his seventieth birthday)”, Russian Math. Surveys, 38:6 (1983), 145–153  mathnet  crossref  mathscinet  zmath  adsnasa
    7. R. B. Stekol'shchik, “Invariant elements in a modular lattice”, Funct. Anal. Appl., 18:1 (1984), 73–75  mathnet  crossref  mathscinet  zmath  isi
    8. Mark Haiman, “Proof theory for linear lattices”, Advances in Mathematics, 58:3 (1985), 209  crossref
    9. Herbert Gross, Christian Herrmann, Remo Moresi, “The classification of subspaces in Hermitean vector spaces”, Journal of Algebra, 105:2 (1987), 516  crossref
    10. R. B. Stekol'shchik, “Perfect elements in the modular structure associated with the extended Dynkin diagram $\widetilde{e}_6$”, Funct. Anal. Appl., 23:3 (1989), 251–254  mathnet  crossref  mathscinet  zmath  isi
    11. C.M.ichael Ringel, “The Auslander bijections: how morphisms are determined by modules”, Bull. Math. Sci, 2013  crossref
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