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Uspekhi Mat. Nauk, 1976, Volume 31, Issue 5(191), Pages 71–88 (Mi umn3844)  

This article is cited in 9 scientific papers (total in 10 papers)

Lattices, representations, and algebras connected with them. I

I. M. Gel'fand, V. A. Ponomarev


Abstract: In this article the authors have attempted to follow the style which one of them learned from P. S. Aleksandrov in other problems (the descriptive theory of functions and topology).
Let $L$ be a modular lattice. By a representation of $L$ in $A$-module $M$, where $A$ is a ring, we mean a morphism from $L$ into the lattice $\mathscr L(A,M)$ of submodules of $M$. In this article we study representations of finitely generated free modular lattices $D^r$. We are principally interested in representations in the lattice $\mathscr L(K,V)$ of linear subspaces of a space $V$ over a field $K$ ($V=K^n$).
An element $a$ in a modular lattice $L$ is called perfect if $a$ is sent either to $O$ or to $V$ under any indecomposable representation $\rho\colon L\to\mathscr L(K,V)$. The basic method of studying the lattice $D^r$ is to construct in it two sublattices $B^+$ and $B^-$, each of which consists of perfect elements.
Certain indecomposable representations $\rho^+_{t,l }$(respectively, $\rho^-_{t,l})$) are connected with the sublattices $B^+$ (respectively, $B^-$). Almost all these representations (except finitely many of small dimension) possess the important property of complete irreducibility. A representation $\rho\colon L\to\mathscr L(K,V)$ is called completely irreducible if the lattice $\rho(L)$ is isomorphic to the lattice of linear subspaces of a projective space over the field $\mathbf Q$ of rational numbers of dimension $n-1$, where $n=\dim_KV$. In this paper we construct a certain special $K$-algebra $A^r$ and study the representations $\rho_A\colon D^r\to\mathscr L_R(A^r)$ of $D^r$ into the lattice of right ideals of $A^r$. We conjecture that the lattice of right homogeneous ideals of the $\mathbf Q$-algebra $A^r$ describes (up to the relation of linear equivalence) the essential part of $D^r$.

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English version:
Russian Mathematical Surveys, 1976, 31:5, 67–85

Bibliographic databases:

UDC: 519.4
MSC: 16G30, 06C05, 14N20, 16D25
Received: 09.04.1976

Citation: I. M. Gel'fand, V. A. Ponomarev, “Lattices, representations, and algebras connected with them. I”, Uspekhi Mat. Nauk, 31:5(191) (1976), 71–88; Russian Math. Surveys, 31:5 (1976), 67–85

Citation in format AMSBIB
\Bibitem{GelPon76}
\by I.~M.~Gel'fand, V.~A.~Ponomarev
\paper Lattices, representations, and algebras connected with them.~I
\jour Uspekhi Mat. Nauk
\yr 1976
\vol 31
\issue 5(191)
\pages 71--88
\mathnet{http://mi.mathnet.ru/umn3844}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=498705}
\zmath{https://zbmath.org/?q=an:0358.06020|0369.06006}
\transl
\jour Russian Math. Surveys
\yr 1976
\vol 31
\issue 5
\pages 67--85
\crossref{https://doi.org/10.1070/RM1976v031n05ABEH004188}


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    This publication is cited in the following articles:
    1. I. M. Gel'fand, V. A. Ponomarev, “Model algebras and representations of graphs”, Funct. Anal. Appl., 13:3 (1979), 157–166  mathnet  crossref  mathscinet  zmath
    2. 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
    3. Christian Herrmann, “Rahmen und erzeugende quadrupel in modularen verbänden”, Algebra univers, 14:1 (1982), 357  crossref  mathscinet  zmath  isi
    4. A. A. Tsyl'ke, “Perfect elements of free modular lattices”, Funct. Anal. Appl., 16:1 (1982), 73–74  mathnet  crossref  mathscinet  zmath  isi
    5. 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
    6. R. B. Stekol'shchik, “Invariant elements in a modular lattice”, Funct. Anal. Appl., 18:1 (1984), 73–75  mathnet  crossref  mathscinet  zmath  isi
    7. Mark Haiman, “Proof theory for linear lattices”, Advances in Mathematics, 58:3 (1985), 209  crossref
    8. A. A. Klyachko, “Equivariant bundles on toral varieties”, Math. USSR-Izv., 35:2 (1990), 337–375  mathnet  crossref  mathscinet  zmath
    9. Christian Herrmann, Marcel Wild, “Acyclic modular lattices and their representations”, Journal of Algebra, 136:1 (1991), 17  crossref
    10. C.M.ichael Ringel, “The Auslander bijections: how morphisms are determined by modules”, Bull. Math. Sci, 2013  crossref
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