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

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

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
Related articles on Google Scholar: Russian articles, English articles

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
2. I. M. Gel'fand, V. A. Ponomarev, “Lattices, representations, and algebras connected with them. II”, Russian Math. Surveys, 32:1 (1977), 91–114
3. I. M. Gel'fand, V. A. Ponomarev, “Representations of graphs. Perfect subrepresentations”, Funct. Anal. Appl., 14:3 (1980), 177–190
4. Christian Herrmann, “Rahmen und erzeugende quadrupel in modularen verbänden”, Algebra univers, 14:1 (1982), 357
5. A. A. Tsyl'ke, “Perfect elements of free modular lattices”, Funct. Anal. Appl., 16:1 (1982), 73–74
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
7. R. B. Stekol'shchik, “Invariant elements in a modular lattice”, Funct. Anal. Appl., 18:1 (1984), 73–75
8. Mark Haiman, “Proof theory for linear lattices”, Advances in Mathematics, 58:3 (1985), 209
9. Herbert Gross, Christian Herrmann, Remo Moresi, “The classification of subspaces in Hermitean vector spaces”, Journal of Algebra, 105:2 (1987), 516
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
11. C.M.ichael Ringel, “The Auslander bijections: how morphisms are determined by modules”, Bull. Math. Sci, 2013
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