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

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

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
2. I. M. Gel'fand, V. A. Ponomarev, “Representations of graphs. Perfect subrepresentations”, Funct. Anal. Appl., 14:3 (1980), 177–190
3. Christian Herrmann, “Rahmen und erzeugende quadrupel in modularen verbänden”, Algebra univers, 14:1 (1982), 357
4. A. A. Tsyl'ke, “Perfect elements of free modular lattices”, Funct. Anal. Appl., 16:1 (1982), 73–74
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
6. R. B. Stekol'shchik, “Invariant elements in a modular lattice”, Funct. Anal. Appl., 18:1 (1984), 73–75
7. Mark Haiman, “Proof theory for linear lattices”, Advances in Mathematics, 58:3 (1985), 209
8. A. A. Klyachko, “Equivariant bundles on toral varieties”, Math. USSR-Izv., 35:2 (1990), 337–375
9. Christian Herrmann, Marcel Wild, “Acyclic modular lattices and their representations”, Journal of Algebra, 136:1 (1991), 17
10. C.M.ichael Ringel, “The Auslander bijections: how morphisms are determined by modules”, Bull. Math. Sci, 2013
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