Lattices, representations, and algebras connected with them.~I

I. M. Gel'fand ; V. A. Ponomarev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 31 (1976) no. 5, pp. 67-85

Résumé

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 $D^r$.