Lattices, representations, and algebras connected with them. I

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 $D^r$.