Weyl groups, lattices and geometric manifolds

By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction of hyperbolic manifolds of very small volume in up to eight dimensions.     

A note on semigroups, groups and geometric lattices

Let G be a closed, additive semigroup in a Hausdorff topological vector space. Then G is a group if and only if it satisfies natural convexity conditions of algebraic or geometric-topological type. This yields a characterization of the geometric lattices among the discrete, additive semigroups of Euclidean d-space \mathbbE^d{\mathbb{E}^{d}} and, more generally, of direct sums of subspaces and lattices in \mathbbE^d{\mathbb{E}^{d}}.     

A note on indecomposable lattices

Presented by Bjarni Jónsson.     

A note on congruence lattices of slim semimodular lattices

Recently, G. Grätzer has raised an interesting problem: Which distributive lattices are congruence lattices of slim semimodular lattices? We give an eight element slim distributive lattice that cannot be represented as the congruence lattice of a slim semimodular lattice. Our lattice demonstrates the difficulty of the problem.     

A note on planar semimodular lattices

In an earlier paper, we constructed all finite, planar, semimodular lattices in three simple steps from the direct product of two finite chains. In this note we prove that one of the three steps can be eliminated. The research of the first author was supported by the NSERC of Canada.     

A note on the multidimensional Weyl fractional operator

The purpose of the present paper is to establish a connection theorem involving the multidimensional Weyl fractional operator and the classical multidimensional Laplace transform. This provides an extension of a result due to Raina and Koul [6].     

A note on odd unimodular Euclidean lattices

We prove that the theta series of any odd unimodular Euclidean lattice is not congruent to 1 modulo any odd prime p. Received: 18 March 2005     

A note on atomistic weak congruence lattices

It is proved that a codistributive element in an atomistic algebraic lattice has a complement, implying that kernels of the related homomorphisms coincide. Some applications to weak congruence lattices of algebras are presented. In particular, necessary and sufficient conditions under which the weak congruence lattice of an algebra is atomistic are given.     

A note on lattices in semi-stable representations

Let p be a prime, K a finite extension over \mathbb Q_p{{\mathbb Q}_p} and G = Gal([`(K)] /K){G = {\rm Gal}(\overline K /K)} . We extend Kisin’s theory on j{\varphi} -modules of finite E(u)-height to give a new classification of G-stable \mathbb Z1_p{{\mathbb Z}1_p} -lattices in semi-stable representations.