Semigroup extensions of isometry groups of flat spacetimes compactified over lightlike lattices

We examine some peculiarities of the subset of lattice preserving elements in a pseudo-Euclidean group, when the lattice under consideration contains a lightlike vector, or more generally, when the restriction of a pseudo-Euclidean metric to the real linear enveloppe of the lattice is not definite. For the case of a Lorentzian metric, it is shown in detail that the isometry group of the spacetime compactified over such a lattice admits a natural extension to a semigroup. We explain why such an extension is not available for spacelike lattices. Furthermore, we argue that for any Lagrangian defined on such a lightlike compactified spacetime, the elements of the semigroup relate sectors of the theory belonging to different discrete compactification radii, and hence connect different superselection sectors of the theory. This mapping occurs as a one-way process owing to the non-invertibility of the semigroup elements on the lattice. These structures might therefore be of relevance to matrix theory.