Carsten Dietzel, Modular noetherian right l-groups

Abstract:

Right l-groups are groups that are equipped with a lattice structure that is invariant under right-multiplication. Those with a strong order unit and distributive, noetherian lattice structure have been characterized by Chouraqui and Rump - these are exactly the structure groups of non-degenerate, involutive set-theoretic solutions to the Yang-Baxter equation.

What can be said about the more general class of modular noetherian right l-groups with strong order unit? Rump proved that each such group has a distinguished strong order interval that is a modular geometric lattice. By this result, we know part of the local lattice structure in this class of groups.

After giving an overview of these results, emphasis will be put on the class of desarguesian right l-groups, that is, modular noetherian right l-groups with a desarguesian strong order interval. I will present two classes of desarguesian right l-groups that can be obtained from matrix rings and skew polynomial rings, respectively. It turns out that these constructions are, in fact, instances of a representation theorem for desarguesian right l-groups of dimension 4 or greater.