Right l-groups, geometric Garside groups, and solutions of the quantum Yang–Baxter equation

Abstract

Right lattice-ordered groups are introduced and studied as a general framework for Garside groups and related groups without a Garside element. Every right l-group G has a (two-sided) partially ordered subgroup N(G) which generalizes the quasi-centre of an Artin–Tits group. The group N(G) splits into copies of Z if G is noetherian. The positive cone of a right l-group is described as a structure that is known from algebraic logic: a pair of left and right self-similar hoops. For noetherian right l-groups G, modularity of the lattice structure is characterized in terms of an operation on the set X(G−) of atoms. It is proved that modular Garside groups are equivalent to finite projective spaces with a non-degenerate labelling. A concept of duality for X(G−) is introduced and applied in the distributive case. This gives a one-to-one correspondence between noetherian right l-groups with duality and non-degenerate unitary set-theoretic solutions of the quantum Yang–Baxter equation. The description of Garside groups via Garside germs is extended to right l-groups, which yields a one-sided enhancement and a new proof of Dvurečenskij's non-commutative extension of Mundici's correspondence between abelian l-groups and MV-algebras.