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

JOURNAL OF ALGEBRA

卷:439

Article

Rump, Wolfgang

关键词: Right l-group;    Self-similar;    L-algebra;    Garside group;    Artin-Tits group;    Yang-Baxter equation;    Geometric lattice;   

DOI  :  10.1016/j.jalgebra.2015.04.045

来源: Elsevier

【 摘 要 】

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 Dvurecenskij's non-commutative extension of Mundici's correspondence between abelian l-groups and MV-algebras. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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