Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs

Building on a result by W.Rump, we show how to exploit the right-cyclic law (xy)(xz) = (yx)(yz) in order to investigate the structure groups and monoids attached with (involutive nondegenerate) set-theoretic solutions of the Yang–Baxter equation. We develop a sort of right-cyclic calculus, and use it to obtain short proofs for the existence both of the Garside structure and of the I-structure of such groups. We describe finite quotients that play for the considered groups the role that Coxeter groups play for Artin– Tits groups. The Yang–Baxter equation (YBE) is a fundamental equation occurring in integrable models in statistical mechanics and quantum field theory [26]. Among its many solutions, some simple ones called set-theoretic turn out to be connected with several interesting algebraic structures. In particular, a group and a monoid are attached with every set-theoretic solution of YBE [16], and the family of all groups and monoids arising in this way is known to have rich properties: as shown by T.Gateva–Ivanova and M.Van den Bergh in [20] and by E. Jespers and J.Okniński in [24], they admit an I-structure, meaning that their Cayley graph is isometric to that of a free Abelian group, and, as shown by F.Chouraqui in [5], they admit a Garside structure, (roughly) meaning that they are groups of fractions of monoids in which divisibility relations are lattice orders. It was shown by W.Rump in [27] that (involutive nondegenerate) set-theoretic solutions of YBE are in one-to-one correspondence with algebraic structures consisting of a set equipped with a binary operation ∗ that obeys the right-cyclic law (xy)(xz) = (yx)(yz) and has bijective left-translations. In this paper, we merge the ideas stemming from the right-cyclic law (RC-law) and those coming from Garside theory to give easy alternative proofs of earlier results and derive new results. The key technical point is the connection between the RC-law and the least common right-multiple operation. A nice point is that one never needs to restrict to squarefree solutions of YBE, that is, those satisfying ρ(s, s) = (s, s). The main benefit of the current approach is to provide a simple and complete solution to the problem of finding a Garside germ for every group associated with a set-theoretic solution of YBE, namely finding a finite quotient of the group that encodes the whole structure in the way a finite Coxeter group encodes the associated Artin–Tits group. The precise statement (Proposition 5.2) says that, if (S, ∗) is an RC-quasigroup with cardinality n and class d (a certain numerical parameter attached with every finite RC-quasigroup), then starting from the canonical presentation of the associated group and adding the RC-torsion relations s = 1 1991 Mathematics Subject Classification. 20F38, 20N02, 20M10, 20F55, 06F05, 16T25.