2022/12/14 by Ilaria Colazzo, Colazzo, Ilaria, Eric Jespers +7
Mathematics · #16S36 #16T25 #16W22 #20M25 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · doi:10.48550/arxiv.2212.07361
openalex publication_date 2022/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is proven that finite idempotent left non-degenerate set-theoretic solutions (X,r) of the Yang-Baxter equation on a set X are determined by a left simple semigroup structure on X (in particular, a finite union of isomorphic copies of a group) and some maps q and φx on X, for x∈ X. This structure turns out to be a group precisely when the associated structure monoid is cancellative and all the maps φx are equal to an automorphism of this group. Equivalently, the structure algebra K[M(X,r)] is right Noetherian, or in characteristic zero it has to be semiprime. The structure algebra always is a left Noetherian representable algebra of Gelfand--Kirillov dimension one. To prove these results it is shown that the structure semigroup S(X,r) has a decomposition in finitely many cancellative semigroups Su indexed by the diagonal, each Su has a group of quotients Gu that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that X equals the diagonal is fully described by a single permutation on X.