From Braces to Hecke algebras & Quantum Groups

Abstract

We examine links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify new quantum groups associated to set-theoretic solutions coming from braces. We also construct a novel class of quantum discrete integrable systems and we derive symmetries for the corresponding periodic transfer matrices.

Set theoretic Yang-Baxter equation, braces and Drinfeld twists

“…In [7,8] key links between set theoretical solutions coming from braces and quantum integrable systems and the associated quantum algebras were uncovered. More precisely:…”

“…Note that in [12] Etingof, Schedler and Soloviev constructed quantum groups associated to set-theoretic solutions, however in [7] a different construction is used coming from parameter dependent solutions of the Yang-Baxter equation and thus the corresponding quantum groups differ from those in [12].…”

“…Our main focus here is the study of involutive set theoretic solutions coming from braces, based on the findings presented in [7,8]. More precisely, our aim is to identify the explicit form of so called admissible Drinfeld twists for brace solutions of the YBE.…”

“…In the analysis of [7,8] Baxterized R-matrices coming from set theoretic solutions of the Yang-Baxter equation were identified, being of the form R(λ) = r+ 1 λ P, where r is the set theoretic solution of the Yang-Baxter equation and P is the permutation operator. Interestingly the r-matrix does not contain any free parameter (deformation parameter), and consequently the R-matrix has no semi-classical analogue.…”

Right Nilpotency of Braces of Cardinality $p^4$

“…Braces were introduced by W. Rump in 2007 [Rum07a], as a generalisation of Jacobson radical rings, in order to help study solutions of the Yang-Baxter equation. Braces have been studied extensively since -connections to many concepts such as integral group rings [Sys12], Garside groups [Cho10], groups with bijective 1-cocycles [CJDR10,ESS99], quantum groups [ESS99,DS19] and trusses [Brz19] have been found, to name a few. In 2016, skew braces were introduced by L. Guarnieri and L. Vendramin in [GV16] as a generalisation of braces in order to study non-involutive solutions to the Yang-Baxter equation.…”