Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. We generalize Rump's braces to the non-commutative setting and use this new ...structure to study not necessarily involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation. Based on results of Bachiller and Catino and Rizzo, we develop an algorithm to enumerate and construct classical and non-classical braces of small size up to isomorphism. This algorithm is used to produce a database of braces of small size. The paper contains several open problems, questions and conjectures.
Abstract
Motivated by the proof of Rump of a conjecture of Gateva–Ivanova on the decomposability of square-free solutions to the Yang–Baxter equation, we present several other decomposability ...theorems based on the cycle structure of a certain permutation associated with the solution.
Factorizations of skew braces Jespers, E.; Kubat, Ł.; Van Antwerpen, A. ...
Mathematische annalen,
12/2019, Letnik:
375, Številka:
3-4
Journal Article
Recenzirano
Odprti dostop
We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang–Baxter equation. We study factorization of skew left braces ...through strong left ideals and we prove analogs of Itô’s theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang–Baxter equation. Finally, we classify skew braces that contain no non-trivial proper characteristic ideals.
are equivalent by twist and hence they have the same Hilbert series. These algebras appear in the classification of pointed Hopf algebras and in the study of the quantum cohomology ring of flag ...manifolds.>
We develop a theory of extensions for involutive and nondegenerate solutions of the set-theoretic Yang–Baxter equation and use it to produce new families of solutions. As an application we construct ...an infinite family of counterexamples to a conjecture of Gateva-Ivanova related to the retractability of square-free solutions.
We begin the study of PBW deformations of graded algebras relevant to the theory of Hopf algebras. One of our examples is the Fomin–Kirillov algebra
E
3
. Another one appeared in a paper of García ...Iglesias and Vay. As a consequence of our methods, we determine when the deformations are semisimple and we are able to produce PBW bases and polynomial identities for these deformations.
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