A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation

By Wolfgang Rump. 2005

291 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/159082dc3541a6f6f940d4feedf994edbc3e6339

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Braces and the Yang–Baxter Equation (2012, background): “Recently many results on set-theoretic solutions appeared in several papers by Etingof, Schedler and Soloviev [11], Gateva-Ivanova [13], Gateva-Ivanova and Van den Bergh [16], Lu, Yan and Zhu [24], Rump [31, 33, 36], Jespers and Okniński [20, 21], Cedó, Jespers and Okniński [7], Cedó, Jespers an”
Retractability of set theoretic solutions of the Yang-Baxter equation (2009, unknown): “G R ] 2 3”
Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation (2023, methodology): “A simple algebraic formula for the correspondence between finite right nilpotent Fp-braces and finite nilpotent pre-Lie algebras over the field Fp is obtained by Smoktunowicz [51], which agrees with the correspondence using Lazard’s correspondence between finite Fp-braces and pre-Lie algebras propos”
Involutive Yang-Baxter groups (2008, background): “A cycle set A with an abelian group structure is called linear [16, 17] In [17], Rump introduced a new entity, called a brace.”
Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova ☆ (2015, methodology): “The theory of involutive solutions was further developed in several papers such as [11, 12], [18,19], [13], [3], [14,15], [4,5], [8] and [7].”
Solutions of the Yang-Baxter equation associated with a left brace (2015, background): “A subclass of this type of solutions, the non-degenerate involutive ones, has received a lot of attention in the last years [6, 7, 8, 11, 13, 14, 15, 16, 18, 19, 21, 22, 23].”
Set-theoretic solutions of the Yang–Baxter equation, RC-calculus, and Garside germs (2014, unknown): “The author wishes to thank Tatiana Gateva-Ivanova, Eddy Godelle, Eric Jespers, Gilbert Levitt, Jan Okniński, and, specially, Wolfgang Rump for discussions about the content of this paper.”
A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation (2004, background): “It follows from a theorem of Rump, [27], that every square-free solution (X, r), where X is a finite set, is decomposable into a disjoint union X = Y ⋃ Z of two nonempty r-invariant subsets Y, Z. Suppose | Y |= k, | Z |= m, k +m = n.”
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, background): “This database (available from the authors immediately on request), and Rump’s identification between such braidings and cycle sets in the finite setting, allowed us to write a computer program for calculating the homologies Hn(X,Z) for small n and X.”
Garside Groups and Yang–Baxter Equation (2009, unknown): “Moreover, he shows that an extension to infinite X is false [19].”
Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation (2016, background): “This class of solutions has received a lot of attention in recent years, see for example [8, 9, 12, 13, 14, 17, 18, 19, 20, 21, 25, 26, 29, 31, 39].”
On skew braces (with an appendix by N. Byott and L. Vendramin) (2017, methodology): “Linear cycle sets were introduced by Rump in [43].”