Braces, radical rings, and the quantum Yang–Baxter equation

By Wolfgang Rump. 2007

486 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Organic Chemistry

View Paper: https://www.semanticscholar.org/paper/2bfa4ccfe679d689db2d759186c8e5851713a623

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In Algebra and Number Theory

In Organic Chemistry

Citations in the corpus

Skew left braces of nilpotent type (2018, background): “To study Gateva–Ivanova’s strong conjecture Rump introduced two sequences of left ideals [23].”
Counterexample to a conjecture about braces (2015, background): “Is G an IYB group? In order to study the class of non-degenerate involutive set-theoretic solutions, Rump in [18] introduced a new algebraic structure called brace.”
Solutions of the Yang-Baxter equation associated with a left brace (2015, background): “edwith eachsolution (X,r) of this type two groups which are fundamental for its study: the structure group G(X,r), and the permutation group G(X,r). In order to study this class of solutions, Rump in [23] introduced a new algebraic structure called brace. Recall that a left brace is a set B with two”
Involutive Yang-Baxter groups (2008, methodology): “This leads us to the following natural question: Does every IYB group admit a non-injective IYB morphism? Equivalently, in the frame work of braces [17]: if A is a finite brace, does there then exist a brace B with non-zero socle so that the adjoint groups of A and B are isomorphic? Note that in [17”
Extensions, matched products, and simple braces (2015, background): “ebras and quantum groups (see [8]). One class of solutions has received a lot of attention recently, the non-degenerate involutive set-theoretic solutions (see [5], and the references there). Rump in [10] introduced a new algebraic structure called brace to study this class of solutions. A left brac”
Trusses: Between braces and rings (2017, methodology): “solutions of the Yang-Baxter equation that form the subject matter of the paper the reader is hereby presented with. This text is motivated by the emergence of algebraic systems known as braces [11], [12], [13] and skew braces [14], which are tantamount to the existence of non-degenerate solutions o”
Classification of braces of order p3 (2014, background): “Bijective 1-cocycle, Radical ring, Yang-Baxter equation, Involutive non-degenerate settheoretical solution MSC:81R50, 16T25, 20D15 1 Introduction Braces are algebraic structures introduced by Rump in [5] in connection with his work on the set-theoretic solutions of the Yang-Baxter equation. A left b”
On skew braces (with an appendix by N. Byott and L. Vendramin) (2017, background): “In [44], Rump introduced braces, a new algebraic structure that turns out to be equivalent to bijective 1-cocycles; see [13, 27, 47].”
Opposite skew left braces and applications (2019, methodology): “n [GV17] to construct nondegenerate, not necessarily involutive set-theoretic solutions to the Yang-Baxter Equation. They were developed as a generalization to the concept of braces deﬁned by Rump in [Rum07] to ﬁnd involutive solutions to the YBE. As ﬁrst pointed out in [Bac16, Remark 2.6] and devel”
Set-theoretic solutions of the Yang–Baxter equation, RC-calculus, and Garside germs (2014, background): “It should be mentioned that some formulas admit counterparts in the world of braces [28, 29], which are equivalent to linear RC-systems, defined to be RC-systems equipped with a compatible abelian group operation (by [27, Prop.”
Semi-braces and the Yang-Baxter equation (2017, background): “In particular, if B is a semi-brace with (B,+) a group, then B is known as skew brace (Guarnieri and Vendramin, 2017) if in addition (B,+) is abelian then B is a brace (Rump, 2007).”
Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks (2016, methodology): “on-degenerateinvolutive solutions. Initially, in [25, 27] they were studied using two associated groups, the structure group G(X,r) and the permutation group G(X,r) of a solution (X,r). Then, Rump in [36] adapted the results in [25, 27] to a more natural setting by introducing a new algebraic struct”
Bi-skew braces and Hopf Galois structures (2019, unknown): “Rump [18] showed that a brace is two-sided if and only if (B,+, ·) is a radical ring.”
Computing (2021, unknown): “This section highlights some of the recent computational approaches to the design and evaluation of the physical environment.”