scholarly journals Corrigendum and addendum to “The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation”

2020 ◽  
Vol 373 (6) ◽  
pp. 4517-4521 ◽  
Author(s):  
Eric Jespers ◽  
Łukasz Kubat ◽  
Arne Van Antwerpen
Keyword(s):  
Baxter Equation ◽  
And Algebra ◽  
Theoretic Solution

scholarly journals Left semi-braces and solutions of the Yang–Baxter equation

2019 ◽  
Vol 31 (1) ◽  
pp. 241-263 ◽  
Author(s):  
Eric Jespers ◽  
Arne Van Antwerpen
Keyword(s):  
Positive Integer ◽  
Polynomial Identity ◽  
Baxter Equation ◽  
Polynomial Algebras ◽  
Finite Set ◽  
Theoretic Solution ◽  
Kirillov Dimension ◽  
Do So

Abstract Let {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions {r_{B}} that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.

Set-theoretic solutions to the Yang–Baxter equation, skew-braces, and related near-rings

2019 ◽  
Vol 18 (08) ◽  
pp. 1950145
Author(s):  
Wolfgang Rump
Keyword(s):  
Residue Field ◽  
Ring Structure ◽  
Baxter Equation ◽  
Exponential Map ◽  
Exponential Form ◽  
Degenerate Solution ◽  
Terminal Object ◽  
Theoretic Solution ◽  
Initial Object ◽  
Central Residue

Skew-braces have been introduced recently by Guarnieri and Vendramin. The structure group of a non-degenerate solution to the Yang–Baxter equation is a skew-brace, and every skew-brace gives a set-theoretic solution to the Yang–Baxter equation. It is proved that skew-braces arise from near-rings with a distinguished exponential map. For a fixed skew-brace, the corresponding near-rings with exponential form a category. The terminal object is a near-ring of self-maps, while the initial object is a near-ring which gives a complete invariant of the skew-brace. The radicals of split local near-rings with a central residue field [Formula: see text] are characterized as [Formula: see text]-braces with a compatible near-ring structure. Under this correspondence, [Formula: see text]-braces are radicals of local near-rings with radical square zero.

scholarly journals Geometry and Algebra of the Deltoid Map

2021 ◽  
Vol 128 (1) ◽  
pp. 25-39
Author(s):  
Joshua P. Bowman
Keyword(s):  
And Algebra
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