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

  • Eric Jespers ORCID logo EMAIL logo and Arne Van Antwerpen ORCID logo
Published/Copyright: October 5, 2018
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 1

Abstract

Let r:X2→X2 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⁢〈x∈X∣x⁢y=u⁢v⁢ if ⁢r⁢(x,y)=(u,v)〉 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 rB 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.

Keywords: Yang–Baxter equation; semi-brace; monoid algebras; monoids; groups
MSC 2010: 20M25; 16T25; 20E22; 16S36

Communicated by Manfred Droste

Funding statement: The first author is supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Belgium). The second author is supported by Fonds voor Wetenschappelijk Onderzoek (Vlaanderen).

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Received: 2018-02-28
Revised: 2018-08-09
Published Online: 2018-10-05
Published in Print: 2019-01-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón–Zygmund operators and applications
  3. Infinite-dimensional triangularizable algebras
  4. Expansion for the product of matrices in groups
  5. Asphericity of positive free product length 4 relative group presentations
  6. Unitary representations with non-zero Dirac cohomology for complex E6
  7. On middle cohomology of special Artin–Schreier varieties and finite Heisenberg groups
  8. Normal elements of noncommutative Iwasawa algebras over SL3(ℤ_p)
  9. Regularity properties of Schrödinger equations in vector-valued spaces and applications
  10. Statistics of Hecke eigenvalues for GL(𝑛)
  11. Bilinear Calderón–Zygmund operators on products of variable Hardy spaces
  12. On the Reidemeister spectrum of an Abelian group
  13. Dense free subgroups of automorphism groups of homogeneous partially ordered sets
  14. Left semi-braces and solutions of the Yang–Baxter equation
  15. On the automorphism group of a symplectic half-flat 6-manifold
https://doi.org/10.1515/forum-2018-0059
Keywords for this article
Yang–Baxter equation; semi-brace; monoid algebras; monoids; groups

Articles in the same Issue

  1. Frontmatter
  2. Sharp maximal estimates for multilinear commutators of multilinear strongly singular Calderón–Zygmund operators and applications
  3. Infinite-dimensional triangularizable algebras
  4. Expansion for the product of matrices in groups
  5. Asphericity of positive free product length 4 relative group presentations
  6. Unitary representations with non-zero Dirac cohomology for complex E6
  7. On middle cohomology of special Artin–Schreier varieties and finite Heisenberg groups
  8. Normal elements of noncommutative Iwasawa algebras over SL3(ℤ_p)
  9. Regularity properties of Schrödinger equations in vector-valued spaces and applications
  10. Statistics of Hecke eigenvalues for GL(𝑛)
  11. Bilinear Calderón–Zygmund operators on products of variable Hardy spaces
  12. On the Reidemeister spectrum of an Abelian group
  13. Dense free subgroups of automorphism groups of homogeneous partially ordered sets
  14. Left semi-braces and solutions of the Yang–Baxter equation
  15. On the automorphism group of a symplectic half-flat 6-manifold
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