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Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses

IIlaria Colazzo, Eric Jespers, Arne Van Antwerpen (UGent) and Charlotte Verwimp
(2022) JOURNAL OF ALGEBRA. 610. p.409-462
Author
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Abstract
To determine and analyze arbitrary left non-degenerate set-theoretic solutions of the Yang-Baxter equation (not necessarily bijective), we introduce an associative algebraic structure, called a YB-semitruss, that forms a subclass of the category of semitrusses as introduced by Brzeziński. Fundamental examples of YB-semitrusses are structure monoids of left non-degenerate set-theoretic solutions and (skew) left braces. Gateva-Ivanova and Van den Bergh introduced structure monoids and showed their importance (as well as that of the structure algebra) for studying involutive non-degenerate solutions. Skew left braces were introduced by Guarnieri, Vendramin and Rump to deal with bijective non-degenerate solutions. Hence, YB-semitrusses also yield a unified treatment of these different algebraic structures. The algebraic structure of YB-semitrusses is investigated and as a consequence it is proven, for example, that any finite left non-degenerate set-theoretic solution of the Yang-Baxter equation is right non-degenerate if and only if it is bijective. Furthermore, it is shown that some finite left non-degenerate solutions can be reduced to non-degenerate solutions of smaller size. The structure algebra of a finitely generated YB-semitruss is an algebra defined by homogeneous quadratic relations. We prove that it often is a left Noetherian algebra of finite Gelfand-Kirillov dimension that satisfies a polynomial identity, but in general it is not right Noetherian.
Keywords
Yang-Baxter equation, Structure semigroup, Structure algebra, Semitruss, Set-theoretic solution, QUADRATIC ALGEBRAS, SEMI-BRACES, MONOIDS, SEMIGROUPS

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MLA
Colazzo, IIlaria, et al. “Left Non-Degenerate Set-Theoretic Solutions of the Yang-Baxter Equation and Semitrusses.” JOURNAL OF ALGEBRA, vol. 610, 2022, pp. 409–62, doi:10.1016/j.jalgebra.2022.07.019.
APA
Colazzo, Ii., Jespers, E., Van Antwerpen, A., & Verwimp, C. (2022). Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses. JOURNAL OF ALGEBRA, 610, 409–462. https://doi.org/10.1016/j.jalgebra.2022.07.019
Chicago author-date
Colazzo, IIlaria, Eric Jespers, Arne Van Antwerpen, and Charlotte Verwimp. 2022. “Left Non-Degenerate Set-Theoretic Solutions of the Yang-Baxter Equation and Semitrusses.” JOURNAL OF ALGEBRA 610: 409–62. https://doi.org/10.1016/j.jalgebra.2022.07.019.
Chicago author-date (all authors)
Colazzo, IIlaria, Eric Jespers, Arne Van Antwerpen, and Charlotte Verwimp. 2022. “Left Non-Degenerate Set-Theoretic Solutions of the Yang-Baxter Equation and Semitrusses.” JOURNAL OF ALGEBRA 610: 409–462. doi:10.1016/j.jalgebra.2022.07.019.
Vancouver
1.
Colazzo Ii, Jespers E, Van Antwerpen A, Verwimp C. Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses. JOURNAL OF ALGEBRA. 2022;610:409–62.
IEEE
[1]
Ii. Colazzo, E. Jespers, A. Van Antwerpen, and C. Verwimp, “Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses,” JOURNAL OF ALGEBRA, vol. 610, pp. 409–462, 2022.
@article{01J0XD7AT25NFVKGXZ7R7GH3GR,
  abstract     = {{To determine and analyze arbitrary left non-degenerate set-theoretic solutions of the Yang-Baxter equation (not necessarily bijective), we introduce an associative algebraic structure, called a YB-semitruss, that forms a subclass of the category of semitrusses as introduced by Brzeziński. Fundamental examples of YB-semitrusses are structure monoids of left non-degenerate set-theoretic solutions and (skew) left braces. Gateva-Ivanova and Van den Bergh introduced structure monoids and showed their importance (as well as that of the structure algebra) for studying involutive non-degenerate solutions. Skew left braces were introduced by Guarnieri, Vendramin and Rump to deal with bijective non-degenerate solutions. Hence, YB-semitrusses also yield a unified treatment of these different algebraic structures. The algebraic structure of YB-semitrusses is investigated and as a consequence it is proven, for example, that any finite left non-degenerate set-theoretic solution of the Yang-Baxter equation is right non-degenerate if and only if it is bijective. Furthermore, it is shown that some finite left non-degenerate solutions can be reduced to non-degenerate solutions of smaller size. The structure algebra of a finitely generated YB-semitruss is an algebra defined by homogeneous quadratic relations. We prove that it often is a left Noetherian algebra of finite Gelfand-Kirillov dimension that satisfies a polynomial identity, but in general it is not right Noetherian.}},
  author       = {{Colazzo, IIlaria and Jespers, Eric and Van Antwerpen, Arne and Verwimp, Charlotte}},
  issn         = {{0021-8693}},
  journal      = {{JOURNAL OF ALGEBRA}},
  keywords     = {{Yang-Baxter equation,Structure semigroup,Structure algebra,Semitruss,Set-theoretic solution,QUADRATIC ALGEBRAS,SEMI-BRACES,MONOIDS,SEMIGROUPS}},
  language     = {{eng}},
  pages        = {{409--462}},
  title        = {{Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses}},
  url          = {{http://doi.org/10.1016/j.jalgebra.2022.07.019}},
  volume       = {{610}},
  year         = {{2022}},
}
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