Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses
Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses
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Yang-Baxter 方程和半桁架的左非简并集合论解
DOI：
10.1016/j.jalgebra.2022.07.019
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发表时间：
2022
影响因子：
0.9
通讯作者：
Colazzo I
中科院分区：
文献类型：
--
作者：
Colazzo I
为了确定和分析 Yang-Baxter 方程（不一定是双射）的任意左非简并集合论解，我们引入了一种关联代数结构，称为 YB-半桁架，它形成了 Brzeziński 引入的半桁架类别的子类。 YB-半桁的基本示例是左非简并集合论解和（倾斜）左括号的结构幺半群。 Gateva-Ivanova 和 Van den Bergh 引入了结构幺半群，并展示了它们（以及结构代数）对于研究内卷非简并解的重要性。 Guarnieri、Vendramin 和 Rump 引入了斜左括号来处理双射非退化解。因此，YB-半桁也产生了对这些不同代数结构的统一处理。研究了 YB-半桁的代数结构，并由此证明，例如，Yang-Baxter 方程的任何有限左非简并集合论解是右非简并当且仅当它是双射的。此外，还表明一些有限左非简并解可以简化为更小尺寸的非简并解。有限生成的 YB-半桁架的结构代数是由齐次二次关系定义的代数。我们证明它通常是满足多项式恒等式的有限 Gelfand-Kirillov 维数的左诺特代数，但一般来说它不是右诺特代数。
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.
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大括号和杨-巴克斯特方程
DOI：
--
发表时间：
2012
期刊：
影响因子：
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作者：
F. Cedó;E. Jespers;J. Okniński
通讯作者：
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DOI：
10.1016/j.jalgebra.2016.05.024
发表时间：
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期刊：
arXiv: Group Theory
影响因子：
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作者：
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影响因子：
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影响因子：
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作者：
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影响因子：
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作者：
F. Catino;I. Colazzo;Paola Stefanelli
通讯作者：
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