The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation
The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation
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DOI：
10.1090/tran/7837
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发表时间：
2018-12
影响因子：
1.3
通讯作者：
E. Jespers;L. Kubat;A. V. Antwerpen
中科院分区：
文献类型：
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作者：
E. Jespers;L. Kubat;A. V. Antwerpen
对于 Yang-Baxter 方程的有限对合非简并解 ( X , r ) (X,r) ，已知结构幺半群 M ( X , r ) M(X,r) 是 I 型幺半群，并且域 K K 上的结构代数 K [ M ( X , r ) ] K[M(X,r)] 与可交换多项式代数共享许多性质；特别是，它是一个具有有限 Gelfand-Kirillov 维数的 Noetherian PI 域。在本文中，我们处理任意有限（左）非简并解。尽管幺半群 M ( X , r ) M(X,r) 和代数 K [ M ( X , r ) ] K[M(X,r)] 的结构比对合情况复杂得多，但我们提供了一些深刻的见解。在这种一般情况下，使用 M ( X , r ) M(X,r) 的 Lebed 和 Vendramin 的实现作为半直积中的正则子幺半群 A ( X , r ) ⋊ Sym ( X ) A(X,r)\rtimes \operatorname {Sym} (X) ，其中 A ( X , r ) A(X,r) 是与 ( X 相关的机架解的结构幺半群, r ) (X,r) ，我们证明 K [ M ( X , r ) ] K[M(X,r)] 是中心仿射子代数上的有限模。特别地，K [ M ( X , r ) ] K[M(X,r)] 是有限 Gelfand-Kirillov 维数的诺特 PI 代数，其边界为 | X | |X| 。我们还用 K [ M ( X , r ) ] K[M(X,r)] 的环理论术语来表征，当 ( X , r ) (X,r) 是对合解时。该表征特别为有关 M <mml:mo stretchy=" 的抵消性的 Gateva-Ivanova 猜想提供了肯定的答案
For a finite involutive non-degenerate solution ( X , r ) (X,r) of the Yang–Baxter equation it is known that the structure monoid M ( X , r ) M(X,r) is a monoid of I-type, and the structure algebra K [ M ( X , r ) ] K[M(X,r)] over a field K K shares many properties with commutative polynomial algebras; in particular, it is a Noetherian PI-domain that has finite Gelfand–Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid M ( X , r ) M(X,r) and the algebra K [ M ( X , r ) ] K[M(X,r)] is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of M ( X , r ) M(X,r) as a regular submonoid in the semidirect product A ( X , r ) ⋊ Sym ( X ) A(X,r)\rtimes \operatorname {Sym} (X) , where A ( X , r ) A(X,r) is the structure monoid of the rack solution associated to ( X , r ) (X,r) , we prove that K [ M ( X , r ) ] K[M(X,r)] is a finite module over a central affine subalgebra. In particular, K [ M ( X , r ) ] K[M(X,r)] is a Noetherian PI-algebra of finite Gelfand–Kirillov dimension bounded by | X | |X| . We also characterize, in ring-theoretical terms of K [ M ( X , r ) ] K[M(X,r)] , when ( X , r ) (X,r) is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of M <mml:mo stretchy="