Finite Idempotent Set-Theoretic Solutions of the Yang-Baxter Equation
Finite Idempotent Set-Theoretic Solutions of the Yang-Baxter Equation
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Yang-Baxter方程的有限幂等集论解
DOI：
10.1093/imrn/rnad183
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发表时间：
2023
影响因子：
1
通讯作者：
Colazzo I
中科院分区：
文献类型：
--
作者：
Colazzo I
证明了集上的Yang-Baxter方程的有限幂等左非退化集论解是由上的左单半群结构（特别是群的同构副本的有限并）和上的映射决定的.当相应的Yang-Baxter单射可消且所有的映射都等于这个群的一个自同构时，这个结构就成为一个群。等价地，杨-巴克斯特代数是右诺特代数，或者在特征零点它必须是半素的。Yang-Baxter代数始终是Gelfand-Kirillov维数为1的左Noether可表示代数。为了证明这些结果，我们证明了Yang-Baxter半群分解为许多对角标可消半群，每个可消半群有一个有限乘无限循环群，这些群的并具有左单半群的结构.的情况下，等于对角线是完全描述了一个单一的置换。
It is proven that finite idempotent left non-degenerate set-theoretic solutionsof the Yang–Baxter equation on a setare determined by a left simple semigroup structure on(in particular, a finite union of isomorphic copies of a group) and some mapsandon, for. This structure turns out to be a group precisely when the associated Yang–Baxter monoidis cancellative and all the mapsare equal to an automorphism of this group. Equivalently, the Yang–Baxter algebrais right Noetherian, or in characteristic zero it has to be semiprime. The Yang–Baxter algebra is always a left Noetherian representable algebra of Gelfand–Kirillov dimension one. To prove these results, it is shown that the Yang–Baxter semigrouphas a decomposition in finitely many cancellative semigroupsindexed by the diagonal, eachhas a group of quotientsthat is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case thatequals the diagonal is fully described by a single permutation on.