Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation
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DOI：
10.1016/j.aim.2016.09.024
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发表时间：
2015-09
影响因子：
1.7
通讯作者：
V. Lebed;L. Vendramin
中科院分区：
文献类型：
--
作者：
V. Lebed;L. Vendramin
本文讨论了杨-巴克斯特方程的左非退化集合论解（= LND解），这是一个包含群、架和圈集的庞大的代数结构类。对于每个这样的解决方案，存在相关联的架子（即，一个自我分配的结构），它抓住了它的主要属性。我们认为两个（上）同调理论的LND解决方案，其中之一是以前已知的，在一个简化的形式，只为biracks。一个明确的同构这些理论之间的描述。对于组和机架，我们恢复他们的经典（上）同调，而循环集，我们得到新的建设。对于某种类型的LND解，包括quandles和非退化循环集，（共）同调分裂为退化和归一化部分。我们表示2-上循环我们的理论在群上同调，并在循环集的情况下，建立与扩展的连接。这导致建设周期集有趣的性质。
This paper deals with left non-degenerate set-theoretic solutions to the Yang–Baxter equation (= LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution there is associated a shelf (i.e., a self-distributive structure) which captures its major properties. We consider two (co)homology theories for LND solutions, one of which was previously known, in a reduced form, for biracks only. An explicit isomorphism between these theories is described. For groups and racks we recover their classical (co)homology, whereas for cycle sets we get new constructions. For a certain type of LND solutions, including quandles and non-degenerate cycle sets, the (co)homologies split into the degenerate and the normalized parts. We express 2-cocycles of our theories in terms of group cohomology, and, in the case of cycle sets, establish connexions with extensions. This leads to a construction of cycle sets with interesting properties.