论文标题
Rota-Baxter操作员的偏斜牙套：共同体表征和一些例子
Skew braces from Rota--Baxter operators: a cohomological characterisation and some examples
论文作者
论文摘要
L. Guo，H。Lang和Y. Sheng最近引入了组的Rota-Baxter操作员。 V. G. Bardakov和V. Gubarev表明，对于每个Rota-Baxter操作员，一个人都可以将偏斜的支撑联系起来。组$ g $上的偏斜括号可以通过某些伽马功能从$ g $到其自动形态组$ \ operatatorName {aut}（g）$来表征，这些功能是由功能方程定义的。对于从rota-baxter运算符获得的偏斜括号，相应的伽马函数在内部自动形态组$ \ permatatorName {inn}（g）$ g $中的值。 在本文中，我们对$ g $的伽马功能进行了表征，其值来自rota-baxter操作员的$ \ operatatorName {inn}（g）$，就合适的第二个同学组中某个元素而消失。 利用这种表征，我们能够展示其相应的伽马函数在内部自动形态组中的值的偏斜牙套的示例，但无法从rota-baxter operator获得。 对于可以从Rota-baxter运算符获得的伽马函数，我们展示了如何从前者那里获得后者，从而利用了合适的中央组扩展的知识。
Rota-Baxter operators for groups were recently introduced by L. Guo, H. Lang, and Y. Sheng. V. G. Bardakov and V. Gubarev showed that with each Rota-Baxter operator one can associate a skew brace. Skew braces on a group $G$ can be characterised in terms of certain gamma functions from $G$ to its automorphism group $\operatorname{Aut}(G)$, that are defined by a functional equation. For the skew braces obtained from a Rota-Baxter operator the corresponding gamma functions take values in the inner automorphism group $\operatorname{Inn}(G)$ of $G$. In this paper, we give a characterisation of the gamma functions on a group $G$, with values in $\operatorname{Inn}(G)$, that come from a Rota-Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group. Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota--Baxter operator. For gamma functions that can be obtained from a Rota-Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits.
