Rota-Baxter算子的斜花括号：上同调刻画及一些例子
Skew braces from Rota--Baxter operators: a cohomological characterisation and some examples
摘要：
群的Rota-Baxter算子最近由L. Guo，H. Lang和Y.盛V. G. Bardakov和V. Gubarev证明了每个Rota-Baxter算子都可以与斜括号相关联。群$G$上的斜括号可以用从$G$到它的自同构群$\operatorname{Aut}(G)$的某些Γ函数来表征，这些伽马函数由函数方程定义。对于从Rota-Baxter算子获得的斜括号，相应的Γ函数取$G$的内自同构群$\operatorname{Inn}(G)$中的值。本文给出了群$G$上的Γ函数的一个特征，其值在$\operatorname{Inn}(G)$中，且来自Rota-Baxter算子，其特征是在一个适当的第二上同调群中某个元素为零.利用这一特点，我们能够展示斜括号的例子，其相应的Γ函数采取值的内自同构组，但不能从一个Rota-巴克斯特运营商。对于Γ函数，可以从一个Rota-Baxter运营商，我们展示了如何得到后者从前者，利用知识，一个合适的中心群扩展分裂。
英文摘要：
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.
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