论文标题
在$λ$ - 巨型形态偏斜的括号
On $λ$-homomorphic skew braces
论文作者
论文摘要
对于偏斜的左撑杆$（g，\ cdot，\ circ）$，地图$λ：（g，circ）团体同态。然后，$λ$也可以看作是$（g，\ cdot）$到$ \ mathrm {aut} \; （g，\ cdot）$，总的来说，这可能不是同态。我们研究偏斜的左括号$（g，\ cdot，\ circ）$，$λ：（g，\ cdot）\ to \ mathrm {aut} \; （g，\ cdot）$是同态。这种偏斜的左括号将称为$λ$ -HOMORPHIC。我们制定了必要和充分的条件，在给定同构$λ：（g，\ cdot）\ to \ mathrm {aut} \; （g，\ cdot）$产生偏斜的左括号，实际上是$λ$ -Homorphic。作为一个应用程序，当$（g，\ cdot）$是免费组或一个免费的阿贝利安组时，我们构造偏斜的牙套。我们证明，任何$λ$ - 巨型形态偏斜的左括号都是微不足道的偏斜支撑物的微不足道偏斜支撑的延伸。特别重点是$λ$ - 巨型形态偏斜的左支架，其图像为$λ$是循环的。获得了二级自由的阿贝尔群体上的这种偏斜左括号的完整表征。
For a skew left brace $(G, \cdot, \circ)$, the map $λ: (G, \circ) \to \mathrm{Aut} \;(G, \cdot),~~a \mapsto λ_a$, where $λ_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $λ$ can also be viewed as a map from $(G, \cdot)$ to $\mathrm{Aut}\; (G, \cdot)$, which, in general, may not be a homomorphism. We study skew left braces $(G, \cdot, \circ)$ for which $λ: (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ is a homomorphism. Such skew left braces will be called $λ$-homomorphic. We formulate necessary and sufficient conditions under which a given homomorphism $λ: (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ gives rise to a skew left brace, which, indeed, is $λ$-homomorphic. As an application, we construct skew left braces when $(G, \cdot)$ is either a free group or a free abelian group. We prove that any $λ$-homomorphic skew left brace is an extension of a trivial skew brace by a trivial skew brace. Special emphasis is given on $λ$-homomorphic skew left brace for which the image of $λ$ is cyclic. A complete characterization of such skew left braces on the free abelian group of rank two is obtained.
