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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2408.06589 |
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| _version_ | 1866911986380439552 |
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| author | Nasybullov, T. Novikov, I. |
| author_facet | Nasybullov, T. Novikov, I. |
| contents | If $A=(A,\oplus,\odot)$ is a $λ$-homomorphic brace with $(A,\oplus)=\mathbb{Z}^2$, then the operations in this brace are given by formulas \begin{align*}\begin{pmatrix}a_1\\a_2\end{pmatrix}\oplus\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1+b_1\\a_2+b_2\end{pmatrix},&&\begin{pmatrix}a_1\\a_2\end{pmatrix}\odot\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1\\a_2\end{pmatrix}+φ^{a_1}ψ^{a_2}\begin{pmatrix}b_1\\b_2\end{pmatrix}, \end{align*} where $φ,ψ\in{\rm GL}_2(\mathbb{Z})$ are cpecific matrices which depend on $A$. Not every pair $(φ,ψ)$ lead to a brace. In the present paper we find all possible pairs $(φ,ψ)$ of matrices from ${\rm GL}_2(\mathbb{Z})$ which lead to $λ$-homomorphic braces with $(A,\oplus)=\mathbb{Z}^2$. The obtained result gives the full classification of $λ$-homomorphic braces on $\mathbb{Z}^2$ which was started by Bardakov, Neshchadim and Yadav in [J. Pure App. Algebra, V. 226, N. 6, 2022, 106961]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06589 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of $λ$-homomorphic braces on $\mathbb{Z}^2$ Nasybullov, T. Novikov, I. Group Theory 16T25, 81R50 If $A=(A,\oplus,\odot)$ is a $λ$-homomorphic brace with $(A,\oplus)=\mathbb{Z}^2$, then the operations in this brace are given by formulas \begin{align*}\begin{pmatrix}a_1\\a_2\end{pmatrix}\oplus\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1+b_1\\a_2+b_2\end{pmatrix},&&\begin{pmatrix}a_1\\a_2\end{pmatrix}\odot\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1\\a_2\end{pmatrix}+φ^{a_1}ψ^{a_2}\begin{pmatrix}b_1\\b_2\end{pmatrix}, \end{align*} where $φ,ψ\in{\rm GL}_2(\mathbb{Z})$ are cpecific matrices which depend on $A$. Not every pair $(φ,ψ)$ lead to a brace. In the present paper we find all possible pairs $(φ,ψ)$ of matrices from ${\rm GL}_2(\mathbb{Z})$ which lead to $λ$-homomorphic braces with $(A,\oplus)=\mathbb{Z}^2$. The obtained result gives the full classification of $λ$-homomorphic braces on $\mathbb{Z}^2$ which was started by Bardakov, Neshchadim and Yadav in [J. Pure App. Algebra, V. 226, N. 6, 2022, 106961]. |
| title | Classification of $λ$-homomorphic braces on $\mathbb{Z}^2$ |
| topic | Group Theory 16T25, 81R50 |
| url | https://arxiv.org/abs/2408.06589 |