Decomposition of matrices from $SL_ 2(K[x, y])$
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arXiv
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2024
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| _version_ | 1866929616739893248 |
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| author | Chapovskyi, Y. Kozachok, O. Petravchuk, A. |
| author_facet | Chapovskyi, Y. Kozachok, O. Petravchuk, A. |
| contents | Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{K}[x,y]$ the polynomial ring. The group $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ of all matrices with determinant equal to $1$ over $\mathbb{K}[x,y]$ can not be generated by elementary matrices. The known counterexample was pointed out by P.M. Cohn. Conversely, A.A.Suslin proved that the group $\text{SL}_{r}\left(\mathbb{K}[x_{1},\dots,x_{n}]\right)$ is generated by elementary matrices for $r\ge 3$ and arbitrary $n\geq 2$, the same is true for $n=1$ and arbitrary $r.$ It is proven that any matrix from $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ with at least one entry of degree $\le 2$ is either a product of elementary matrices or a product of elementary matrices and of a matrix similar to the one pointed out by P. Cohn. For any matrix $\begin{pmatrix}\begin{array}{cc} f & g\\ -Q & P \end{array}\end{pmatrix}\in\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$, we obtain formulas for the homogeneous components $P_i , Q_i$ for the unimodular row $(-Q, P) $ as combinations of homogeneous components of the polynomials $f, g, $ respectively, with the same coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03688 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decomposition of matrices from $SL_ 2(K[x, y])$ Chapovskyi, Y. Kozachok, O. Petravchuk, A. Group Theory Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{K}[x,y]$ the polynomial ring. The group $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ of all matrices with determinant equal to $1$ over $\mathbb{K}[x,y]$ can not be generated by elementary matrices. The known counterexample was pointed out by P.M. Cohn. Conversely, A.A.Suslin proved that the group $\text{SL}_{r}\left(\mathbb{K}[x_{1},\dots,x_{n}]\right)$ is generated by elementary matrices for $r\ge 3$ and arbitrary $n\geq 2$, the same is true for $n=1$ and arbitrary $r.$ It is proven that any matrix from $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ with at least one entry of degree $\le 2$ is either a product of elementary matrices or a product of elementary matrices and of a matrix similar to the one pointed out by P. Cohn. For any matrix $\begin{pmatrix}\begin{array}{cc} f & g\\ -Q & P \end{array}\end{pmatrix}\in\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$, we obtain formulas for the homogeneous components $P_i , Q_i$ for the unimodular row $(-Q, P) $ as combinations of homogeneous components of the polynomials $f, g, $ respectively, with the same coefficients. |
| title | Decomposition of matrices from $SL_ 2(K[x, y])$ |
| topic | Group Theory |
| url | https://arxiv.org/abs/2412.03688 |