一阶线性q-差分系统的等单值形变
Isomonodromic Deformations for Linear $q$-Difference Systems of Degree One
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中文总结 AI 辅助
本文针对首项系数为对角矩阵的一阶线性q-差分系统构造等单值变换，得到离散局部tau函数，其极限对应$\boldsymbol{\text{P}^1}$上特定亚纯联络的等单值形变。
中文摘要 AI 辅助
我们针对形如$Y(qz)=A(z)Y(z)$的一阶线性q-差分系统构造等单值变换，其中$A(z)=A_0+z A_1$且首项系数为对角矩阵。这些变换会移动首项系数$A_1$的特征值及$\text{det}A(z)$的根，将相容性提升至$A(z)$的右特征对，得到离散局部tau函数。所得形变方程保持Birkhoff连接矩阵，且在$q\to1$极限下退化为$\boldsymbol{\text{P}^1}$上具有无穷远点处Poincaré秩为1的不规则奇点的亚纯联络的等单值形变。
英文摘要
We construct isomonodromy transformations for linear $q$-difference systems of the form $Y(qz)=A(z)Y(z)$, where $A(z)=A_0+z A_1$ has diagonal leading coefficient. These transformations shift eigenvalues of the leading coefficient $A_1$ together with roots of $\det A(z)$. They lift compatibility to the right eigenpairs of $A(z)$, yielding a discrete local tau function. The resulting deformation equations preserve the Birkhoff connection matrix, and reduce in their $q\to 1$ limit to the isomonodromic deformation of a meromorphic connection on $\mathbb P^1$ with an irregular singularity of Poincaré rank one at $\infty$.
