$q$-形变交比：模不变量与Coxeter花彩
The $q$-deformed cross-ratio: modular invariants and Coxeter friezes
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中文总结 AI 辅助
本文引入$q$-形变交比，基于$q$-形变有理数，证明其在模群下不变，并与$q$-形变Coxeter花彩关联，展开得到模不变量序列并计算前两个系数。
中文摘要 AI 辅助
我们引入并研究了$\u005cmathbb P^1(\u005cmathbb Q)$上交比的标量$q$-形变。我们的构造基于Morier-Genoud和作者提出的$q$-形变有理数概念。$q$-交比在$\u005cmathrm{PSL}(2,\u005cmathbb{Z})$作用下不变，而$\u005cmathrm{PGL}(2,\u005cmathbb{Z})$中行列式为$-1$的元素通过$q\u005cmapsto q^{-1}$作用。一个主要结果是它与有理多边形相关的$q$-形变Coxeter花彩的联系。在$q=e^h$处的展开产生了一个代数独立的模不变量和相对不变量序列，尽管该序列不能分离模轨道。我们明确计算了该展开的前两个非零非常数系数。
英文摘要
We introduce and study a scalar $q$-deformation of the cross-ratio on $\mathbb P^1(\mathbb Q)$. Our construction is based on the notion of $q$-deformed rational numbers due to Morier-Genoud and the author. The $q$-cross-ratio is invariant under $\mathrm{PSL}(2,\mathbb{Z})$, while elements of determinant $-1$ of $\mathrm{PGL}(2,\mathbb{Z})$ act by $q\mapsto q^{-1}$. A principal result is its relation to $q$-deformed Coxeter friezes associated with rational polygons. The expansion at $q=e^h$ yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.
