AI 中文总结
研究旋转双扩张上舒昂平方在特定轨迹的限制，通过正交分解和复曲面矩映射等方法分析其径向与角向分量性质，给出相关估计和分类，不依赖非共振假设，使不变量算法化。
AI 中文摘要
设\(\g=\fb\oplus V\oplus\fb^{*}\)为一个梅迪纳 - 勒瓦伊旋转双扩张，其中阿贝尔李代数\(\fb\)通过标量旋转作用于定向欧几里得二维平面的正交和。我们分析舒昂平方\(r\mapsto[r,r]\)在块秩一轨迹\(\Rone\subset\fb\wedge V\)及其非退化开部分\(\Rplus\)上的限制。在每个标记切片\(S_{H}\subset\Rplus\)上，舒昂平方正交分解为径向和角向分量。径向分量通过标准复曲面矩映射分解，其像为单纯锥，满足具有内在最优常数的尖锐二次强制估计，且受限舒昂映射是恰当的，具有紧致半代数仿射纤维。角向分量在联合本征坐标中变为洛朗单项式，标记消去准则确定其符号指数配置，其积分格控制微分秩、紧致迷向、其分量群以及定义复化像的扎里斯基闭包的洛朗二项式理想。史密斯标准型使每个不变量算法化。所有陈述对任意\(\dim\fb\)都成立，且不施加非共振假设。基础不相交支撑分解的一个推论对\(\Rone\)上的经典杨 - 巴克斯特方程进行了分类：对于任何其\(\fb\wedge V\)分量位于\(\Rone\)中的二向量，三角性迫使由其旋转权重检测到的每个块消失，而与\(\Lambda^{2}\fb\)、\(\Lambda^{2}V\)或中心楔项无关。
英文摘要
Let $\g=\fb\oplus V\oplus\fb^{*}$ be a Medina-Revoy rotational double extension in which an abelian Lie algebra $\fb$ acts by scalar rotations on an orthogonal sum of oriented Euclidean two-planes. We analyse the restriction of the Schouten square $r\mapsto[r,r]$ to the block-rank-one locus $\Rone\subset\fb\wedge V$ and to its nondegenerate open part $\Rplus$. On every marked slice $S_{H}\subset\Rplus$ the Schouten square splits orthogonally into a radial and an angular component. The radial component factors through the standard toric moment map: its image is a simplicial cone, it satisfies a sharp quadratic coercive estimate with an intrinsic optimal constant, and the restricted Schouten map is proper with compact semialgebraic affine fibres. The angular component becomes Laurent monomial in joint eigencoordinates; a marker-cancellation criterion determines its signed exponent configuration, whose integral lattice controls the differential rank, the compact isotropy, its component group, and the Laurent binomial ideal defining the Zariski closure of the complexified image. Smith normal form makes each invariant algorithmic. All statements hold for arbitrary $\dim\fb$ and impose no nonresonance assumption. A corollary of the underlying disjoint-support decomposition classifies the classical Yang-Baxter equation on $\Rone$: for any bivector whose $\fb\wedge V$ component lies in $\Rone$, triangularity forces every block detected by its rotation weight to vanish, regardless of the $Λ^{2}\fb$, $Λ^{2}V$, or central-wedge terms.
