We show that discrete $W_m$ lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler-Gel'fand-Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structure defined on the space of geometric invariants by lifting them to a pair of pre-symplectic forms on the space of arc length parametrized polygons. The simplicity of the expressions of the pre-symplectic forms makes the proof of compatibility straightforward. We also study their kernels and possible integrable systems associated to the pair.

中文翻译：

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Centro-Affine ℝm 中扭曲多边形的可积演化

我们使用 Adler-Gel'fand-Dikii 流离散化的几何实现作为中心仿射空间中弧长参数化多边形的局部演化，表明离散的 $W_m$ 晶格是双汉密尔顿的。我们通过将它们提升到弧长参数化多边形空间上的一对前辛形式来证明定义在几何不变量空间上的两个已知哈密顿结构的兼容性。前辛形式表达式的简单性使得兼容性证明变得简单明了。我们还研究了它们的内核和与这对相关的可能的可集成系统。