Integrable pentagram-type maps on polyhedra via partial difference operators

Bao Wang

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arXiv · 2606.26526

Integrable pentagram-type maps on polyhedra via partial difference operators

Bao Wang

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Abstract

This paper introduces a family of natural generalizations of the pentagram map from polygons to (twisted) polyhedra and proves their integrability through the partial difference operators. A canonical special case, which corresponds to the discrete Laplace transformation of discrete conjugate nets, is investigated in detail. We first establish a canonical bijection between the projective equivalence classes of these polyhedra in $\mathbb{RP}^3$ and the spectral data of doubly periodic partial difference operators modulo the gauge actions. Furthermore, we prove the complete integrability of these pentagram-type maps by explicitly identifying them with the refactorization maps on the Poisson-Lie group of pseudo partial difference operators. This algebraic identification naturally yields an explicit Lax representation and an $r$-matrix induced Poisson bracket for the geometric dynamics.

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Bao Wang. 2026-06-25. Integrable pentagram-type maps on polyhedra via partial difference operators. https://arxiv.org/abs/2606.26526

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