Integrable pentagram-type maps on polyhedra via partial difference operators

2026/06/25 by Bao Wang · 1 voice
Physics and Astronomy · #nlin.SI #math-ph

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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 \mathbbRP³ 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.

Citations

• Generalized pentagram maps via Q-nets and refactorization mappings
• On non-commutative leapfrog map
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• Two-dimensional algebro-geometric difference operators
• Transformations of quadrilateral lattices
• Geometric discretisation of the Toda system

Discussions

• Bao Wang: Integrable pentagram-type maps on polyhedra via partial difference operators https://arxiv.org/abs/2606.26526 https://arxiv.org/pdf/2606.26526 https://arxiv.org/html/2606.26526 [bsky, 0 points, 0 comments]

Related

• Pentagram maps and refactorization in Poisson-Lie groups
• The Pentagram map: a discrete integrable system
• Generalized pentagram maps via Q-nets and refactorization mappings
• Quasiperiodic Motion for the Pentagram Map
• The Algebraic Dynamics of the Pentagram Map