Integrability of n-dimensional dynamical systems of type E₇⁽¹⁾ and E₈⁽¹⁾

2004/09/26 by Tomoyuki Takenawa, Takenawa, Tomoyuki
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons #nlin.SI

paper · pdf · doi:10.48550/arxiv.nlin/0409051

openalex publication_date 2004/09/26 · arxiv created 2007/05/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose an n-dimensional analogue of elliptic difference Painlevé equation. Some Weyl group acts on a family of rational varieties obtained by successive blow-ups at m points in \mppⁿ(\mc), and in many cases they include the affine Weyl groups with symmetric Cartan matrices as subgroups. It is shown that the dynamical systems obtained by translations of these affine Weyl groups possess commuting flows and that their degrees grow quadratically. For the E₇⁽¹⁾ and E₈⁽¹⁾ cases, existence of preserved quantities is investigated. The elliptic difference case is also studied.

Citations

• Elliptic curves and birational representation of Weyl groups

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