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Affine Weyl groups, discrete dynamical systems and Painlevé equations. (English) Zbl 0952.37031
The authors propose a class of discrete dynamical systems associated with affine root systems, by constructing new representations of the affine Weyl group. The constructed representation provides naturally a discrete dynamical system from the lattice part of the affine Weyl group. It is shown how discrete dynamical systems could be applied to the problem of symmetry of nonlinear differential systems. As an application, the authors consider examples of difference and differential systems of Painlevé type.
MSC:
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K30Relations of infinite-dimensional systems with algebraic structures
17B67Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
34M55Painlevé and other special equations; classification, hierarchies
37K35Lie-Bäcklund and other transformations