Second order polynomial Hamiltonian systems with ${\tilde W}(E_6^{(1)}),{\tilde W}(E_7^{(1)})$ and $W(E_8^{(1)})$-symmetry

Details Second order polynomial Hamiltonian systems with ${\tilde W}(E_6^{(1)}),{\tilde W}(E_7^{(1)})$ and $W(E_8^{(1)})$-symmetry Second order polynomial Hamiltonian systems with ${\tilde W}(E_6^{(1)}),{\tilde W}(E_7^{(1)})$ and $W(E_8^{(1)})$-symmetry

2007-05-22

arxiv.org/abs/0705.3137v4

Mathematics

Algebraic Geometry

27 pages, 7 figures

Scientific paper

We find and study a six (resp. seven, eight)-parameter family of polynomial Hamiltonian systems of second order, respectively. This system admits the affine Weyl group symmetry of type $E_6^{(1)}$ (resp. $E_7^{(1)}, E_8^{(1)}$) as the group of its B{\"a}cklund transformations. Each system is the first example which gave second-order polynomial Hamiltonian system with ${\tilde W}(E_6^{(1)})$ (resp. ${\tilde W}(E_7^{(1)}), W(E_8^{(1)})$)-symmetry. We also show that its space of initial conditions $S$ is obtained by gluing eight (resp. nine, ten) copies of ${\Bbb C}^2$ via the birational and symplectic transformations.

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