XXVI Workshop on Geometric Methods in Physics 1-7.07.2007

Ian Marshall

Hidden Poisson structures related to Hill's equation

As argued by George Wilson, the Miura map $v\mapsto u$ may be seen as part of a chain of field extensions $\CC\!<\!u\!>\subset\CC\!<\!v\!>\subset\CC\!<\!\phi,\psi\!>$. By (i) incorporating monodromy in the space of solutions to the linear problem $\psi''+u\psi=0$ and (ii) extending the notion of Poisson symmetry to that of Poisson Lie group type, the result of Wilson is shown to be an example of Poisson reduction. The result is extended to the lattice analogue by direct means.