Abstract
We discuss the lift of Poisson structures associated with auxiliary linear problems for the differential and difference Lax equations to the space of wave functions. Due to a peculiar symmetry breaking, the corresponding differential and difference Galois groups become Poisson Lie Groups.
Highlights
It is well known that the transition from Classical to Quantum Mechanics is best described in terms of the deformation of the associated algebras of observables
In both dierence q-dierence operators and q-dierence cases the resulting bracket is Poisson covariant with respect to the right action of group of constant matrices; generically, it is isomorphic to the dierence Galois group [4]
The explicit formulae will be presented in a separate publication
Summary
It is well known that the transition from Classical to Quantum Mechanics is best described in terms of the deformation of the associated algebras of observables. There exist not non-trivial examples when a natural group action does preserve the Poisson brackets. A natural Poisson structure for the potentials is related to the famous Virasoro algebra It is natural to ask whether there is a natural Poisson structure on wave functions the space of of a Schroedinger operator. This question is of practical interest in application to a family of KdV-like equation. The space of second order dierential operators carries a natural Poisson struQctuuerest(iionnci:deEnxttaelnlyd, this this is the PoissonVirasoro algebra). The Poisson structure associated with rst order matrix difference operators reveals similar anomalies which are the main subject of the present talk
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