Quantum geometry and non-Lie algebras in basic mechanical systems

Three closely interlaced topics are considered:
- quantum geometry as obstruction to quantum uncertainty,
- non-Lie algebras in fundamental mechanical systems,
- the necessity of quantum geometry for quantum reduction.
We describe how to deform the classical symplectic geometry of the phase space in such a way that the given quantum integrable system becomes asymptotically equivalent to a deformed classical integrable system, thus preventing the quantum uncertainty and supporting the famous Bohr's correspondence and complementarity principles of the “old” quantum mechanics.
Interesting examples of quantum integrable systems arise from even elementary mechanics: the resonance oscillator (or the Penning trap, or the Zeeman-Stark effect, or the dyonic “atom”) in a perturbing field. The leading parts of these systems posses non-Lie algebras of symmetries with quadratic, cubic and other polynomial commutation relations depending on physical parameters.
In all these systems the reduction by the leading part, in order to be made at the quantum level, needs a deformation of the classical Kähler structure on the reduced phase space. The deformed quantum structure is generated by the nonlinear commutation relations of the symmetry algebra. In the routine Lie algebra case (linear commutation relations) the quantum deformation is absent.
All objects in the above procedures are introduced explicitly and imply effective and global, geometrically invariant formulas for asymptotical solutions of spectral and evolution problems.