Abstract
We use quantum tori Lie algebras (QTLA), which are a one-parameter family of sub-algebras ofgl ∞, to describe local and non-local versions of the Toda systems. It turns out that the central charge of QTLA is responsible for the non-locality. There are two regimes in the local systems-conformal for irrational values of the parameter and non-conformal and integrable for its rational values. We also consider infinite-dimensional analogs of rigid tops. Some of these systems give rise to “quantized” (magneto-)hydrodynamic equations of an ideal fluid on a torus. We also consider infinite dimensional versions of the integrable Euler and Clebsch cases.
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Hoppe, J., Olshanetsky, M. & Theisen, S. Dynamical systems on quantum tori Lie algebras. Commun.Math. Phys. 155, 429–448 (1993). https://doi.org/10.1007/BF02096721
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DOI: https://doi.org/10.1007/BF02096721