Abstract
A deformation of the classical trigonometric BCn Sutherland system is derived via Hamiltonian reduction of the Heisenberg double of SU(2n). We apply a natural Poisson–Lie analogue of the Kazhdan–Kostant–Sternberg type reduction of the free particle on SU(2n) that leads to the BCn Sutherland system. We prove that this yields a Liouville integrable Hamiltonian system and construct a globally valid model of the smooth reduced phase space wherein the commuting flows are complete. We point out that the reduced system, which contains 3 independent coupling constants besides the deformation parameter, can be recovered (at least on a dense submanifold) as a singular limit of the standard 5-coupling deformation due to van Diejen. Our findings complement and further develop those obtained recently by Marshall on the hyperbolic case by reduction of the Heisenberg double of SU(n,n).
Highlights
Models amenable to exact treatment provide key paradigms for our understanding of natural phenomena and form a fertile field of research crossing the border of physics and mathematics
In Subsection 4.1 we prove that the reduced phase space is smooth, as formulated in Theorem 4.4
The smoothness of the reduced phase space and the completeness of the reduced free flows follows immediately if we can show that the gauge group Gμ acts in such a way on Φ−+1(μ) that the isotropy group of every point is just the finite center of the symmetry group
Summary
Models amenable to exact treatment provide key paradigms for our understanding of natural phenomena and form a fertile field of research crossing the border of physics and mathematics. We shall deal with a reduction of the Heisenberg double of SU(2n) and derive a Liouville integrable Hamiltonian system related to Marshall’s one in a way similar to the connection between the original trigonometric Sutherland system and its hyperbolic variant. This is essentially analytic continuation, it should be noted that the resulting systems are qualitatively different in their dynamical characteristics and global features. Marshall [20] obtained similar results for an analogous deformation of the hyperbolic BCn Sutherland Hamiltonian His deformed Hamiltonian differs from (1.1) above in some important signs and in the relevant domain of the ‘position variables’ p. Appendix A deals with the connection to van Diejen’s system; the other 3 appendices contain important details relegated from the main text
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