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Transverse structures to orbits in algebraic groups. (English) Zbl 0901.58014

Summary: We study nonlinear Poisson brackets associated with orbits through nilpotent conjugacy classes in \(gl (n,\mathbb{C})\) and formulate some conjectures. We determine the degree of the transverse Poisson structure through such nilpotent elements in \(gl(n, \mathbb{C})\) for \(n \leq 7\). This is accomplished by using Dirac’s bracket formula. The conjectures are proved in some special cases.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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