The structure of \(q\)-symplectic geometry. (English) Zbl 0829.58015

de Gosson, M.

J. Math. Pures Appl., IX. Sér. 71, No. 5, 429-453 (1992).

Felix Klein defined a geometry by specifying a manifold and a Lie group acting on that manifold. J. Leray has shown in the first chapter of his treatise ‘Lagrangian analysis and quantum mechanics’ (1981; Zbl 0483.35002) that for every \(q= 1, 2, \dots, +\infty\) the \(q\)-fold covering group \(\text{Sq}_q (n)\) of the symplectic group \(\text{Sp} (n)\) acts on the \(2q\)-fold covering space \(\Lambda_{2q} (n)\) of the lagrangian Grassmannian \(\Lambda (n)\); each of the groups \(\text{Sp}_q (n)\) thus defines a geometry on \(\Lambda_{2q} (n)\), which Leray calls \(q\)- symplectic geometry.
The aim of this article is to show that the algebraic and topological structures of \(\text{Sp}_q (n)\) and \(\Lambda_q (n)\) can be described by using a modified Maslov index, which will be defined as a function \(\Lambda_\infty (n)\times \Lambda_\infty (n)\to \mathbb{Z}\), exempted of any transversality assumption. It will lead us ultimately to an explicit description of the action of \(\text{Sp}_q (n)\) on \(\Lambda_{2q} (n)\), that is, of the structure of \(q\)-symplectic geometry.

Reviewer: M.de Gosson (Sens, France)

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
57S05	Topological properties of groups of homeomorphisms or diffeomorphisms

Keywords:

manifold; Lie group; \(q\)-fold covering group; symplectic group; \(q\)- symplectic geometry; Maslov index

Citations:

Zbl 0483.35002