The structure of \(q\)-symplectic geometry
Felix Klein defined a geometry by specifying a manifold and a Lie group acting on that manifold. \textit{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.
scientific article; zbMATH DE number 811423