Continuous selection of Lagrangian subspaces

TL;DR

This paper investigates continuous selection methods for maximal isotropic subspaces associated with skew-symmetric bilinear forms, with applications to Lie algebra structures and Grassmannian topology.

Contribution

It introduces a framework for continuous selection of Lagrangian subspaces and applies it to analyze Vergne polarizing subalgebras in solvable Lie algebras.

Findings

Established continuity properties of Vergne polarizing subalgebras

Connected isotropic subspaces to Schubert cells in Grassmannians

Provided new insights into the topology of Lie algebra substructures

Abstract

We study continuous selections of the set-valued map that takes every skew-symmetric bilinear form on a vector space to its corresponding set of maximal isotropic subspaces. Applications are made to establishing continuity properties of the Vergne polarizing subalgebras of completely solvable Lie algebras in terms of Schubert cells of suitable Grassmann manifolds.