Stratification of moduli spaces of Lie algebras, similar matrices and bilinear forms

TL;DR

This paper uses stratification techniques to analyze the structure of moduli spaces for complex Lie algebras, similar matrices, and bilinear forms, revealing their geometric and algebraic properties.

Contribution

It introduces a stratification approach to describe moduli spaces of matrices and bilinear forms, providing explicit descriptions for low-dimensional cases.

Findings

Complete stratification of similarity classes of matrices as projective orbifolds.

Explicit stratification of bilinear forms up to dimension 3.

Connection between Lie algebra moduli and geometric stratification methods.

Abstract

In this paper, the authors apply a stratification of moduli spaces of complex Lie algebras to analyzing the moduli spaces of nxn matrices under scalar similarity and bilinear forms under the cogredient action. For similar matrices, we give a complete description of a stratification of the space by some very simple projective orbifolds of the form P^n/G, where G is a subgroup of the symmetric group sigma_{n+1} acting on P^n by permuting the projective coordinates. For bilinear forms, we give a similar stratification up to dimension 3.