Gaudin subalgebras and wonderful models

TL;DR

This paper explores the geometric structure of principal Gaudin subalgebras, revealing they form a smooth projective variety isomorphic to a well-known compactification related to Coxeter groups.

Contribution

It establishes a geometric characterization of principal Gaudin subalgebras as a smooth projective variety, linking algebraic and geometric structures in Coxeter arrangements.

Findings

Principal Gaudin subalgebras form a smooth projective variety.

This variety is isomorphic to the De Concini-Procesi compactification.

The work connects algebraic structures with geometric models.

Abstract

Gaudin hamiltonians form families of r-dimensional abelian Lie subalgebras of the holonomy Lie algebra of the arrangement of reflection hyperplanes of a Coxeter group of rank r. We consider the set of principal Gaudin subalgebras, which is the closure in the appropriate Grassmannian of the set of spans of Gaudin hamiltonians. We show that principal Gaudin subalgebras form a smooth projective variety isomorphic to the De Concini-Procesi compactification of the projectivized complement of the arrangement of reflection hyperplanes.