Cyclotomic discriminantal arrangements and diagram automorphisms of Lie algebras

TL;DR

This paper introduces cyclotomic discriminantal arrangements linked to affine hyperplane arrangements, establishing their algebraic structures and demonstrating the non-vanishing of Bethe vectors in related cyclotomic Gaudin models.

Contribution

It identifies a new class of arrangements and connects them to Lie algebra substructures, advancing understanding of cyclotomic Gaudin models and their algebraic properties.

Findings

Established correspondence between arrangements and Lie algebra complexes

Proved Bethe vectors in cyclotomic Gaudin models are nonzero

Linked hyperplane arrangements to diagram automorphisms of Lie algebras

Abstract

Recently a new class of quantum integrable models, the cyclotomic Gaudin models, were described in arXiv:1409.6937, arXiv:1410.7664. Motivated by these, we identify a class of affine hyperplane arrangements that we call cyclotomic discriminantal arrangements. We establish correspondences between the flag and Aomoto complexes of such arrangements and chain complexes for nilpotent subalgebras of Kac-Moody type Lie algebras with diagram automorphisms. As a byproduct, we show that the Bethe vectors of cyclotomic Gaudin models associated to diagram automorphisms are nonzero.