Borel-de Siebenthal theory for affine reflection systems

TL;DR

This paper develops a classification theory for maximal closed subroot systems in affine reflection systems, unifying various Lie algebra root systems and extending previous results to higher nullity toroidal Lie algebras.

Contribution

It introduces a Borel-de Siebenthal theory for affine reflection systems, providing a complete classification of their maximal closed subroot systems, especially for nullity $k$ toroidal Lie algebras.

Findings

Classifies maximal closed subroot systems via triples $(q,(b_i),H)$

Establishes a one-to-one correspondence with these triples for nullity $k$

Generalizes previous $k=1$ results to higher nullity cases

Abstract

We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(k\times k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras.