On root systems in spaces with degenerate metric

TL;DR

This paper introduces root systems in spaces with degenerate metrics, showing their Cartan matrices and reflection groups are affine, and provides a simple algorithm to determine the root system structure of affine algebras.

Contribution

It defines root systems in Carroll spaces with degenerate metrics and characterizes their affine Cartan matrices and reflection groups, offering a straightforward algorithm for affine algebra root structures.

Findings

Root systems in Carroll spaces with degenerate metrics are affine.

Cartan matrices and reflection groups are affine.

A simple algorithm determines the root system structure of affine algebras.

Abstract

A root systems in Carroll spaces with degenerate metric are defined. It is shown that their Cartan matrices and reflection groups are affine. With the help of the geometric consideration the root system structure of affine algebras is determined by a sufficiently simple algorithm.