Explicit determination of a 727-dimensional root space of the hyperbolic Lie algebra $E_{10}$

TL;DR

This paper explicitly determines a 727-dimensional root space of the hyperbolic Lie algebra E10 at a specific level, revealing complex string-theoretic structures within the algebra.

Contribution

It introduces a novel string theoretic method to explicitly compute and analyze a high-dimensional root space of E10, advancing understanding of its structure.

Findings

Explicit basis for the 727-dimensional root space obtained

Revealed complex transversal and longitudinal string states

Demonstrated the applicability of string theory methods to hyperbolic algebras

Abstract

The 727-dimensional root space associated with the level-2 root $\bLambda_1$ of the hyperbolic Kac--Moody algebra $E_{10}$ is determined using a recently developed string theoretic approach to hyperbolic algebras. The explicit form of the basis reveals a complicated structure with transversal as well as longitudinal string states present.