$E_{11},K_{11}$ and $EE_{11}$

TL;DR

This paper explores the algebraic structures underlying M-theory, showing that certain roots and weights of Kac-Moody algebras $K_{11}$ and $EE_{11}$ align with those of $E_{11}$, supporting a non-linear realization framework.

Contribution

It demonstrates the correspondence between roots of $K_{11}$ and $EE_{11}$, and links fundamental weights of $EE_{11}$ to $E_{11}$'s brane charge representations, extending to $E_{10}$ and $E_9$.

Findings

Roots of $K_{11}$ match roots of $EE_{11}$.

One fundamental weight of $EE_{11}$ equals the $l_1$ weight of $E_{11}$.

Extension of root correspondence to $E_{10}$ and $E_9$.

Abstract

In the study of conjecture on M-theory as a non-linear realization $E_{11}/K_{11}$ we present arguments for the following: 1)roots of $K_{11}$ coincide with the roots of Kac-Moody algebra $EE_{11}$ with Dynkin diagram given in the paper, 2)one of the fundamental weights of $EE_{11}$ coincides with $l_1$ weight of $E_{11}$, known to contain 11d supergravity brane charges. The statement 1) is extended on $E_{10}$ and $E_9$ algebras.