N-Extended Lorentzian Kac-Moody algebras

TL;DR

This paper introduces a new class of n-extended Lorentzian Kac-Moody algebras, providing their structural properties and methods to analyze their subalgebra decompositions, extending understanding of these complex algebraic structures.

Contribution

It defines n-extended Lorentzian Kac-Moody algebras, derives explicit formulas for their roots and weights, and develops techniques to analyze their subalgebra structures and signatures.

Findings

Derived generic expressions for roots, weights, and Weyl vectors.

Established criteria for algebra decomposition into simpler components.

Identified conditions for the presence of specific principal subalgebras.

Abstract

We investigate a class of Kac-Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac-Moody algebras defined by their Dynkin diagrams through the connection of an $A_n$ Dynkin diagram to the node corresponding to the affine root. The cases $n=1$ and $n=2$ correspond to the well studied over and very extended Kac-Moody algebras, respectively, of which the particular examples of $E_{10}$ and $E_{11}$ play a prominent role in string and M-theory. We construct closed generic expressions for their associated roots, fundamental weights and Weyl vectors. We use these quantities to calculate specific constants from which the nodes can be determined that when deleted decompose the n-extended Lorentzian Kac-Moody algebras into simple Lie algebras and Lorentzian Kac-Moody algebra. The signature of these constants also serves to establish whether the algebras possess $SO(1,2)$ and/or $SO(3)$-principal subalgebras.