Generalized conformal realizations of Kac-Moody algebras

TL;DR

This paper introduces a new method to generate an infinite sequence of Kac-Moody algebras from a single Jordan algebra, extending known constructions and connecting to exceptional Lie algebras and conformal transformations.

Contribution

It generalizes the Kantor-Koecher-Tits construction by linking Jordan algebras to an infinite sequence of Kac-Moody algebras, including exceptional and infinite-dimensional cases.

Findings

Constructs infinite sequences of Kac-Moody algebras from Jordan algebras.

Recovers exceptional Lie algebras f4, e6, e7, e8 for specific cases.

Extends conformal transformations to higher-dimensional and infinite-dimensional settings.

Abstract

We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n=1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of hermitian 3x3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n=2. Moreover, we obtain their infinite-dimensional extensions for n greater or equal to 3. In the case of 2x2 matrices the resulting Lie algebras are of the form so(p+n,q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q).