$S^7$ Current Algebras

TL;DR

This paper introduces $S^7$-algebras as a new class of generalized Kac-Moody algebras, provides free-field representations, and constructs octonionic projective spaces, expanding the algebraic and geometric framework involving octonions.

Contribution

It presents the formulation of $S^7$-algebras, their free-field representations, and the construction of octonionic projective spaces, offering new tools in algebra and geometry.

Findings

$S^7$-algebras are established as generalized Kac-Moody algebras.

Multiple free-field representations of $S^7$-algebras are derived.

Octonionic projective spaces ${ m O}P^N$ are constructed.

Abstract

We present $S^7$-algebras as generalized Kac-Moody algebras. A number of free-field representations is found. We construct the octonionic projective spaces ${\O}P^N$.