From Tensor Algebras to Hyperbolic Kac-Moody Algebras

TL;DR

This paper introduces a new tensor algebra approach to study hyperbolic Kac-Moody algebras, revealing their structure through affine and coset Virasoro symmetries up to level five.

Contribution

It develops a novel method based on tensor algebras and Virasoro symmetries to analyze hyperbolic Kac-Moody algebras, providing explicit decompositions and generating states.

Findings

Complete decomposition of tensor algebra for levels up to 5.

Identification of maximal tensor ground states.

Expression of algebra elements via multi-commutators and DDF states.

Abstract

We propose a novel approach to study hyperbolic Kac-Moody algebras, and more specifically, the Feingold-Frenkel algebra $\mathfrak{F}$, which is based on considering the tensor algebra of level-one states before descending to the Lie algebra by converting tensor products into multiple commutators. This method enables us to exploit the presence of mutually commuting coset Virasoro algebras, whose number grows without bound with increasing affine level. We present the complete decomposition of the tensor algebra under the affine and coset Virasoro symmetries for all levels $\ell\leq 5$, as well as the maximal tensor ground states from which all elements of $\mathfrak{F}$ up to level five can be (redundantly) generated by the joint action of the affine and coset Virasoro generators, and subsequent conversion to multi-commutators, which are then expressed in terms of transversal and longitudinal DDF states. We outline novel directions for future work.