Tensor hierarchy algebra extensions of over-extended Kac--Moody algebras

TL;DR

This paper explores tensor hierarchy algebras as super-extensions of over-extended Kac--Moody algebras, revealing new algebraic structures and proposing a conjecture linking them to Borcherds superalgebras, with implications for extended geometry.

Contribution

It introduces novel superalgebra extensions of over-extended Kac--Moody algebras and formulates a conjecture relating them to Borcherds superalgebras.

Findings

Extension of over-extended algebra by fundamental module generalizes affine Kac--Moody and Virasoro extension.

Contains novel algebraic structures relevant for extended geometry.

Proposes a conjecture connecting superalgebras to Borcherds superalgebras.

Abstract

Tensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental role they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac--Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that contains and generalises the extension of an affine Kac--Moody algebra by a Virasoro derivation $L_1$. A conjecture about the complete superalgebra is formulated, relating it to the corresponding Borcherds superalgebra.