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

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.