Tensor hierarchy algebras and extended geometry I: Construction of the algebra

TL;DR

This paper constructs tensor hierarchy algebras using generators and relations encoded in Dynkin diagrams, extending Lie superalgebras with applications in extended geometry and supergravity.

Contribution

It develops a new definition of tensor hierarchy algebras via Dynkin diagrams and applies it to extended Kac-Moody algebra cases, proving a new identity for representation matrices.

Findings

Constructed tensor hierarchy algebras from Dynkin diagrams with added grey nodes.

Proved a new identity involving representation matrices for integral highest weight representations.

Extended the algebraic framework relevant for extended geometry and supergravity models.

Abstract

Tensor hierarchy algebras constitute a class of non-contragredient Lie superalgebras, whose finite-dimensional members are the "Cartan-type" Lie superalgebras in Kac's classification. They have applications in mathematical physics, especially in extended geometry and gauged supergravity. We further develop the recently proposed definition of tensor hierarchy algebras in terms of generators and relations encoded in a Dynkin diagram (which coincides with the diagram for a related Borcherds superalgebra). We apply it to cases where a grey node is added to the Dynkin diagram of a rank $r+1$ Kac-Moody algebra $\mathfrak{g}^+$, which in turn is an extension of a rank $r$ finite-dimensional semisimple simply laced Lie algebra $\mathfrak{g}$. The algebras are specified by $\mathfrak{g}$ together with a dominant integral weight $\lambda$. As a by-product, a remarkable identity involving representation matrices for arbitrary integral highest weight representations of $\mathfrak{g}$ is proven. An accompanying paper describes the application of tensor hierarchy algebras to the gauge structure and dynamics in models of extended geometry.