Tensor hierarchy algebras and restricted associativity

TL;DR

This paper introduces local algebras with restricted associativity linked to Kac-Moody algebras, leading to new insights into tensor hierarchy algebras and their potential applications in extended geometry.

Contribution

It constructs local algebras with restricted associativity from Kac-Moody data and connects them to tensor hierarchy algebras, revealing new structural insights.

Findings

Identification of generators satisfying tensor hierarchy algebra relations

Construction of local Lie superalgebras from local algebras

Potential applications to extended geometry

Abstract

We study local algebras, which are structures similar to $\mathbb{Z}$-graded algebras concentrated in degrees $-1,0,1$, but without a product defined for pairs of elements at the same degree $\pm1$. To any triple consisting of a Kac-Moody algebra $\mathfrak{g}$ with an invertible and symmetrisable Cartan matrix, a dominant integral weight of $\mathfrak{g}$ and an invariant symmetric bilinear form on $\mathfrak{g}$, we associate a local algebra satisfying a restricted version of associativity. From it, we derive a local Lie superalgebra by a commutator construction. Under certain conditions, we identify generators which we show satisfy the relations of the tensor hierarchy algebra $W$ previously defined from the same data. The result suggests that an underlying structure satisfying such a restricted associativity may be useful in applications of tensor hierarchy algebras to extended geometry.