Tensor Hierarchy Algebras and Restricted Associativity

Journal article, 2025

We study local algebras, which are structures similar to Z-graded algebras concentrated in degrees -1,0,1, but without a product defined for pairs of elements at the same degree ±1. To any triple consisting of a Kac–Moody algebra g with an invertible and symmetrisable Cartan matrix, a dominant integral weight of g and an invariant symmetric bilinear form on 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.