2022/07/25 by Martin Cederwall, Cederwall, Martin, Jakob Palmkvist +1 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2207.12417
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 \mathfrakg with an invertible and symmetrisable Cartan matrix, a dominant integral weight of \mathfrakg and an invariant symmetric bilinear form on \mathfrakg, 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.