Cartanification of contragredient Lie superalgebras

TL;DR

This paper introduces a new construction called cartanification for certain Lie superalgebras, compares it to existing tensor hierarchy algebras, and proves a conjecture relating the two, with potential applications in physics.

Contribution

It defines the cartanification of Lie superalgebras, compares it to tensor hierarchy algebras, and proves conditions for their isomorphism, confirming a previous conjecture.

Findings

Established conditions for isomorphism between $W$ and $B^W$

Proved a conjecture relating cartanification and tensor hierarchy algebras

Suggested applications in extended geometry in physics

Abstract

Let $B$ be a $\mathbb{Z}$-graded Lie superalgebra equipped with an invariant $\mathbb{Z}_2$-symmetric homogeneous bilinear form and containing a grading element. Its local part (in the terminology of Kac) $B_{-1} \oplus B_{0} \oplus B_{1}$ gives rise to another $\mathbb{Z}$-graded Lie superalgebra, recently constructed in arXiv:2207.12417, that we here denote $B^W$ and call the cartanification of $B^W$, since it is of Cartan type in the cases where it happens to finite-dimensional. In cases where $B$ is given by a generalised Cartan matrix, we compare $B^W$ to the tensor hierarchy algebra $W$ constructed from the same generalised Cartan matrix by a modification of the generators and relations. We generalise this construction and give conditions under which $W$ and $B^W$ are isomorphic, proving a conjecture in arXiv:2207.12417. We expect that the algebras with restricted associativity underlying the cartanifications will be useful in applications of tensor hierarchy algebras to the field of extended geometry in physics.