Cartanification of contragredient Lie superalgebras

2023/09/25 by Martin Cederwall, Cederwall, Martin, Jakob Palmkvist +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2309.14423

Abstract

Let B be a ℤ-graded Lie superalgebra equipped with an invariant ℤ₂-symmetric homogeneous bilinear form and containing a grading element. Its local part (in the terminology of Kac) B_-1 ⊕ B₀ ⊕ B₁ gives rise to another 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.

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• Non-associative structures in extended geometry

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• Tensor hierarchy algebras and restricted associativity
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