The tensor hierarchy algebra

Abstract

We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for D between 3 and 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D-2-p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.

32 pages

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Cited by in corpus (35)

• The gauge structure of Exceptional Field Theories and the tensor hierarchy
• The Embedding Tensor, Leibniz-Loday Algebras, and Their Higher Gauge Theories
• Generalised diffeomorphisms for E${}_9$
• Beyond E11
• Extended geometries
• Tensor hierarchies and Leibniz algebras
• A master exceptional field theory
• E${}_9$ exceptional field theory II. The complete dynamics
• On supersymmetric E11 exceptional field theory
• Superalgebras, constraints and partition functions
• Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions
• Exceptional geometry and Borcherds superalgebras
• $L_{\infty }$ algebras for extended geometry from Borcherds superalgebras
• Tensor hierarchy algebras and extended geometry II: Gauge structure and dynamics
• Towards an M5-Brane Model II: Metric String Structures
• Forms and algebras in (half-)maximal supergravity theories
• Tensor hierarchy algebras and extended geometry I: Construction of the algebra
• $L_{\infty }$ algebras and Tensor Hierarchies in Exceptional Field Theory and Gauged Supergravity
• Tensor hierarchy algebra extensions of over-extended Kac--Moody algebras
• Generators and relations for Lie superalgebras of Cartan type
• Infinity-enhancing of Leibniz algebras
• On the different formulations of the E11 equations of motion
• Symmetries of M-theory and free Lie superalgebras
• Teleparallelism in the algebraic approach to extended geometry
• SL(5) supersymmetry
• All Chern-Simons Invariants of 4D, N = 1 Gauged Superform Hierarchies
• From Lie algebra crossed modules to tensor hierarchies
• Extended geometry of magical supergravities
• Nonabelian embedding tensors
• The teleparallel complex
• $L_{\infty }$ algebras for extended geometry
• Unfolding $E_{11}$
• Generators and relations for (generalised) Cartan type superalgebras
• Algebraic structures in exceptional geometry
• Dg Loday-Pirashvili modules over Lie algebras