Abstract
Tensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental rôle they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac–Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that contains and generalises the extension of an affine Kac–Moody algebra by a Virasoro derivation L_1. A conjecture about the complete superalgebra is formulated, relating it to the corresponding Borcherds superalgebra.
Highlights
Tensor hierarchy algebras [1,2] constitute a class of simple non-contragredient Lie superalgebras, whose finite-dimensional members are those of Cartan type in the classification of Kac [3]
It had already been clear that certain classes of Lie superalgebras were relevant for the gauge structure and dynamics of extended geometry [5,6,9,10], but it turns out that the tensor hierarchy algebras provide all information necessary, so that they may be viewed as basic building blocks of extended geometry
The hope of understanding extended geometry for infinite-dimensional, in particular hyperbolic, structure groups is the main motivation of the present work, and we hope that it will lead to a reformulation of gravity or supergravity where the hyperbolic Belinskii–Khalatnikov–Lifshitz group [33,34,35] emerges in extreme situations, but is an integral part of the formulation of the theory
Summary
Tensor hierarchy algebras [1,2] constitute a class of simple non-contragredient Lie superalgebras, whose finite-dimensional members are those of Cartan type in the classification of Kac [3]. The hope of understanding extended geometry for infinite-dimensional, in particular hyperbolic, structure groups is the main motivation of the present work, and we hope that it will lead to a reformulation of gravity or supergravity where the hyperbolic Belinskii–Khalatnikov–Lifshitz group [33,34,35] emerges in extreme situations, but is an integral part of the formulation of the theory This may eventually put the E10 [35] and E11 [11,36,37] proposals on a firm ground, and provide a mechanism for the emergence of space(-time).
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