From Lie algebra crossed modules to tensor hierarchies |
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Affiliation: | 1. Euler Institute (EIMI) & Steklov Institute, Fontanka River Embankment 27, 191023, St-Petersburg, Russia;2. Aristotle University of Thessaloniki, Department of Mathematics, 541 24, Thessaloniki, Greece;3. 109 Holly Dr, Lansdale, PA 19446, USA;1. School of Mathematical Sciences, Narsi Monji Institute of Management Sciences (NMIMS), V.L. Mehta Road, Vile Parle West, Mumbai 400056, India;2. School of Mathematical and Statistical Sciences, IIT Mandi, Mandi 175005 (HP), India;1. Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA;2. Mathematics Institute, Zeeman Building, Coventry CV4 7AL, UK |
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Abstract: | The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing – the embedding tensor – between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra – its associated tensor hierarchy – whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ‘lie-ization’ of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories. |
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Keywords: | Differential graded Lie (super)-algebra Leibniz algebra Lie algebra crossed module |
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