Infinity-enhancing of Leibniz algebras
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Infinity-enhancing of Leibniz algebras Sylvain Lavau1
· Jakob Palmkvist2,3
Received: 9 February 2020 / Revised: 13 July 2020 / Accepted: 31 July 2020 © Springer Nature B.V. 2020
Abstract We establish a correspondence between infinity-enhanced Leibniz algebras, recently introduced in order to encode tensor hierarchies (Bonezzi and Hohm in Commun Math Phys 377:2027–2077, 2020), and differential graded Lie algebras, which have been already used in this context. We explain how any Leibniz algebra gives rise to a differential graded Lie algebra with a corresponding infinity-enhanced Leibniz algebra. Moreover, by a theorem of Getzler, this differential graded Lie algebra canonically induces an L ∞ -algebra structure on the suspension of the underlying chain complex. We explicitly give the brackets to all orders and show that they agree with the partial results obtained from the infinity-enhanced Leibniz algebras in Bonezzi and Hohm (Commun Math Phys 377:2027–2077, 2020).
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Embedding tensors and Leibniz algebras . . . . . . . . . . . . . 2.1 From differential graded Lie algebras to Leibniz algebras . . 2.2 Embedding tensors and Lie-Leibniz triples . . . . . . . . . . 2.3 From Leibniz algebras to differential graded Lie algebras . . 2.4 Construction from a universal Z-graded Lie superalgebra . . 3 Infinity-enhanced Leibniz algebras . . . . . . . . . . . . . . . . 4 The L ∞ -algebra induced by the dgLa . . . . . . . . . . . . . . . 4.1 L ∞ -algebras and Getzler’s theorem . . . . . . . . . . . . . 4.2 Comparing the L ∞ -algebra structures . . . . . . . . . . . . 4.3 Example: the (1, 0) superconformal model in six dimensions References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Sylvain Lavau [email protected] Jakob Palmkvist [email protected]
1
IMJ-PRG, Université Paris Diderot, Paris, France
2
Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden
3
School of Science and Technology, Örebro University, Örebro, Sweden
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S. Lavau, J. Palmkvist
1 Introduction Leibniz algebras (also known as Loday algebras or Leibniz-Loday algebras [2]) have during the last years attracted attention for their applications to gauge theories where the gauge variation δx y of one parameter y with respect to another x is not antisymmetric under the interchange of x and y (but where the symmetrization leads to a parameter that acts trivially on the fields). Such situations occur for example in the embedd
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