Infinity-enhancing of Leibniz algebras

Abstract

We establish a correspondence between infinity-enhanced Leibniz algebras, recently introduced in order to encode tensor hierarchies, 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 $L_{\infty }$-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 arXiv:1904.11036.

24 pages. v2: Minor changes. Comments and explanations added. v3: Clarifications on functorial aspects added. Version to appear in Letters in Mathematical Physics

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• Nonabelian embedding tensors
• Nonlinear realisations of Lie superalgebras
• Dg Loday-Pirashvili modules over Lie algebras