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Leibniz algebras with derivations

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Abstract

In this paper, we consider Leibniz algebras with derivations. A pair consisting of a Leibniz algebra and a distinguished derivation is called a LeibDer pair. We define a cohomology theory for LeibDer pair with coefficients in a representation. We study central extensions of a LeibDer pair. In the next, we generalize the formal deformation theory to LeibDer pairs in which we deform both the Leibniz bracket and the distinguished derivation. It is governed by the cohomology of LeibDer pair with coefficients in itself. Finally, we consider homotopy derivations on sh Leibniz algebras and 2-derivations on Leibniz 2-algebras. The category of 2-term sh Leibniz algebras with homotopy derivations is equivalent to the category of Leibniz 2-algebras with 2-derivations.

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Acknowledgements

The author would like to thank the referee for his/her valuable comments on the earlier version that have improved the paper. The author also thanks Indian Institute of Technology Kanpur for financial support.

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  1. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, Uttar Pradesh, 208016, India

    Apurba Das

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  1. Apurba Das
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Correspondence to Apurba Das.

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Communicated by Tom Lada.

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Das, A. Leibniz algebras with derivations. J. Homotopy Relat. Struct. 16, 245–274 (2021). https://doi.org/10.1007/s40062-021-00280-w

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