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arXiv · 2409.18599

Deformation maps in proto-twilled Leibniz algebras

Apurba Das·Suman Majhi·Ramkrishna Mandal
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Abstract

This paper aims to find a unified approach to studying the cohomology theories of various operators on Leibniz algebras. We first introduce deformation maps in a proto-twilled Leibniz algebra to do this. Such maps generalize various well-known operators (such as homomorphisms, derivations, crossed homomorphisms, Rota-Baxter operators, modified Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators etc) defined on Leibniz algebras and embedding tensors on Lie algebras. We define the cohomology of a deformation map unifying the existing cohomologies of all the operators mentioned above. Then we construct a curved $L_\infty$-algebra whose Maurer-Cartan elements are precisely deformation maps in a given proto-twilled Leibniz algebra. In particular, we get the Maurer-Cartan characterizations of modified Rota-Baxter operators, twisted Rota-Baxter operators and Reynolds operators on a Leibniz algebra. Finally, given a proto-twilled Leibniz algebra and a deformation map $r$, we construct two governing $L_\infty$-algebras, the first one controls the deformations of the operator $r$ while the second one controls the simultaneous deformations of both the proto-twilled Leibniz algebra and the operator $r$.

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BibTeXRIS

Apurba Das, Suman Majhi, Ramkrishna Mandal. 2024-09-27. Deformation maps in proto-twilled Leibniz algebras. https://arxiv.org/abs/2409.18599

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