A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras

Abstract

In this paper, we review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an $L_{\infty }$-algebra whose Maurer-Cartan elements are relative Rota-Baxter Lie algebras. Then using the twisting method, one can obtain the $L_{\infty }$-algebra that controls deformations of a relative \RB Lie algebra. Meanwhile, the cohomologies of relative Rota-Baxter Lie algebras can also be defined with the help of the twisted $L_{\infty }$-algebra. Using the controlling algebra approach, one can also introduce the notion of homotopy relative Rota-Baxter Lie algebras with close connection to pre-Lie${}_{\infty }$-algebras. Finally, we briefly review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras of nonzero weights.

arXiv admin note: substantial text overlap with arXiv:2008.06714

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