Deformations and homotopy theory of relative Rota-Baxter Lie algebras

Abstract

We determine the \emph{$L_{\infty }$-algebra} that controls deformations of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying LieRep pair by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Consequently, we define the {\em cohomology} of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota-Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a \emph{homotopy} relative Rota-Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to \emph{pre-Lie${}_{\infty }$-algebras}.

arXiv comment31 pages, to appear in Comm. Math. Phys

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• Representations and cohomologies of relative Rota-Baxter Lie algebras and applications
• The controlling $L_{\infty }$-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples
• The $L_{\infty }$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators
• Twisted Rota-Baxter operators on 3-Lie algebras and NS-3-Lie algebras
• Deformations and cohomology theory of Rota-Baxter $3$-Lie algebras of arbitrary weights
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
• $\mathcal{O}$-operators on Lie $\infty$-algebras with respect to Lie $\infty$-actions
• Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O}N$-structures