Deformations and Homotopy Theory of Relative Rota-Baxter Lie Algebras

Lazarev, Andrey and Sheng, Yunhe and Tang, Rong (2021) Deformations and Homotopy Theory of Relative Rota-Baxter Lie Algebras. Communications in Mathematical Physics, 383. 595–631. ISSN 0010-3616

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Abstract

We determine the L∞-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 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 homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie∞-algebras.