Deformations and Homotopy Theory of Relative Rota–Baxter Lie Algebras

Abstract

We determine the $$L_\infty $$ 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 $$\mathsf {Lie}\mathsf {Rep}$$ Lie Rep 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$$_\infty $$ ∞ -algebras.

The Controlling $$L_\infty $$-Algebra, Cohomology and Homotopy of Embedding Tensors and Lie–Leibniz Triples

“…L ∞ -algebras were constructed using the above method to study simultaneous deformations of morphisms between Lie algebras in [13,14], and to study simultaneous deformations of relative Rota-Baxter Lie algebras in [30].…”

$\mathcal{O}$-operators on Lie $\infty$-algebras with respect to Lie $\infty$-actions

“…The current paper also deals with O-operators on Lie ∞-algebras, but with a different approach which uses Lie ∞-actions instead of representations of Lie ∞-algebras. Our definition is therefore different from the one given in [6] but there is a relationship between them.…”

“…As we said before, the two O-operator definitions, ours and the one in [6], are different. However, since there is a close connection between Lie ∞-actions and representations of Lie ∞-algebras, the two definitions can be related.…”

“…In [9], a homotopy version of O-operators on symmetric graded Lie algebras was introduced. This was the first step towards the definition of a O-operator on a Lie ∞-algebra with respect to a representation on a graded vector space that was given in [6]. The current paper also deals with O-operators on Lie ∞-algebras, but with a different approach which uses Lie ∞-actions instead of representations of Lie ∞-algebras.…”

“…Representations of Lie ∞-algebras on graded vector spaces were introduced in [7]. In [6], the authors consider a representation Φ of a Lie ∞-algebra E on a graded vector space V and define an O-operator (homotopy relative Rota-Baxter operator) on E with respect to Φ as a degree zero element T of Hom( S(V ), E) satisfying a family of suitable identities. Inspired by the notion of an action of a Lie ∞-algebra on a graded manifold [8], we define an action of a Lie ∞-algebra (E, M E ) on a Lie ∞-algebra (V, M V ) as a Lie ∞-morphism Φ between E and Coder( S(V )) [1], the symmetric DGLA of coderivations of S(V ).…”