Deformations of relative Rota–Baxter operators on Leibniz algebras

Abstract

In this paper, we introduce the cohomology theory of relative Rota–Baxter operators on Leibniz algebras. We use the cohomological approach to study linear and formal deformations of relative Rota–Baxter operators. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order [Formula: see text] deformations of a relative Rota–Baxter operator are also characterized in terms of the cohomology theory.

Pre-Leibniz algebras

“…This cohomology in a certain sense splits the Loday-Pirashvili cohomology of Leibniz algebra. We also find the relation between the cohomology of a relative Rota-Baxter operator introduced in [24,25] and the cohomology of the induced pre-Leibniz algebra.…”

“…In [20] Loday and Pirashvili introduced a cohomology theory for Leibniz algebras with coefficients in a representation. Given a vector space g, Balavoine [4] constructs a graded Lie algebra (known as Balavoine's graded Lie algebra) on the space of multilinear maps on g whose Maurer-Cartan elements correspond to Leibniz algebra structures on g. Recently, Rota-Baxter operators and relative Rota-Baxter operators on Leibniz algebras are introduced and their relation with Leibniz Yang-Baxter equation and Leibniz bialgebras are discovered in [22,24,25]. Like a Rota-Baxter operator on a Lie algebra induces a pre-Lie algebra structure, a (relative) Rota-Baxter operator on a Leibniz algebra gives rise to a pre-Leibniz algebra (already introduced in [24] by the name of Leibniz-dendriform algebra) structure.…”

Weighted relative Rota-Baxter operators on Leibniz algebras and Post-Leibniz algebra structures

“…Recenely, cohomology and deformation theory of relative Rota-Baxter operators are studied in [7,13]. Relative Rota-Baxter operators on Leibniz algebras, their deformation theory, and relation with Leibniz Yang-Baxter equation, Leibniz bialgebras are extensively considered in [12,14]. Such operators induce Leibniz-dendriform algebras (also called pre-Leibniz algebras) which are split object for Leibniz algebras.…”

Twisted relative Rota-Baxter operators on Leibniz algebras and NS-Leibniz algebras

“…In general, deformation theory was developed for algebras over binary quadratic operads by Balavoine [1]. Recently, deformations of relative Rota-Baxter operators (also called O-operators) are developed in [18,6,19].…”