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In this paper, we introduce twisted relative Rota-Baxter operators on a Leibniz algebra as a generalization of twisted Poisson structures. We define the cohomology of a twisted relative Rota-Baxter operator $K$ as the Loday-Pirashvili cohomology of a certain Leibniz algebra induced by $K$ with coefficients in a suitable representation. Then we consider formal deformations of twisted relative Rota-Baxter operators from cohomological points of view. Finally, we introduce and study NS-Leibniz algebras as the underlying structure of twisted relative Rota-Baxter operators.
In this paper, first we introduce the notion of a twisted Rota-Baxter operator on a 3-Lie algebra $g$ with a representation on $V$. We show that a twisted Rota-Baxter operator induces a 3-Lie algebra structure on $V$, which represents on $g$. By this
In this paper, first we introduce the notion of a quadratic Lie-Yamaguti algebra and show that the invariant bilinear form in a quadratic Lie-Yamaguti algebra induces an isomorphism between the adjoint representation and the coadjoint representation.
Based on the differential graded Lie algebra controlling deformations of an $n$-Lie algebra with a representation (called an n-LieRep pair), we construct a Lie n-algebra, whose Maurer-Cartan elements characterize relative Rota-Baxter operators on n-L
We generalize the notion of a Rota-Baxter operator on groups and the notion of a Rota-Baxter operator of weight 1 on Lie algebras and define and study the notion of a Rota-Baxter operator on a cocommutative Hopf algebra $H$. If $H=F[G]$ is the group
In this paper, we first construct a graded Lie algebra which characterizes Rota-Baxter operators on an anti-flexible algebra as Maurer-Cartan elements. Next, we study infinitesimal deformations of bimodules over anti-flexible algebras. We also consid