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Register a new userTwisted Rota-Baxter operators on 3-Lie algebras and NS-3-Lie algebras
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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 fact, we define the cohomology of a twisted Rota-Baxter operator and study infinitesimal deformations of a twisted Rota-Baxter operator using the second cohomology group. Then we introduce the notion of an NS-3-Lie algebra, which produces a 3-Lie algebra with a representation on itself. We show that a twisted Rota-Baxter operator induces an NS-3-Lie algebra naturally. Thus NS-3-Lie algebras can be viewed as the underlying algebraic structures of twisted Rota-Baxter operators on 3-Lie algebras. Finally we show that a Nijenhuis operator on a 3-Lie algebra gives rise to a representation of the deformed 3-Lie algebra and a 2-cocycle. Consequently, the identity map will be a twisted Rota-Baxter operator on the deformed 3-Lie algebra. We also introduce the notion of a Reynolds operator on a 3-Lie algebra, which can serve as a special case of twisted Rota-Baxter operators on 3-Lie algebras.
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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-LieRep pairs. The notion of an n-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extensions of order m deformations to order m+1 deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on n-LieRep pairs and those on (n+1)-LieRep pairs by certain linear functions.
In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an $L_infty$-algebra, whose Maurer-Cartan elements give rise to new twilled 3-Lie algebras by twisting. In particular, we recover the Lie $3$-algebra whose Maurer-Cartan elements are O-operators (also called relative Rota-Baxter operators) on 3-Lie algebras. Then we introduce the notion of generalized matched pairs of 3-Lie algebras using generalized representations of 3-Lie algebras, which will give rise to twilled 3-Lie algebras. The usual matched pairs of 3-Lie algebras correspond to a special class of twilled 3-Lie algebras, which we call strict twilled 3-Lie algebras. Finally, we use O-operators to construct explicit twilled 3-Lie algebras, and explain why an $r$-matrix for a 3-Lie algebra can not give rise to a double construction 3-Lie bialgebra. Examples of twilled 3-Lie algebras are given to illustrate the various interesting phenomenon.
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, we define a class of 3-algebras which are called 3-Lie-Rinehart algebras. A 3-Lie-Rinehart algebra is a triple $(L, A, rho)$, where $A$ is a commutative associative algebra, $L$ is an $A$-module, $(A, rho)$ is a 3-Lie algebra $L$-module and $rho(L, L)subseteq Der(A)$. We discuss the basic structures, actions and crossed modules of 3-Lie-Rinehart algebras and construct 3-Lie-Rinehart algebras from given algebras, we also study the derivations from 3-Lie-Rinehart algebras to 3-Lie $A$-algebras. From the study, we see that there is much difference between 3-Lie algebras and 3-Lie-Rinehart algebras.
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. Then we introduce the notions of relative Rota-Baxter operators on Lie-Yamaguti algebras and pre-Lie-Yamaguti algebras. We prove that a pre-Lie-Yamaguti algebra gives rise to a Lie-Yamaguti algebra naturally and a relative Rota-Baxter operator induces a pre-Lie-Yamaguti algebra. Finally we study symplectic structures on Lie-Yamaguti algebra, which give rise to relative Rota-Baxter operators as well as pre-Lie-Yamaguti algebras. As applications, we study phase spaces of Lie-Yamaguti algebras, and show that there is a one-to-one correspondence between phase spaces of Lie-Yamaguti algebras and Manin triples of pre-Lie-Yamaguti algebras.
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