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Register a new userManin triples of 3-Lie algebras induced by involutive derivations
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For any $n$-dimensional 3-Lie algebra $A$ over a field of characteristic zero with an involutive derivation $D$, we investigate the structure of the 3-Lie algebra $B_1=Altimes_{ad^*} A^* $ associated with the coadjoint representation $(A^*, ad^*)$. We then discuss the structure of the dual 3-Lie algebra $B_2$ of the local cocycle 3-Lie bialgebra $(Altimes_{ad^*} A^*, Delta)$. By means of the involutive derivation $D$, we construct the $4n$-dimensional Manin triple $(B_1oplus B_2,$ $ [ cdot, cdot, cdot]_1,$ $ [ cdot, cdot, cdot]_2,$ $ B_1, B_2)$ of 3-Lie algebras, and provide concrete multiplication in a special basis $Pi_1cupPi_2$. We also construct a sixteen dimensional Manin triple $(B, [ cdot, cdot, cdot])$ with $dim B^1=12$ using an involutive derivation on a four dimensional 3-Lie algebra $A$ with $dim A^1=2$.
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In this paper, we study the structure of 3-Lie algebras with involutive derivations. We prove that if $A$ is an $m$-dimensional 3-Lie algebra with an involutive derivation $D$, then there exists a compatible 3-pre-Lie algebra $(A, { , , , }_D)$ such that $A$ is the sub-adjacent 3-Lie algebra, and there is a local cocycle $3$-Lie bialgebraic structure on the $2m$-dimensional semi-direct product 3-Lie algebra $Altimes_{ad^*} A^*$, which is associated to the adjoint representation $(A, ad)$. By means of involutive derivations, the skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra $Altimes_{ad^*}A^*$, a class of 3-pre-Lie algebras, and eight and ten dimensional local cocycle 3-Lie bialgebras are constructed.
We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.
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 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, 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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