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The $n$-Lie bialgebras are studied. In Section 2, the $n$-Lie coalgebra with rank $r$ is defined, and the structure of it is discussed. In Section 3, the $n$-Lie bialgebra is introduced. A triple $(L, mu, Delta)$ is an $n$-Lie bialgebra if and only if $Delta$ is a conformal $1$-cocycle on the $n$-Lie algebra $L$ associated to $L$-modules $(L^{otimes n}, rho_s^{mu})$, $1leq sleq n$, and the structure of $n$-Lie bialgebras is investigated by the structural constants. In Section 4, two-dimensional extension of finite dimensional $n$-Lie bialgebras are studied. For an $m$ dimensional $n$-Lie bialgebra $(L, mu, Delta)$, and an $ad_{mu}$-invariant symmetric bilinear form on $L$, the $m+2$ dimensional $(n+1)$-Lie bialgebra is constructed. In the last section, the bialgebra structure on the finite dimensional simple $n$-Lie algebra $A_n$ is discussed. It is proved that only bialgebra structures on the simple $n$-Lie algebra $A_n$ are rank zero, and rank two.
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
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
For n even, we prove Pozhidaevs conjecture on the existence of associative enveloping algebras for simple n-Lie algebras. More generally, for n even and any (n+1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(
Let $(mathfrak{g}, [cdot,cdot], delta_mathfrak{g})$ be a fixed Lie bialgebra, $E$ be a vector space containing $mathfrak{g}$ as a subspace and $V$ be a complement of $mathfrak{g}$ in $E$. A natural problem is that how to classify all Lie bialgebraic
We study (quasi-)twilled pre-Lie algebras and the associated $L_infty$-algebras and differential graded Lie algebras. Then we show that certain twisting transformations on (quasi-)twilled pre-Lie algbras can be characterized by the solutions of Maure