Related papers: Symplectic, product and complex structures on 3-Li…
We consider principal fibre bundles with a given connection and construct almost complex structures on the total space if the adjoint bundle is isomorphic to the tangent bundle of the base. We derive the integrability condition. If the…
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, \{ , , ,…
The PostLie algebra is an enriched structure of the Lie algebra that has recently arisen from operadic study. It is closely related to pre-Lie algebra, Rota-Baxter algebra, dendriform trialgebra, modified classical Yang-Baxter equations and…
We present two constructions of complex symplectic structures on Lie algebras with large abelian ideals. In particular, we completely classify complex symplectic structures on almost abelian Lie algebras. By considering compact quotients of…
In this paper, we introduce the concept of (weak) NL bialgebras. These structures consist of a Lie bialgebra $(\g,[\cdot,\cdot],\delta)$ equipped with a Nijenhuis structure on the Lie algebra $(\g,[\cdot,\cdot])$, satisfying specific…
After endowing with a 3-Lie-Rinehart structure on Hom 3-Lie algebras, we obtain a class of special Hom 3-Lie algebras, which have close relationships with representations of commutative associative algebras. We provide a special class of…
We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any…
For a pre-Lie-Yamaguti algebra $A$, by using its sub-adjacent Lie-Yamaguti algebra $A^c$, we are able to construct a semidirect product Lie-Yamaguti algebra via a representation of $A^c$. The investigation of such semidirect Lie-Yamaguti…
We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$.…
In this paper, we introduce the notions of a $3$-$Lie_\infty$-algebra and a 3-Lie 2-algebra. The former is a model for a 3-Lie algebra that satisfy the fundamental identity up to all higher homotopies, and the latter is the categorification…
For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations…
3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. The paper concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such…
In this paper, we investigate the mathematical structure of Nijenhuis Lie triple systems, an extension of classical Lie triple systems augmented with the Nijenhuis operator. Our study focuses on the cohomology of Nijenhuis Lie triple…
First we use a new approach to give a graded Lie algebra whose Maurer-Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket we define the notion of a Nijenhuis operator on a pre-Lie algebra which…
We identify a direct correspondence between the crossed product construction which plays a crucial role in the theory of Type III von Neumann algebras, and the extended phase space construction which restores the integrability of non-zero…
We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively.…
We introduce the notion of 3-Hom-Lie-Rinehart algebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the first…
This paper studies two types of 3-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycles and double constructions respectively, and are therefore called the local cocycle…
We study symplectic structures on filiform Lie algebras -- nilpotent Lie algebras of the maximal length of the descending central sequence. In the present article we classify the Lie algebras with the structure relations of the following…
The space of Lie algebra cohomology is usually described by the dimensions of components of certain degree even for the adjoint module as coefficients when the spaces of cochains and cohomology can be endowed with a Lie superalgebra…