Related papers: 3-Lie Bialgebras
We introduce the notion of weak Lie 2-bialgebra. Roughly, a weak Lie 2-bialgebra is a pair of compatible 2-term $L_\infty$-algebra structures on a vector space and its dual. The compatibility condition is described in terms of the big…
Orthogonal Lie algebras in dimension 4 are identified as current Lie algebras, thus producing a natural decomposition for them over any field.
We give explicit formulas proving restrictedness of the following Lie (super)algebras: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, and (under…
We present Bianchi's proof on the classification of real (and complex) $3$-dimensional Lie algebras in a coordinate free version from a strictly representation theoretic point of view. Nearby we also compute the automorphism groups and from…
We approach the classification of Lie bialgebra structures on simple Lie algebras from the viewpoint of descent and non-abelian cohomology. We achieve a description of the problem in terms faithfully flat cohomology over an arbitrary ring…
Some examples are given of finite dimensional Lie bialgebras whose brackets and cobrackets are determined by pairs of $r$-matrices.
Several open questions are discussed. The topics include cohomology of current and related Lie algebras, algebras represented as the sum of subalgebras, structures and phenomena peculiar to characteristic $2$, and variations on themes of…
Starting with Lie's classification of finite-dimensional transitive Lie algebras of vector fields on $\mathbb C^2$ we construct Lie algebras of vector fields on the bundle $\mathbb C^2 \times \mathbb C$ by lifting the Lie algebras from the…
The object of investigations are almost contact B-metric structures on 3-dimensional Lie groups considered as smooth manifolds. There are established the existence and some geometric characteristics of these manifolds in all basic classes.…
A morphism Lie algebra is a triple $(\mathfrak{g}, \mathfrak{h}, \phi)$ consisting of two Lie algebras $\mathfrak{g}, \mathfrak{h}$ and a Lie algebra homomorphism $\phi : \mathfrak{g} \rightarrow \mathfrak{h}$. We define representations and…
One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial…
In this paper, we introduce the notion of generalized representation of a $3$-Lie algebra, by which we obtain a generalized semidirect product $3$-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of…
We discuss a generalization of N=6 three-algebras to N=5 three-algebras in connection to anti-Lie triple systems and basic Lie superalgebras of type II. We then show that the structure constants defined in anti-Lie triple systems agree with…
Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras. On them the quantum Lie…
A complex fuzzy Lie algebra is a fuzzy Lie algebra whose membership function takes values in the unit circle in the complex plane. In this paper, we deine the complex fuzzy Lie subalgebras and complex fuzzy ideals of Lie algebras. Then, we…
In this paper, we introduce the notion of a pre-Lie 2-algebra, which is a categorification of a pre-Lie algebra. We prove that the category of pre-Lie 2-algebras and the category of 2-term pre-Lie$_\infty$-algebras are equivalent. We…
This is a short presentation of some classical results on finite dimensional complex Lie algebras (classification of nilpotent Lie algebras, deformations and perturbations, contractions and rigidity). We present some applications to…
We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple…
In this paper, we study Lie 2-bialgebras, with special attention to coboundary ones, with the help of the cohomology theory of $L_\infty$-algebras with coefficients in $L_\infty$-modules. We construct examples of strict Lie 2-bialgebras…
Structures of dual Lie bialgebras on the one sided Witt algebra, the Witt algebra and the Virasoro algebra are investigated. As a result, we obtain some infinite dimensional Lie algebras.