Related papers: Deformations and rigidity in varieties of Lie alge…
We thoroughly explore the class of k-step nilpotent Lie algebras associated with a simple graph looking for k-step nilpotent Lie algebras which are rigid in the variety of at most k-step nilpotent Lie algebras. We find out that, besides the…
The classification of complex of real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example the nilpotent Lie algebras are classified only up to the dimension 7. Moreover, to recognize a given…
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 the first section we recall some basic notions on Lie algebras. In a second time we study the algebraic variety of complex $n$-dimensional Lie algebras. We present different notions of deformations : Gerstenhaber deformations,…
In this article, we develop a systematic cohomological framework for the study of the rigidity of nilpotent Lie foliations with respect to solvable deformations. We introduce the deformation complex associated to a pair of Lie algebras…
We give a characterization of the $2$-step nilpotent Lie algebras whose corresponding Lie groups admit a left invariant complex structure. This is done by considering separately the cases when the complex structure is 2-step or 3-step…
We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation.…
We say that a Lie algebra $\gfr$ is quasi-state rigid if every Ad-invariant continuous Lie quasi-state on it is the directional derivative of a homogeneous quasimorphism. Extending work of Entov and Polterovich, we show that every reductive…
We generalize a result on the Heisenberg Lie algebra that gives restrictions to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some additional properties. For the class of 2-step nilpotent Lie algebras coming from…
We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step…
In this paper we focus on algebraic aspects of contractions of Lie and Leibniz algebras. The rigidity of algebras plays an important role in the study of their varieties. The rigid algebras generate the irreducible components of this…
We classify the (n-5)-filiform Lie algebras which have the additional property of a non-abelian derived subalgebra. We show that this property is strongly related with the structure of the Lie algebra of derivations; explicitely we show…
We present a deformation theory approach to the classification of kinematical Lie algebras in 3+1 dimensions and present calculations leading to the classifications of all deformations of the static kinematical Lie algebra and of its…
In the present paper we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either…
We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex…
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 characterize those graphs which correspond to a rigid 2-step nilpotent Lie algebra in the variety of at most 2-step nilpotent Lie algebras.
In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras. We suggest that with appropriate combinations of both procedures one may construct new Lie algebras.…
The classification of graded non-alternating Hamiltonian Lie algebras over perfect field of characteristic 2 is obtained. It is shown that the filtered deformations of such algebras correspond to non-alternating Hamiltonian forms with…
We study the problem of the infinitesimal deformations of all real, simple, finite-dimensional Filippov (or n-Lie) algebras, considered as a class of n-Leibniz algebras characterized by having an n-bracket skewsymmetric in its n-1 first…