A pair of sequences of nilpotent Lie algebras denoted by $N_{n,11}$ and $N_{n,19}$ are introduced. Here $n$ denotes the dimension of the algebras that are defined for $n\ge 6$; the first term in the sequences are denoted by 6.11 and 6.19, respectively, in the standard list of six-dimensional Lie algebras. For each of $N_{n,11}$ and $N_{n,19}$ all possible solvable extensions are constructed so that $N_{n,11}$ and $N_{n,19}$ serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating...

The KV-homology theory is a new framework which yields interesting properties of lagrangian foliations. This short note is devoted to relationships between the KV-homology and the KV-cohomology of a lagrangian foliation. Let us denote by ${}_F$ (resp. $V^F$) the KV-algebra (resp. the space of basic functions) of a lagrangian foliation F. We show that there exists a pairing of cohomology and homology to $V^F$. That is to say, there is a bilinear map $H^q{(}_F,V^F)\times H_q{(}_F,V^F)\to V^F$, which is invariant under F-preserving symplectic diffeomorphisms....

Let A be an ultraprime Banach algebra. We prove that each approximately commuting continuous linear (or quadratic) map on A is near an actual commuting continuous linear (resp. quadratic) map on A. Furthermore, we use this analysis to study how close are approximate Lie isomorphisms and approximate Lie derivations to actual Lie isomorphisms and Lie derivations, respectively.

The natural filtration of the infinite-dimensional simple modular Lie superalgebra M over a field of characteristic p > 2 is proved to be invariant under automorphisms by discussing ad-nilpotent elements. Moreover, an intrinsic property is obtained and all the infinite-dimensional simple modular Lie superalgebras M are classified up to isomorphisms. As an application, a property of automorphisms of M is given.

A Lie algebra $𝔤$ is called two step nilpotent if $𝔤$ is not abelian and $[𝔤,𝔤]$ lies in the center of $𝔤$. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension $8$ over the...

Let $${𝒪}_1$$ and $${𝒪}_2$$ be adjoint nilpotent orbits in a real semisimple Lie algebra. Write $${𝒪}_1$$ ≥ $${𝒪}_2$$ if $${𝒪}_2$$ is contained in the closure of $${𝒪}_1$$ . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E 8. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible...

Let $L_n=K[x_1^{\pm 1},...,x_n^{\pm 1}]$ be a Laurent polynomial algebra over a field $K$ of characteristic zero, $W_n:={\mathrm{Der}}_K\left(L_n\right)$ the Lie algebra of $K$-derivations of the algebra $L_n$, the so-called Witt Lie algebra, and let $\mathrm{Vir}$ be the Virasoro Lie algebra which is a $1$-dimensional central extension of the Witt Lie algebra. The Lie algebras $W_n$ and $\mathrm{Vir}$ are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: ${\mathrm{Aut}}_{\mathrm{Lie}}\left(\mathrm{Vir}\right)\simeq {\mathrm{Aut}}_{\mathrm{Lie}}\left(W_1\right)\simeq \{\pm 1\}⋉K^{*}$, and give a short proof that ${\mathrm{Aut}}_{\mathrm{Lie}}\left(W_n\right)\simeq {\mathrm{Aut}}_{\mathrm{K}-\mathrm{alg}}\left(L_n\right)\simeq {\mathrm{GL}}_n\left(\mathbb{Z}\right)⋉K^{*n}$.