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...

The aim of this work is to obtain the structure of c-covers of c-capable Lie algebras. We also obtain some results on the existence of c-covers and, under some assumptions, we prove the absence of c-covers of Lie algebras.

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 ${\mathrm{Aut}}_{\mathrm{Lie}}\left(W_n\right)\simeq {\mathrm{Aut}}_{\mathrm{K}-\mathrm{alg}}\left(L_n\right)\simeq {\mathrm{GL}}_n\left(\mathbb{Z}\right)⋉K^{*n}$.

We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several examples and also analyse when small deformations remain complex parallelisable.

In this paper we obtain the description of the Leibniz algebras whose subalgebras are ideals.

2000 Mathematics Subject Classification: Primary: 17A32; Secondary: 16R10, 16P99, 17B01, 17B30, 20C30Let F be a field of characteristic zero. In this paper we study the variety of Leibniz algebras 3N determined by the identity x(y(zt)) ≡ 0. The algebras of this variety are left nilpotent of class not more than 3. We give a complete description of the vector space of multilinear identities in the language of representation theory of the symmetric group Sn and Young diagrams. We also show that the...

An invertible linear map $\varphi$ on a Lie algebra $L$ is called a triple automorphism of it if $\varphi \left(\right[x,[y,z]\left]\right)=\left[\varphi \right(x),[\varphi \left(y\right),\varphi \left(z\right)\left]\right]$ for $\forall x,y,z\in L$. Let $𝔤$ be a finite-dimensional simple Lie algebra of rank $l$ defined over an algebraically closed field $F$ of characteristic zero, $𝔭$ an arbitrary parabolic subalgebra of $𝔤$. It is shown in this paper that an invertible linear map $\varphi$ on $𝔭$ is a triple automorphism if and only if either $\varphi$ itself is an automorphism of $𝔭$ or it is the composition of an automorphism of $𝔭$ and an extremal map of order $2$.