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

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 $2$.