Related papers: A normal form for admissible characters in the sen…

Parabolic subalgebras of semi-simple Lie algebras decompose as $\frak{p}=\frak{m}\oplus\frak{n}$ where $\frak{m}$ is a Levi factor and $\frak{n}$ the corresponding nilradical. By Richardsons theorem, there exists an open orbit under the…

Let $G$ be a simple algebraic group of classical type over an algebraically closed field $k$. Let $P$ be a parabolic subgroup of $G$ and let $\p = \Lie P$ be the Lie algebra of $P$ with Levi decomposition $\p = {\l}\oplus \u$, where $\u$ is…

This paper gives a classification of parabolic subalgebras of simple Lie algebras over $\CC$ that are complexifications of parabolic subalgebras of real forms for which Lynch's vanishing theorem for generalized Whittaker modules is…

In this paper, we study certain ad-nilpotent subalgebras contained in the non-zero graded portion of a simple Z_n-graded Lie algebra. These subalgebras respect the grading on the Lie algebra and are modules for a Borel subalgebra for the…

We show that semi-simple lie algebras can be characterized by their maximal nilpotent subalgebra, which is the same as the nilpotent radical of a Borel subalgebra.

A parabolic subalgebra $\mathfrak{p}$ of a complex semisimple Lie algebra $\mathfrak{g}$ is called a parabolic subalgebra of abelian type if its nilpotent radical is abelian. In this paper, we provide a complete characterization of the…

We study some elementary properties of the quantum enveloping algebra associated to a parabolic subalgebra $\mathfrak{p}$ of a semisimple Lie algebra $\mathfrak{g}$. In particular we prove an explicit formula for the degree of this algebra,…

Let $\g$ be a locally reductive complex Lie algebra which admits a faithful countable-dimensional finitary representation $V$. Such a Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of $\sl_\infty$,…

We study ad-nilpotent ideals of a parabolic subalgebra of a simple Lie algebra. Any such ideal determines an antichain in a set of positive roots of the simple Lie algebra. We give a necessary and sufficient condition for an antichain to…

We characterize locally semisimple subalgebras $\l$ of $\sl_\infty$, $\so_\infty$, and $\sp_\infty$ which are Levi components of parabolic subalgebras. Given $\l$, we characterize the parabolic subalgebras $\p$ such that $\l$ is a Levi…

We extend the results of Cellini-Papi on the characterizations of nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple Lie algebra. These characterizations are given in terms of elements of the affine Weyl…

We characterize, in a purely algebraic manner, certain linear forms, called stable, on a Lie algebra. As an application, we determine the index of a Borel subalgebra of a semi-simple Lie algebra. Finally, we give an example of a parabolic…

A finite-dimensional Lie algebra $L$ over a field $F$ of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an $A$-algebra if every nilpotent subalgebra is abelian. This paper is a…

Let $\mathfrak g$ be a semisimple Lie algebra, $\mathfrak h\subset\mathfrak g$ a reductive subalgebra such that $\mathfrak h^\perp$ is a complementary $\mathfrak h$-submodule of $\mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains…

We provide explicit formulas for the number of ad-nilpotent ideals of a Borel subalgebra of a complex simple Lie algebra having fixed class of nilpotence.

The main goal of this paper is to prove the following theorem: Let $\frak k$ be an $\frak {sl}_2$-subalgebra of a semisimple Lie algebra $\frak g$, none of whose simple factors is of type $A1$. Then there exists a positive integer $b(\frak…

Let $G$ be a classical linear algebraic group over an algebraically closed field, and let $\mathfrak{n}$ denote the subset of nilpotent elements in its Lie algebra. In this paper we study a partial order on the $G$-orbits in $\mathfrak{n}$…

We consider the conjugation-action of the Borel subgroup of the symplectic or the orthogonal group on the variety of nilpotent complex elements of nilpotency degree $2$ in its Lie algebra. We translate the setup to a…

Let g be a complex simple Lie algebra and e a nilpotent element in g. To a certain nilpotent subalgebra m attached to e, called an admissible subalgebra of g, we associate an endomorphism algebra H. When m is constructed from a good grading…

A (vector space) basis B of a Lie algebra is said to be very nilpotent if all the iterated brackets of elements of B are nilpotent. In this note, we prove a refinement of Engel's Theorem. We show that a Lie algebra has a very nilpotent…