Related papers: Abelian ideals with given dimension in Borel subal…

Let $B$ be a Borel subgroup of a semisimple algebraic group $G$, and let $\mathfrak a$ be an abelian ideal of $\mathfrak b=Lie(B)$. The ideal $\mathfrak a$ is determined by certain subset $\Delta_{\mathfrak a}$ of positive roots, and using…

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…

A characterization of the finite-dimensional Leibniz algebras with an abelian subalgebra of codimension two over a field $\mathbb{F}$ of characteristic $p\neq2$ is given. In short, a finite-dimensional Leibniz algebra of dimension $n$ with…

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Leibniz algebras. We study Leibniz algebras containing abelian subalgebras of codimension 1, solvable and supersolvable Leibniz…

Let $g$ be a simple Lie algebra and $Ab$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $g$. In 2003 (IMRN, no.35, 1889--1913), we constructed a partition of $Ab$ into the subposets $Ab_\mu$, parameterised by the…

In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis and the corresponding multiplication table, which is specified by 27 structure constants. We describe all…

If $\frak g$ is a complex simple Lie algebra, and $k$ does not exceed the dual Coxeter number of $\frak g$, then the k$^{th}$ coefficient of the $dim \frak g$ power of the Euler product may be given by the dimension of a subspace of…

Solvable Lie algebras having at least one Abelian descending central ideal are studied. It is shown that all such Lie algebras can be built up from canonically defined ideals. The nature of such ideals is elucidated and their construction…

For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the…

Let g=g_0+ g_1 be a simple Z_2-graded Lie algebra and let b_0 be a fixed Borel subalgebra of g_0. We describe and enumerate the abelian b_0-stable subalgebras of g_1.

To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained…

In this paper, we study the maximal dimension $\alpha(L)$ of abelian subalgebras and the maximal dimension $\beta(L)$ of abelian ideals of m-dimensional 3-Lie algebras $L$ over an algebraically closed field. We show that these dimensions do…

An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media -…

Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak b$ a fixed Borel subalgebra, and $W$ the Weyl group of $\mathfrak g$. In this note, we study a relationship between the maximal abelian ideals of $\mathfrak b$ and the minimal inversion…

A classification up to automorphism of the inner ideals of the real finite-dimensional simple Lie algebras is given, jointly with precise descriptions in the case of the exceptional Lie algebras.

Let $\mathfrak g$ be a simple Lie algebra and $\mathfrak{Ab}$ the poset of all abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. If $\mathfrak a\in\mathfrak{Ab}$, then the normaliser of $\mathfrak a$ is a standard parabolic…

A tree diagram is a tree with positive integral weight on each edge, which is a notion generalized from the Dynkin diagrams of finite-dimensional simple Lie algebras. We introduce two nilpotent Lie algebras and their extended solvable Lie…

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas…

We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We…

In this article, we present a constructive procedure for determining all ideals of the Borel subalgebra of a complex semisimple Lie algebra from its root system or, equivalently, its Dynkin diagram. The proposed algorithmic approach has…