Lie Algebras Of Finite Dimension Research Articles

Thomae’s function on a Lie group

Let $\mathfrak g$ be a simple complex Lie algebra of finite dimension. This paper gives an inequality relating the order of an automorphism of $\mathfrak g$ to the dimension of its fixed-point subalgebra, and characterizes those automorphisms of $\mathfrak g$ for which equality occurs. This is amounts to an inequality/equality for Thomae's function on the group of automorphisms of $\mathfrak g$. The result has applications to characters of zero weight spaces, graded Lie algebras, and inequalities for adjoint Swan conductors.

Camina Lie algebras

Let [Formula: see text] be a Lie algebra and [Formula: see text] be a proper ideal of [Formula: see text]. Then [Formula: see text] is called a Camina pair if [Formula: see text] for all [Formula: see text]. Also, [Formula: see text] is called a Camina Lie algebra if [Formula: see text] is a Camina pair. In this paper, we give some properties of Camina Lie algebras. Moreover, we show that a nilpotent Camina Lie algebra of finite dimension over an algebraically closed field is nilpotent with nilindex at most [Formula: see text].

A Characterization of Stem Algebras in Terms of Central Derivations

Let L be a Lie algebra, and Der z (L) denote the set of all central derivations of L, that is, the set of all derivations of L mapping L into the center. In this paper, by using the notion of isoclinism, we study the center of Der z (L) for nilpotent Lie algebras with nilindex 2. We also give a characterization of stem Lie algebras by their central derivations. In fact we show that for non-abelian nilpotent Lie algebras of finite dimension and any nilpotent Lie algebra with nilindex 2 (not finite dimensional in general), Der z (L) is abelian if and only if L is a stem Lie algebra.

On semicomplete Lie algebras

Let L be a Lie algebra and \(L^2\) be its derived algebra. A derivation \(\alpha \) of L is called \(\mathrm {ID}\)-derivation if \(\alpha (x)\in L^2\) for each \(x\in L\). The set of all \(\mathrm {ID}\)-derivations of L is denoted by \(\mathrm {ID}(L)\). A Lie algebra L is called semicomplete Lie algebra if and only if \(\mathrm {ID}(L)=\mathrm {ad}(L),\) where \(\mathrm {ad}(L)\) is the set of inner derivations of L. In this paper, we give some results on semicomplete Lie algebras and characterization of semicomplete nilpotent Lie algebras of finite dimension.

A remark on the Schur multiplier of nilpotent Lie algebras

In this article, we indicate that the Schur multiplier of every nilpotent Lie algebra of finite dimension at least 2 is non-zero. Also, we present a criterion for nilpotent Lie algebras lacking any covers with respect to the variety of nilpotent Lie algebras of class at most c.

Simple Classical Lie Algebras in Characteristic 2 and Their Gradations, II

This paper is a continuation of [2]. We prove Conjecture 5.1 of [2] which gives a characterization of simple Lie algebras of finite dimension of type B2ℓ, C2ℓ, D2ℓ+1, E7 and E8 in terms of some gradations of these algebras over a field of characteristic 2.

Some criteria for detecting capable Lie algebras

In virtue of a recent bound obtained in [P. Niroomand, F.G. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra 39 (2011) 1293–1297], we classify all capable nilpotent Lie algebras of finite dimension possessing a derived subalgebra of dimension one. Indirectly, we find also a criterion for detecting noncapable Lie algebras. The final part contains a construction, which shows that there exist capable Lie algebras of arbitrary big corank (in the sense of Berkovich–Zhou).

On related varieties to the commuting variety of a semisimple Lie algebra

Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x,y) in g×g such that x and y are nilpotents and are in the same Borel subalgebra. The main result of this paper is that N is a closed and irreducible subvariety of g×g and its normalization morphism is bijective.

On the tensor square of non-abelian nilpotent finite-dimensional Lie algebras

For every finite p-group G of order p n with derived subgroup of order p m , Rocco [N.R. Rocco, On a construction related to the nonabelian tensor square of a group, Bol. Soc. Brasil. Mat. 1 (1991), pp. 63–79] proved that the order of tensor square of G is at most p n(n−m). This upper bound has been improved recently by the author [P. Niroomand, On the order of tensor square of non abelian prime power groups (submitted)]. The aim of this article is to obtain a similar result for a non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any given n-dimensional non-abelian nilpotent Lie algebra L with derived subalgebra of dimension m we have dim(L ⊗ L) ≤ (n − m)(n − 1) + 2. Furthermore for m = 1, the explicit structure of L is given when the equality holds.

Fair-Sized Projective Modules

We investigate a condition on particular chains of ideals that allows us to determine properties of infinitely generated modules over noetherian rings. The results apply to semilocal noetherian rings, integral group rings of finite groups and universal enveloping algebras of solvable Lie algebras of finite dimension.

A method to obtain the lie group associated with a nilpotent lie algebra

According to Ado and Cartan Theorems, every Lie algebra of finite dimension can be represented as a Lie subalgebra of the Lie algebra associated with the general linear group of matrices. We show in this paper a method to obtain the simply connected Lie group associated with a nilpotent Lie algebra, by using unipotent matrices. Two cases are distinguished, according to the nilpotent Lie algebra is or not filiform.

Localization and Catenarity in Iterated Differential Operator Rings

ABSTRACT Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and R[Θ1, δ1] ··· [Θ n , δ n ] an iterated differential operator k-algebra over R such that δ j (Θ i ) ∈ R[Θ1, δ1] ··· [Θ i−1, δ i−1]; 1 ≤ i < j ≤ n. If R is Noetherian Δ-hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the Δ-prime ideals of R, the local ring A P is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers μ i of Bass. Let g be a nilpotent Lie algebra of finite dimension n acting on R by derivations and U(g) the enveloping algebra of g. Then the crossed product of R by U(g) is an iterated differential operator k-algebra as above. In this particular case, our results are known if k has characteristic zero.

Lie algebras associated with triangular configurations

A new class of Lie algebras of finite dimension, those which are associated with a certain combinatorial configuration made up by triangles of weighted and non-directed edges, is introduced and a characterization theorem for them is proved. Moreover, some subclasses of such Lie algebras are classified.

Combinatorial structures associated with Lie algebras of finite dimension

Given a Lie algebra of finite dimension, with a selected basis of it, we show in this paper that it is possible to associate it with a combinatorial structure, of dimension 2, in general. In some particular cases, this structure is reduced to a weighted graph. We characterize such graphs, according to they have 3-cycles or not.

Sur les suites d'algèbres de Lie de dérivations

If ${\frak g}$ is a Lie algebra of finite dimension, let $ ({\frak g}_n) $ be the sequence of Lie algebras of derivations associated to $ {\frak g}:{\frak g}_0={\frak g}, {\frak g}_{n+1}=\hbox {Der}\,{\frak g}_n $ . If the center of $ {\frak g} $ is null, E. Schenkman showed that this sequence ends finitely for a complete Lie algebra $ {\frak g}_{\infty } $ (i.e. a Lie algebra with null center and inner derivations). In the first part of this work we construct $ {\frak g}_{\infty } $ . In the second part are studied the sequences for which the center of each $ {\frak g}_n $ is different from zero. One shows, if the dimensions of $ {\frak g}_n $ are bounded, that a limit $ {\frak g}_{\infty } $ exists too: it is the direct product of the ground field by a complete Lie algebra equal to its derived ideal. Note that the proofs used here are valid for a field of characteristic 0 only.

KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank

Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.