DEKOMPOSISI LEVI ALJABAR LIE AFFINE FROBENIUS aff(2, R)

Dalam artikel ini dipelajari aljabar Lie affine Frobenius aff(2, R) berdimensi 6. Aljabar Lie aff(2, R) dapat didekomposisi menggunakan dekomposisi Levi menjadi aljabar Lie linear khusus semisederhana sl(2, R) berdimensi 3, subaljabar Lie komutatif R ⊂ R2 berdimensi 2, dan split torus T berdimensi 1 sedemikian sehingga aff(2, R) = sl(2, R) ⊕ R ⊕ T. Karena aljabar Lie sl(2, R) semisederhana maka bracket Lie-nya dapat dinyatakan sebagai [sl(2, R), sl(2, R)] = sl(2, R). Selanjutnya, misalkan g = R⊕T sehingga aff(2, R) = sl(2, R) ⊕ g. Diperoleh bahwa [sl(2, R), g] ⊆ g dan [g, g] ⊆ g. Dalam hal ini, g adalah solvable radical dari aff(2, R).Kata Kunci: Aljabar Lie affine, Aljabar Lie Semisederhana, Dekomposisi Levi

Almost inner derivations of Lie algebras II

We continue the algebraic study of almost inner derivations of Lie algebras over a field of characteristic zero and determine these derivations for free nilpotent Lie algebras, for almost abelian Lie algebras, for Lie algebras whose solvable radical is abelian and for several classes of filiform nilpotent Lie algebras. We find a family of [Formula: see text]-dimensional characteristically nilpotent filiform Lie algebras [Formula: see text], for all [Formula: see text], all of whose derivations are almost inner. Finally, we compare the almost inner derivations of Lie algebras considered over two different fields [Formula: see text] for a finite-dimensional field extension.

On the nilpotency of the solvable radical of a finite group isospectral to a simple group

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that, except for one specific case, the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.

Finite groups have more conjugacy classes

We prove that for every ${\epsilon>0}$ there exists a ${\delta>0}$ such that every group of order ${n\geq 3}$ has at least ${\delta\log_{2}n/{(\log_{2}\log_{2}n)}^{3+\epsilon}}$ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order n has more than ${\log_{3}n}$ conjugacy classes. We answer Bertram’s question in the affirmative for groups with a trivial solvable radical.

On Vinberg ( − 1,1) rings

We give a description of a 2-torsion free Vinberg ([Formula: see text]) ring [Formula: see text]. If every nonzero root space of [Formula: see text] for [Formula: see text] is one-dimensional where [Formula: see text] is a split abelian Cartan subring of [Formula: see text] which is nil on [Formula: see text] then [Formula: see text] is a Lie ring isomorphic to [Formula: see text]. This generalizes the known result obtained by Myung for the case that [Formula: see text] is a 2-torsion free Vinberg ([Formula: see text]) ring and is power associative. We also give a condition that a Levi factor [Formula: see text] of [Formula: see text] be an ideal of [Formula: see text] when the solvable radical of [Formula: see text] is nilpotent. We apply these results for reductive case of [Formula: see text].

The Ado theorem for finite Lie conformal algebras with Levi decomposition

We prove that a finite torsion-free conformal Lie algebra with a splitting solvable radical has a finite faithful conformal representation.

Group Gradings on Finite Dimensional Lie Algebras

We study gradings by non-commutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if L is graded by a non-abelian finite group G, then the solvable radical R of L is G-graded and there exists a Levi subalgebra B=H1⊕ ⋯ ⊕ Hm homogeneous in G-grading with graded simple summands H1,…,Hm. All Supp Hi (i=1,…,m) are commutative subsets of G.

ON A FORMULA FOR THE PI-EXPONENT OF LIE ALGEBRAS

We prove that one of the conditions in Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite-dimensional semisimple Lie algebra acts by derivations on a finite-dimensional Lie algebra over a field of characteristic 0, then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial G-identities of a finite-dimensional Lie algebra with a rational action of a connected reductive affine algebraic group G by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for H-module Lie algebras whose solvable radical is nilpotent, assuming only the H-invariance of the radical, i.e. under weaker assumptions on the H-action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for G-codimensions of all finite-dimensional Lie G-algebras whose solvable radical is nilpotent, for an arbitrary group G.

CHARACTERIZATION OF SOLVABLE GROUPS AND SOLVABLE RADICAL

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.