We show that the dimension of the minimal nilpotent coadjoint orbit for a complex simple Lie algebra is equal to twice the dual Coxeter number minus two. Let g be a finite dimensional complex simple Lie algebra. We fix a Cartan subalgebra h, a root system ∆ ⊂ h and a set of positive roots ∆+ ⊂ ∆. Let ρ be half the sum of all positive roots. Denote by θ the highest root and normalize the Killing...

Let n be a locally nilpotent infinite-dimensional Lie algebra over C. U(n) and S(n) its respective universal enveloping symmetric algebra. Consider the Jacobson topology on primitive spectrum of U(n), Poisson S(n). We provide homeomorphism between corresponding topological spaces (at level points, it gives bijection ideals S(n)). also show that all from an open set in properly chosen are genera...

This paper describes an algorithm for computing maximal tori of the reductive centralizer of a nilpotent element of an exceptional complex symmetric space. It illustrates also a good example of the use of Computer Algebra Systems to help answer important questions in the field of pure mathematics. Such tori play a fundamental rôle in several problems such as: classification of nilpotent orbits ...

A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphism. In this paper, we present combinatorial methods for finding such bounds. U...

The commutator of Lie algebra is extended to a product of fuzzy Lie subspaces over fuzzy field. An action of fuzzy field is introduced as an extension of scalar multiplication in Lie algebra. The notion of derived series and descending central series of fuzzy Lie ∼ideal over fuzzy field and the concepts of ∼nilpotent and ∼solvable fuzzy Lie ∼ideals over fuzzy field are studied. Mathematics Subj...

In a recent paper Borel and Serre proved the theorem: If 8 is a Lie algebra of characteristic 0 and 8 has an automorphism of prime period without fixed points f^O, then 8 is nilpotent.1 In this note we give a proof valid also for characteristic p^O. By the same method we can prove several other similar results on automorphisms and derivations. Our method is based on decompositions of the Lie al...

Any Lie algebra equipped with a symplectic form can be equipped with an affine structure. On the other hand there exist (2p + 1)-dimensional Lie algebras with contact form and no affine structure. But each nilpotent contact Lie algebra is a one-dimensional central extension of a symplectic algebra. The aim of this work is to study how we can extend, under certain conditions, the symplectic stuc...

This paper deals with the variety of commutative nonassociative algebras satisfying the identity Lx + γLx3 = 0, γ ∈ K. In [2] it is proved that if γ = 0, 1 then any finitely generated algebra is nilpotent. Here we generalize this result by proving that if γ 6= −1, then any such algebra is locally nilpotent. Our results require characteristic 6= 2, 3.

In this note we shall investigate a topological version of the problem of Kurosh: “Is any algebraic algebra locally finite?” Kaplansky’s theorem concerning the local nilpotence of nil PI-algebras is well-known. We will prove a generalization of Kaplansky’s theorem to the class of locally compact rings. We use in the proof a theorem of A. I. Shirshov [8] concerning the height of a finitely gener...