We show that any CPA-structure (commutative post-Lie algebra structure) on a perfect Lie algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all ...CPA-structures on complete Lie algebras. As a special case we obtain the CPA-structures of parabolic subalgebras of semisimple Lie algebras.
We study disemisimple Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra g is disemisimple if and only if its ...solvable radical coincides with its nilradical and is a prehomogeneous s-module for a Levi subalgebra s of g. We use the classification of prehomogeneous s-modules for simple Lie algebras s given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend this result to disemisimple Lie algebras having no simple quotients of type A.
We study Lie algebras L that are graded by an arbitrary group (G,⁎) and have finite support, X. We show that L is nilpotent of |X|-bounded class if X is arithmetically-free. Conversely: if a finite ...subset Y of G is not arithmetically-free, then Y supports the grading of a non-nilpotent Lie algebra.
We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity r(t) = 0. In the special case of the polynomial
we obtain a ...uniform bound on the nilpotency class of Lie algebras admitting a periodic derivation of order n. We even find an optimal bound on the nilpotency class in characteristic p if p does not divide a certain invariant ρ
n
.
Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by ...geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic p>0. We show that every perfect modular Lie algebra in characteristic p>2 having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation algebra Out(g) of a simple modular Lie algebra g is solvable. We try to summarize the known results on the Zassenhaus conjecture and prove some new results using the classification of simple modular Lie algebras by Premet and Strade for algebraically closed fields of characteristic p>3. As a corollary we obtain that every central simple modular Lie algebra of characteristic p>3 admits only the trivial CPA-structure.
Let G be any group and A be a non-empty subset of G. The h-fold product set of A is defined asAh:={a1⋅a2⋯ah:a1,…,ah∈A}. Nathanson considered the concept of an asymptotic approximate group. Let ...r,l∈Z>0. The set A is said to be an (r,l)-approximate group in G if there exists a subset X in G such that |X|⩽l and Ar⊆XA. The set A is an asymptotic (r,l)-approximate group if the product set Ah is an (r,l)-approximate group for all sufficiently large h. Recently, Nathanson showed that every finite subset A of an abelian group is an asymptotic (r,l′)-approximate group (with the constant l′ explicitly depending on r and A). In this article, our motivations are three-fold:(1)We give an alternate proof of Nathanson's result.(2)From the alternate proof we deduce an improvement in the bound on the explicit constant l′.(3)We generalise the result and show that, in an arbitrary abelian group G, the union of k (unbounded) generalised arithmetic progressions is an asymptotic (r,(4rk)k)-approximate group.
We show that for a given nilpotent Lie algebra g with Z(g)⊆g,g all commutative post-Lie algebra structures, or CPA-structures, on g are complete. This means that all left and all right multiplication ...operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras Fg,c and discover a strong relationship to solving systems of linear equations of type x,u+y,v=0 for generator pairs x,y∈Fg,c. We use results of Remeslennikov and Stöhr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie algebra Fg,c has only central CPA-structures.
Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra ...has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques. We only consider finite-dimensional Lie algebras over a fields of characteristic zero.
We consider finite-dimensional complex Lie algebras admitting a periodic derivation, i.e., a nonsingular derivation which has finite multiplicative order. We show that such Lie algebras are at most ...two-step nilpotent and give several characterizations, such as the existence of gradings by sixth roots of unity, or the existence of a nonsingular derivation whose inverse is again a derivation. We also obtain results on the existence of periodic prederivations. In this context we study a generalization of Engel-4 Lie algebras.
PDF
