arXiv subjects

Wolfgang Alexander Moens

Publications and source records attributed to Wolfgang Alexander Moens.

14 recordsLinked to original sources

arXiv2022

Finite groups admitting a coprime automorphism satisfying an additional polynomial identity

It is known that a finite group with an automorphism $φ$ of coprime order has a soluble radical of $(|φ|,|C_G(φ)|)$-bounded Fitting height and index. We extend this classic result as follows. Let $f(x) = a_0 + a_1 \cdot x + \cdots + a_d \cdot x^d \in \mathbb{Z}[x]$ be a primitive polynomial and let $G$ be a finite group with an automorphism $φ$ of coprime order satisfying $ g^{a_0} \cdot φ(g)^{a_1} \cdots φ^d(g)^{a_d} = 1 $, for all $g \in G$. Then the soluble radical of $G$ has $(d,|C_G(φ)|)$-boundex Fitting height and index. The bounds are made explicit and are particularly good for small values of the degree $d$.

math.GR↗

arXiv2022

Semisimple decompositions of Lie algebras and prehomogeneous modules

We study {\em 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 $\mathfrak{g}$ is disemisimple if and only if its solvable radical coincides with its nilradical and is a prehomogeneous $\mathfrak{s}$-module for a Levi subalgebra $\mathfrak{s}$ of $\mathfrak{g}$. We use the classification of prehomogeneous $\mathfrak{s}$-modules for simple Lie algebras $\mathfrak{s}$ 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 $A$.

math.RT↗

arXiv2021

The Fitting height of finite groups with a fixed-point-free automorphism satisfying an identity

Motivated by classic theorems of Thompson and Berger on the Fitting height of finite groups with a fixed-point-free automorphism of coprime order, we conjecture that, for every non-zero polynomial $f(x) = a_0 + a_1 x + \cdots + a_d x^d \in \mathbb{Z}[x] $, there is an integer $k > 0$ with the following property. Let $G$ be a finite (solvable) group with a fixed-point-free automorphism $α$ satisfying $\gcd(|G|,k)= 1$ and $$\{ g^{a_0} \cdot α(g)^{a_1} \cdot α^2(g)^{a_2} \cdots α^d(g)^{a_d} | g \in G \} = \{1\}.$$ Then the Fitting height of $G$ is at most the number of irreducible factors of $f(x)$. We confirm the conjecture for a large family of polynomials with explicit constants $k$.

math.GR↗

arXiv2020

The nilpotency of finite groups with a fix-point-free automorphism satisfying an identity

We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial $r(t) = a_0 + a_1 \cdot t + \cdots + a_d \cdot t^d \in \mathbb{Z}[t]$ that is irreducible over $\mathbb{Q}$, there exist (explicit) invariants $a,b,c \in \mathbb{N}$ with the following property. Consider a finite group with a fix-point-free automorphism $α:{G}\longrightarrow{G}$ and suppose that for all $x \in G$ we have the equality $x^{a_0} \cdot α(x^{a_1}) \cdot α^2(x^{a_2})\cdots α^d(x^{a_d}) = 1_G.$ Then $G$ is solvable and of the form $A \cdot (B \rtimes (C \times D))$, where $A$ is an $a$-group, $B$ is a $b$-group, $C$ is a nilpotent $c$-group, and $D$ is a nilpotent group of class at most $d^{2^d}$. Here, a group $H$ is said to be an $a$-group (resp. $b$-group or $c$-group) if the order of every $h \in H$ divides some natural power of $a$ (resp. $b$ or $c$).

math.GR↗

arXiv2020

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

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 ${\rm Out}(\mathfrak{g})$ of a simple modular Lie algebra $\mathfrak{g}$ 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 that every central simple modular Lie algebra of characteristic $p>3$ admits only the trivial CPA-structure.

math.RA↗

arXiv2019

On semilinear sets and asymptotically approximate groups

Let $G$ be any group and $A$ be an arbitrary subset of $G$ (not necessarily symmetric and not necessarily containing the identity). The $h$-fold product set of $A$ is defined as $$A^{h} :=\lbrace a_{1}.a_{2}...a_{h} : a_{1},\ldots,a_n \in A \rbrace.$$ Nathanson considered the concept of an asymptotic approximate group. Let $r,l \in \mathbb{N}$. The set $A$ is said to be an $(r,l)$ approximate group in $G$ if there exists a subset $X$ in $G$ such that $|X|\leqslant l$ and $A^{r}\subseteq XA$. The set $A$ is an asymptotic $(r,l)$-approximate group if the product set $A^{h}$ 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 $A$). 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.

math.NT↗

arXiv2018

Post-Lie algebra structures for nilpotent Lie algebras

We study post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$ for nilpotent Lie algebras. First we show that if $\mathfrak{g}$ is nilpotent such that $H^0(\mathfrak{g},\mathfrak{n})=0$, then also $\mathfrak{n}$ must be nilpotent, of bounded class. For post-Lie algebra structures $x\cdot y$ on pairs of $2$-step nilpotent Lie algebras $(\mathfrak{g},\mathfrak{n})$ we give necessary and sufficient conditions such that $x\circ y=\frac{1}{2}(x\cdot y+y\cdot x)$ defines a CPA-structure on $\mathfrak{g}$, or on $\mathfrak{n}$. As a corollary we obtain that every LR-structure on a Heisenberg Lie algebra of dimension $n\ge 5$ is complete. Finally we classify all post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$ for $\mathfrak{g}\cong \mathfrak{n}\cong \mathfrak{n}_3$, where $\mathfrak{n}_3$ is the $3$-dimensional Heisenberg Lie algebra.

math.RA↗

arXiv2017

Representing Lie algebras using approximations with nilpotent ideals

We prove a refinement of Ado's theorem for Lie algebras over an algebraically-closed field of characteristic zero. We first define what it means for a Lie algebra $L$ to be approximated with a nilpotent ideal, and we then use such an approximation to construct a faithful representation of $L$. The better the approximation, the smaller the degree of the representation will be. We obtain, in particular, explicit and combinatorial upper bounds for the minimal degree of a faithful $L$-representation. The proofs use the universal enveloping algebra of Poincaré-Birkhoff-Witt and the almost-algebraic hulls of Auslander and Brezin.

math.RA↗

arXiv2016

Arithmetically-free group-gradings of Lie algebras

A Lie algebra L is known to be nilpotent if it admits a grading by (Zp, +) with support X not containing 0. It is also known that the class of L can be bounded by some explicit function of |X|. We generalise this and other classical results to gradings of Lie algebras by arbitrary groups with arithmetically-free support. We then apply these results to automorphisms of groups satisfying an identity.

math.RA↗

arXiv2016

Arithmetically-free group-gradings of Lie algebras: II

We study group-graded Lie algebras L with finite support X. We show that L is nilpotent of |X|-bounded class if X is arithmetically-free. Conversely: we show that Y supports the grading of a non-nilpotent Lie algebra if Y is not arithmetically-free.

math.RA↗

arXiv2015

Commutative post-Lie algebra structures on Lie algebras

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 parabolic subalgebras of simple Lie algebras.

math.RA↗

arXiv2015

A note on the structure of graded Lie algebras

Consider a finite-dimensional, complex Lie algebra G and a semi-simple automorphism α. This note aims to give a short and simple proof for explicit upper bounds for the derived length of the radical R and the rank of a Levi complement G/R in terms of the number of eigenvalues of α and the dimension of the space of fixed-points. This is an extension of classical theorems by Kreknin, Shalev and Jacobson.

math.RA↗

arXiv2010

A characterisation of nilpotent Lie algebras by invertible Leibniz-derivations

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.

math.RA↗

arXiv2010

Faithful Lie algebra modules and quotients of the universal enveloping algebra

We describe a new method to determine faithful representations of small dimension for a finite dimensional nilpotent Lie algebra. We give various applications of this method. In particular we find a new upper bound on the minimal dimension of a faithful module for the Lie algebras being counter examples to a well known conjecture of J. Milnor.

math.RT↗