In this paper we study non-abelian extensions of a Lie group $G$ modeled on a locally convex space by a Lie group $N$. The equivalence classes of such extension are grouped into those corresponding to a class of so-called smooth outer actions $S$ of $G$ on $N$. If $S$ is given, we show that the corresponding set $\Ext(G,N)_S$ of extension classes is a principal homogeneous space of the locally smooth cohomology group $H^2_{ss}(G,Z(N))_S$. To each $S$ a locally smooth obstruction class $χ(S)$ in a suitably defined cohomology group $H^3_{ss}(G,Z(N))_S$ is defined. It vanishes if and only if there is a corresponding extension of $G$ by $N$. A central point is that we reduce many problems concerning extensions by non-abelian groups to questions on extensions by abelian groups, which have been dealt with in previous work. An important tool is a Lie theoretic concept of a smooth crossed module $α: H \to G$, which we view as a central extension of a normal subgroup of $G$.
An almost Abelian Lie algebra is a non-Abelian Lie algebra with a codimension 1 Abelian ideal. Most 3-dimensional real Lie algebras are almost Abelian, and they appear in every branch of physics that deals with anisotropic media - cosmology, crystallography etc. In differential geometry and theoretical physics, almost Abelian Lie groups have given rise to some of the simplest solvmanifolds on which various geometric structures such as symplectic, Kähler, spin etc., are currently studied in explicit terms. However, a systematic study of almost Abelian Lie groups and algebras from mathematics pe…
We classify elementary abelian 2 subgroups of compact simple Lie groups of adjoint type. This finishes the classification of elementary abelian $p$ subgroups of compact (or linear algebraic) simple groups of adjoint type.
An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media - cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified a…
We classify abelian subgroups of the automorphism group of any compact simple Lie algebra whose centralizer has the same dimension as the dimension of the subgroup. This leads to a classification of the maximal abelian subgroups of compact simple groups of adjoint type and a classification of the fine group gradings of complex simple Lie algebras.
We classify closed abelian subgroups of a compact simple Lie group of adjoint type and of type E having centralizer of the same dimension as the dimension of the subgroup and describe Weyl groups of maximal abelian subgroups.
We classify closed abelian subgroups of the simple groups $G_2$, $F_4$, $Aut(so(8))$ having centralizer the same dimension as the dimension of the subgroup, as well as finite abelian subgroups of certain spin and half-spin groups having finite centralizer.
For a positive integer $n$, with $n \geq 4$, let $R_{n}$ be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank $n$. We show that the subgroup of Aut$(R_{n})$ generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of $R_{n}$ is dense in Aut$(R_{n})$ with respect to the formal power series topology.
Let G be a p-adic Lie group and Ad be the adjoint representation of G on its Lie algebra. It was claimed in the literature that the kernel K of Ad always has an abelian open normal subgroup. We show by means of a counterexample that this assertion is false; it can even happen that K=G but G has no abelian subnormal subgroup except for the trivial group. The arguments are based on auxiliary results on subgroups of free products with central amalgamation.
In the present paper we study abelian extensions of connected Lie groups $G$ modeled on locally convex spaces by smooth $G$-modules $A$. We parametrize the extension classes by a suitable cohomology group $H^2_s(G,A)$ defined by locally smooth cochains and construct an exact sequence that describes the difference between $H^2_s(G,A)$ and the corresponding continuous Lie algebra cohomology space $H^2_c(\g,\a)$. The obstructions for the integrability of a Lie algebra extensions to a Lie group extension are described in terms of period and flux homomorphisms. We also characterize the extensions w…
Infinite systems of equations in abelian and nilpotent groups
Faculty of Mechanics and Mathematics of Lomonosov Moscow State University（莫斯科国立大学力学与数学系）；Moscow Center for Fundamental and Applied Mathematics（莫斯科基础与应用数学中心）
We call a linear operator on a vector space over a field Jordanable if it has a Jordan canonical form. An almost Abelian Lie algebra has only one non-vanishing Lie bracket, which is given by a linear operator. If the latter is Jordanable then we call the almost Abelian Lie algebra Jordanable. The existence of canonical form allows to describe important structural properties of these Lie algebras in explicit terms. Lie subalgebras, ideals, automorphisms and derivations, as well as quadratic Casimir elements are given in explicitly in a suitable basis.
The optimal constants are found for Lebesgue norm multilinear inequalities of Holder-Brascamp-Lieb type for arbitrary discrete Abelian groups. Previously a criterion for finiteness of the constants had been established for finitely generated Abelian groups, and the optimal constant had been found in the torsion-free case. The main step here is the analysis of finite groups.
In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two. Throughout the paper, we also give several examples to clarify some results.
The purpose of this paper is twofold. Firstly, we generalize the notion of characteristic polynomials of hyperplane and toric arrangements to those of certain abelian Lie group arrangements. Secondly, we give two interpretations for the chromatic quasi-polynomials and their constituents through subspace and toric viewpoints.
In this note we show that if $G$ is a countably infinite abelian group such that $nG=0$ for some integer $n$, then the only locally minimal group topology on $G$ is the discrete one.
We prove that there exist infinitely many a non-abelian strongly real Beauville $p$-group for every prime $p$. Previously only finitely many in the case $p=2$ have been constructed.
For any countable subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $τ$ such that $H$ is the von Neumann radical of $(G,τ)$. In particular, we obtain the positive answer to Comfort's question: any unbounded countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. A bounded infinite Abelian group admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. If, in addition, $G$ is countably infinite, a MinAP group topology can be chosen to be complete.
A topological group G is said to be almost maximally almost-periodic if its von Neumann radical is non-trivial, but finite. In this paper, we prove that every abelian group with an infinite torsion subgroup admits a (Hausdorff) almost maximally almost-periodic group topology. Some open problems are also formulated.
Let G be a simple non-compact linear connected Lie group and H be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces G/H admitting proper actions of discrete non virtually abelian subgroups of G. We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces G/H admitting proper action of a discrete and non virtually abelian subgroup of G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup.
In this paper we introduce a polynomial time algorithm that solves both the conjugacy decision and search problems in free abelian-by-infinite cyclic groups where the input is elements in normal form. We do this by adapting the work of Bogopolski, Martino, Maslakova, and Ventura in \cite{bogopolski2006conjugacy} and Bogopolski, Martino, and Ventura in \cite{bogopolski2010orbit}, to free abelian-by-infinite cyclic groups, and in certain cases apply a polynomial time algorithm for the orbit problem over $\Z^n$ by Kannan and Lipton.
The number of Frattini chief factors or of chief factors which are complemented by a maximal subalgebra of a finite-dimensional Lie algebra $L$ is the same in every chief series for $L$, by \cite[Theorem 2.3]{[11]}. However, this is not the case for the number of chief factors which are simply complemented in $L$. In this paper we determine the possible variation in that number.
On the torus group, on the group of p-adic integers and on the p-adic solenoid, we give a construction of an arbitrary weakly infinitely divisible probability measure using a random element with values in a product of (possibly infinitely many) subgroups of the real numbers. As a special case of our results, we have a new construction of the Haar measure on the p-adic solenoid.
We show that if $G$ is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous ergodic invariant random subgroups. If $G$ is a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous $G$-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary $p$-groups of countably infinite rank.
In this paper we extend and adapt several results on extensions of Lie algebras to topological Lie algebras over topological fields of characteristic zero. In particular we describe the set of equivalence classes of extensions of the Lie algebra by the Lie algebra a disjoint union of affine spaces with translation group H^2(\g,\z(\n))_{[S]}, where [S] denotes the equivalence class of the continuous outer action S \: \g \to \der \n. We also discuss topological crossed modules and explain how they are related to extensions of Lie algebras by showing that any continuous outer action gives rise to…
A $G$-grading on a complex semisimple Lie algebra $L$, where $G$ is a finite abelian group, is called quasi-good if each homogeneous component is 1-dimensional and 0 is not in the support of the grading. Analogous to classical root systems, we define a finite root system $R$ to be some subset of a finite symplectic abelian group satisfying certain axioms. There always corresponds to $R$ a semisimple Lie algebra $L(R)$ together with a quasi-good grading on it. Thus one can construct nice basis of $L(R)$ by means of finite root systems. We classify finite maximal abelian subgroups $T$ in $\Aut(L…
