C-minimal topological groups

Let \(G\) be a connected and simply connected Lie group with Lie algebra \(\mathfrak g \). We say that a subset \(X\) in the set \(\mathfrak g ^\star / G\) of coadjoint orbits is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in \(X\). In this paper, we define a class of solvable Lie groups, and we give an explicit construction of an overgroup \(G^+\) and a quadratic map \(\varphi \) sending each generic orbit in \(\mathfrak g ^\star \) to a \(G^+\)-orbit in \(\mathfrak{g ^+}^\star \), in such a manner that the set \(\varphi (\mathfrak g ^\star _{gen}){/ G^+}\) is convex hull separable. We then call \(G^+\) a weak quadratic overgroup for \(G\). Thanks to this construction, we prove that any nilpotent Lie group, with dimension at most 7 admits such a weak quadratic overgroup. Finally, we produce different examples of solvable Lie groups, having weak quadratic overgroups, but which are not in our class of Lie groups and for which usual constructions fail to hold.

In this paper, we continue with the results in \cite{Pg} and compute the group of quasi-isometries for a subclass of split solvable unimodular Lie groups. Consequently, we show that any finitely generated group quasi-isometric to a member of the subclass has to be polycyclic, and is virtually a lattice in an abelian-by-abelian solvable Lie group. We also give an example of a unimodular solvable Lie group that is not quasi-isometric to any finitely generated group, as well deduce some quasi-isometric rigidity results.

This paper is concerned with the topological structure of solvmanifolds i.e. spaces on which solvable Lie groups of transformations act transitively. The special case of nilmanifolds in which the group of transformations is nilpotent has been studied by Malcev [5]. Examples of two dimensional solvmanifolds are the torus, Klein bottle, cylinder, mobius strip. The central results are THEOREM A. Two compact solvmanifolds having the same fundamental group are isomorphic. THEOREM B. Any solvmanifold is covered a finite number of times by the direct product of a compact solvmanifold and a euclidean space; moreover, the covering group is abelian. THEOREM C. If S is an algebraically subgroup (cf. ?9 for definition) of the solvable Lie group G, then G/S is homeomorphic to the direct product of a compact solvmanifold and a euclidean space. Theorem C applies for example, if S is a closed subgroup which is generated by its connected identity-component together with elements s1, s2, -such that all the eigenvalues of Adsi are positive (i = 1, 2, * ). The assertion in Theorem A, applied to nilmanifolds, was proved by Malcev [5]. The situation in that case is simplified by virtue of the fact that two nilpotent groups which operate transitively and effectively on spaces with same fundamental group are isomorphic-a fact which is not true in the case of general solvmanifolds. Accordingly, there is a natural homeomorphism between nilmanifolds with isomorphic fundamental groups but there is no natural homeomorphism in the case of solvmanifolds. In the absence of uniqueness, therefore, proof of the existence of the homeomorphisin in Theorem A is rather elusive. After a number of algebraic maneuvers, proof of Theorem A is reduced to comparing two fibre bundles over homeomorphic toroids-the fibres being the orbits of transformation groups which are isomorphic as abstract topological groups. The main topological lemma asserts the existence of an equivariant homeomorphism between the two bundles. The major algebraic fact which underlies our method is the theorem of ?5 on uniform subgroups of solvable groups. In Theorem B, the covering need not be univalent-as is illustrated by the mobius band. The sufficiency condition in Theorem C yields as a special case a result of Chevalley: the factor space of a solvable Lie group by a connected subgroup is a product of circles and straight lines. Theorem A yields more generally

We are concerned with conditions under which a locally compact group $G$ has a maximal compact normal subgroup $K$ and whether or not $G/K$ is a Lie group. If $G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group, then $G$ is pro-Lie. If in $G$ there is a collection of closed normal subgroups $\{ {H_\alpha }\}$ such that $\cap {H_\alpha } = e$ and $G/{H_\alpha }$ is a Lie group for each $\alpha$, then $G$ is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group.

We establish analogs of the three Bieberbach theorems for a lattice \(\Gamma\) in a semidirect product \(\mathsf{K}\rtimes\mathsf{K}\) where \(\mathsf{S}\) is a connected, simply connected solvable Lie group and \(\mathsf{K}\) is a compact subgroup of its automorphism group. We first prove that the action of \(\Gamma\) on \(\mathsf{S}\) is metrically equivalent to an action of \(\Gamma\) on a supersolvable Lie group. The latter is shown to be determined by \(\Gamma\) itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group \(\Gamma\) as a lattice in a semidirect product \(\mathsf{S}\rtimes\mathsf{F}\) with \(\mathsf{F}\) being a finite group whose order is bounded by a constant only depending on the dimension of \(\mathsf{S}\). This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup \(\Upsilon \subset \mathsf{K}\rtimes\mathsf{K}\) that the quotient \(\mathsf{S}/\Upsilon\) is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group.

Let F be a discrete uniform subgroup of a connected simply connected solvable Lie group S. It is shown how S is essentially determined by F, using the point of view of group extensions. Let r be a discrete uniform subgroup of a connected simply connected solvable Lie group S. The main purpose of this work is to study how S is essentially determined by T from the point of view of group extensions. In [5], Mal'cev proved that T determines 5 uniquely if S is nilpotent. However, for a general solvable group, the situation seems less favorable. Nevertheless, L. Auslander (Theorem 2 in [1]) has obtained some results when T is the fundamental group of a nilmanifold, and in the subsequent works of Auslander and Tolimieri ([2], [8], [9]), stronger results were proved by using the notation of semisimple splittings, from which one can obtain the generalization of the result in [1] to arbitrary discrete uniform subgroups T. It is the purpose of this paper to revisit and prove directly this generalization by making use of the theory of group extensions as Auslander originally did in [1]. The author is indebted to the referee for pointing out the above results of Auslander and Tolimieri. 1. Let G and H be topological groups. An extension of G by H is a pair (E, rr) consisting of a topological group E which contains G as a closed normal subgroup and a continuous open homomorphism it of E onto H whose kernel is G. Two extensions (Et, 7r,-) of G by H are said to be equivalent if there exists an isomorphism a : Ex-+E2 of topological groups which leaves elements of G fixed and is such that tt2o=ttx. If (E, it) is an extension of G by H, then this determines a homomorphism Tr°:H—>0(G), where 0(G) denotes the group A(G) of all automorphisms of G modulo the inner automorphism group 1(G) of G. If (Ex, irx) and (E2, tt2) are equivalent, then 77^=712. For any homomorphism -0(G), let Ext(G, H, cp) denote the set of all equivalence classes of the extensions (E, it) of G by H Received by the editors July 12, 1971 and, in revised form, November 17, 1971. AMS 1970 subject classifications. Primary 22E15, 22E20, 22E40.

We introduce a Lohe group which is a new class of matrix Lie groups and present a continuous dynamical system for the synchronization of group elements in a Lohe group. The Lohe group includes classical Lie groups such as the orthogonal, unitary, and symplectic groups, and since Lohe groups need not be compact, global existence of ODEs may fail. The proposed dynamical system generalizes the Lohe model (Lohe in J Phys A 43:465301, 2010; Lohe in J Phys A 42:395101–395126, 2009) itself a nonabelian generalization of the Kuramoto model, and alongside we also generalize the analytical framework (Ha and Ryoo in J Stat Phys 163:411–439, 2016) of emergent and unique phase-locked states. For the construction of the phase-locked states, we introduce Lyapunov functions measuring the ensemble diameter and the dissimilarity between two Lohe flows, and derive Gronwall-type differential inequalities for them. The global existence of solutions then become a consequence of the boundedness of these Lyapunov functions. Our sufficient framework for the emergent dynamics is formulated in terms of coupling strength and initial states, and it leads to the global existence of solutions and the formation and uniqueness of a phase-locked asymptotic state. As a concrete example, we demonstrate how our theory can show emergent phenomenon on the Heisenberg group, where all initial configurations tend to a unique phase-locked state exponentially fast.

A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is easy to decide whether a nilmanifold is formal, Kaehlerian or (Hard) Lefschetz. The study of solvmanifolds meets with noticeably greater obstacles than the study of nilmanifolds. Even the construction of solvmanifolds is considerably more difficult than is the case for nilmanifolds. The reason is that there is no simple criterion for the existence of a lattice in a connected and simply-connected solvable Lie group. We consider the question of existence of lattices in solvable Lie groups up to dimension six and examine whether the corresponding solvmanifolds are formal, symplectic, Kaehler or (Hard) Lefschetz.

The well-known theory of square-integrable representations is generalized to the case of primary representations (not necessarily type I) quasi-contained in either the regular representation or the representation induced from a character of the center of a (not necessarily unimodular) locally compact group, and relations with the topology of the primitive ideal space of the group C*-algebra are obtained.The cases of discrete and almost connected groups are examined in more detail, and it is shown that for such groups, square-integrable factor representations must be traceable.For connected Lie groups, these representations can (in principle) be determined up to quasi-equivalence using a complicated construction of L. Pukanszky-for type I simply connected solvable Lie groups, the characterization reduces to that conjectured by C. C. Moore and J. Wolf.In the case of unimodular exponential groups, essentially everything is as in the nilpotent case (including a result on multiplicities in the decomposition of L2(G/T), T a discrete uniform subgroup of G).Finally, it is shown that the same criterion as for type I solvable Lie groups characterizes the squareintegrable representations of certain solvable p-adic groups studied by R. Howe.1. Introduction.The theory of square-integrable irreducible representations of unimodular groups has been understood for some time, and is easily accessible in [8, 14] and [36].Modifications have been introduced to take into account groups with noncompact center (e.g., [24]) and nonunimodular groups (most notably [10]; cf. also [25], [26]), but until recently, we were unaware of any treatment of non-type I square-integrable representations aside from some remarks on cyclic representations in [36], [25], and [26].A few months ago, the author received a copy of [7], which, in Chapter III, does treat a much more general situation, but here the emphasis is on von Neumann algebras and not much is said about groups.The lack of study of general square-integrable representations would not be much of a deficiency were it not for the fact that many discrete groups and Lie groups have

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

In the paper, we investigate three-dimensional solvable Lie groups from the point of view of the generalized Hermitian geometry. The corresponding three-dimensional solvable Lie algebras were firstly classified by G. M. Mubarakzyanov in 1963. Using the classification in somewhat different notations, we construct basic left-invariant metric f-structures of rank 2 on all three-dimensional solvable Lie groups equipped with the standard left-invariant Riemannian metric. It was proved that all the considered f-structures belong to one or several classes of generalized almost Hermitian structures. As a result, it gives the opportunity to present new examples of left-invariant Killling, nearly Kähler, generalized nearly Kähler and Hermitian f-structures on solvable Lie groups.

Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner stability, asymptotic stability, and BIBO (Bounded Input, Bounded Output) stability, are well understood for classical linear control systems in Euclidean spaces. This paper aims to thoroughly address the stability problem for a class of linear control systems defined on matrix Lie groups. This approach generalizes classical models corresponding to the latter when the group is Abelian and non-compact. It is important to note that this generalization leads to a very difficult control system, due to the complexity of the state space and the special dynamics resulting from the drift and control vectors. Several mathematical concepts help us understand and characterize stability in the classical case. We first show how to extend these algebraic, topological, and dynamical concepts from Euclidean space to a connected Lie group of matrices. Building on classical results, we identify a pathway that enables us to formulate conjectures about stability in this broader context. This problem is closely linked to the controllability and observability properties of the system. Fortunately, these properties are well established for both classes of linear systems, whether in Euclidean spaces or on Lie groups. We are confident that these conjectures can be proved in future work, initially for the class of nilpotent and solvable groups, and later for semi-simple groups. This will provide valuable insights that will facilitate, through Jouan’s Equivalence Theorem, the analysis of an important class of nonlinear control systems on manifolds beyond Lie groups. We provide an example involving a three-dimensional solvable Lie group of rigid motions in a plane to illustrate these conjectures.

We study the positive Hermitian curvature flow for left-invariant metrics on 2-step nilpotent Lie groups G with a left-invariant complex structure J . We describe the long-time behavior of the flow under the assumption that $$J[\mathfrak {g}, \mathfrak {g}]$$ J [ g , g ] is contained in the center of $$\mathfrak {g}$$ g . We show that under our assumption the flow $$g_{t}$$ g t exists for all positive t and $$(G,(1+t)^{-1}g_{t})$$ ( G , ( 1 + t ) - 1 g t ) converges, in the Cheeger-Gromov topology, to a 2-step nilpotent Lie group with a non flat semi-algebraic soliton. Moreover, we prove that, in our class of Lie groups, there exists at most one semi-algebraic soliton solution, up to homothety. Similar results were proved by M. Pujia and J. Stanfield for nilpotent complex Lie groups [21, 24]. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.

A finitely generated subgroup${\rm\Gamma}$of a real Lie group$G$is said to be Diophantine if there is${\it\beta}>0$such that non-trivial elements in the word ball$B_{{\rm\Gamma}}(n)$centered at$1\in {\rm\Gamma}$never approach the identity of$G$closer than$|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group$G$is said to be Diophantine if for every$k\geqslant 1$a random$k$-tuple in$G$generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most$5$, or derived length at most$2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

Exponential Radicals of Solvable Lie Groups