Two-Step Nilpotent Lie Groups and Homogeneous Fiber Bundles

In this paper, we construct two-step nilpotent Lie groups from homogeneous fiber bundles over compact symmetric spaces. The structure of the constructed nilpotent groups is expressed in terms of the compact Lie groups involved in the fiber bundles. There are close relations between the geometric properties of the nilpotent groups and the total spaces of the fiber bundles. We will find new examples of nilpotent groups which are weakly symmetric and Riemannian geodesic orbit spaces.     

Localization and completion of nilpotent groups of automorphisms

We study nilpotent subgroups of automorphism groups in the category of groups and the homotopy category of spaces. We establish localization and completion theorems for nilpotent groups of automorphisms of nilpotent groups. We then apply these algebraic theorems to prove analogous results for certain groups of self-homotopy equivalences of spaces.     

André-Quillen cohomology and rational homotopy of function spaces

We develop a simple theory of André-Quillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy self-equivalences of rational nilpotent CW-complexes. This puts certain results of Sullivan in a more conceptual framework.     

Nilpotient Lie群上Herz空间及它的某些应用

§1 .　IntroductionLetGbeaconnectedandsimplyconnectednilpotentLiegroupwith(bi invariant)HaarmeasuredgandLiealgebraG .Theexponetialmapissurjectiveby [12 ],Theorem 3.6 .1.Onecanassociatedasubellipticdistance (g ,h)d′( g ;h)witheachfixedalgeraicbasisa1 ,a2 ,…ad′ofG .Foralli∈ { 1,2 ,… ,d′} ,letAi =dL(ai)denotethegeneratorsoflefttranslationsactingontheclass{ai} .Thisdistancehasthecharacterizationd′( g ;h) =sup{ ｜ψ( g) - ψ(h) ｜ :ψ∈C∞0 (G) ,∑d′i=1｜ (Aiψ) ｜2 ≤ 1} ,(see [9],…     

Morphisms inverted by profinite completion

It is shown that a map between nilpotent spaces becomes an equivalence upon P-profinite completion, where P is a collection of primes, if and only if it is an equivalence with respect to mod p homology for all p in P. Homological criteria for a homomorphism between nilpotent groups to become an isomorphism or an epimorphism under P-profinite completion are given. These results are relativised to nilpotent fibrations and relative groups.     

Caractérisation des distributions gaussiennes sur les groupes nilpotents simplement connexes et les espaces symétriques

For simply connected nilpotent Lie groups, we show that a probability measure is gaussian in the sense of Bernstein (for a definition thereof which in a natural way involves non-commutativity) iff it is a gaussian measure in the classical sense concentrated on an abelian subgroup. Furthermore we carry over the Skitovic̆-Darmois theorem to symmetric spaces of non-compact type.     

On the classification of weakly symmetric Finsler spaces

In this paper, we give the classification of some special types of weakly symmetric Finsler spaces. We first present a general principle to classify weakly symmetric Finsler spaces and also give a method to figure out the Berwald spaces among the class of weakly symmetric Finsler spaces. Then we give an explicit classification of weakly symmetric Finsler spaces with reductive isometric groups as well as the left invariant weakly symmetric Finsler metrics on nilpotent Lie groups of the Heisenberg type. As an application, we obtain a large number of high-dimensional examples of reversible Finsler spaces which are non-Berwaldian and with vanishing S-curvature, a kind of spaces which are sought after in an open problem of Z. Shen.     

Subgroups of bounded residue in ultraproducts of finite groups

We consider a variety of linearity generalizations: finitary linearizability, local linearizability, and so on. The natural partial ordering on the set of classes of generalized linear groups under examination is described in full. Criteria for abstract groups to be finitary linearized in terms of ultraproducts of linear and finite groups are proven. Examples of nilpotent groups are given which have not faithful finitary linear representations by automorphisms of vector spaces over a field. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 531–542, September–October, 1997.     

Naturally reductive homogeneous Finsler spaces

In this paper, we study naturally reductive Finsler metrics. We first give a sufficient and necessary condition for a Finsler metric to be naturally reductive with respect to certain transitive group of isometries. Then we study in detail the left invariant naturally reductive metrics on compact Lie groups and give a method to construct the non-Riemannian ones. Further, we give a classification of left invariant naturally reductive metrics on nilpotent Lie groups. Finally, we give a classification of all the naturally reductive Finsler spaces of dimension less or qual to 4. As applications, we obtain some rigidity theorems about naturally reductive Finsler metrics. Namely, any left invariant non-symmetric naturally reductive Finsler metric on a compact simple Lie group or an indecomposable nilpotent Lie group must be Riemannian. On the other hand, we provide a very convenient method to construct non-symmetric Berwald spaces which are neither Riemannian nor locally Minkowskian, a kind of spaces which are sought after in the book by Bao et al. (An introduction to Riemann–Finsler geometry, GTM 200, 2000).     

Continuity of magnetic Weyl calculus

We investigate continuity properties of the operators obtained by the magnetic Weyl calculus on nilpotent Lie groups, using modulation spaces associated with unitary representations of certain infinite-dimensional Lie groups.     

Decompositions of nilpotent Lie algebras

It is proved that decompositions of nilpotent Lie algebras are global. In the complex case, nilpotency is also a necessary condition for every decomposition to be global. The results obtained are applied to the classification of complex homogeneous spaces of simply connected nilpotent Lie groups.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 27–30, January, 1978.In conclusion, the author would like to thank A. L. Onishchik for his interest in this research.     

Cochains and homotopy type

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E_∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E_∞ algebras is faithful but not full.     

A curve of nilpotent Lie algebras which are not Einstein nilradicals

The only known examples of non-compact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, i.e. nilpotent Lie algebras which are nilradicals of the Lie algebras of Einstein solvmanifolds. Up to now, very few examples of \mathbb N{\mathbb N}-graded nilpotent Lie algebras that cannot be Einstein nilradicals have been found. In particular, in each dimension, there are only finitely many known. We exhibit in the present paper two curves of pairwise non-isomorphic nine-dimensional two-step nilpotent Lie algebras which are not Einstein nilradicals.     

A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory

We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.

     

ON A CLASSIFICATION OF PRIME RINGS

Abstract

We formalize the adjunction of maps of localized nilpotent spaces of the homotopy type of CW-complexes. Using the semilocalization theory of M. Bendersky, in which only the higher homotopy groups are localized, we obtain an adjunction theorem with fewer nilpotency conditions on the spaces. The two main theorems are in the form of a theorem on maps of triads by J. P. May.     

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ?-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.     

Anosov diffeomorphisms on infra-nilmanifolds associated to graphs

Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore, their properties are intensively studied, including which spaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold, that is, a compact quotient of a 1-connected nilpotent Lie group by a discrete group of isometries. This conjecture motivates the problem of describing which infra-nilmanifolds admit an Anosov diffeomorphism. So far, most research was focused on the restricted class of nilmanifolds, which are quotients of 1-connected nilpotent Lie groups by uniform lattices. For example, Dani and Mainkar studied this question for the nilmanifolds associated to graphs, which form the natural generalization of nilmanifolds modeled on free nilpotent Lie groups. This paper further generalizes their work to the full class of infra-nilmanifolds associated to graphs, leading to a necessary and sufficient condition depending only on the induced action of the holonomy group on the defining graph. As an application, we construct families of infra-nilmanifolds with cyclic holonomy groups admitting an Anosov diffeomorphism, starting from faithful actions of the holonomy group on simple graphs.     

The central heights of stability groups of series in vector spaces

We compute the central heights of the full stability groups S of ascending series and of descending series of subspaces in vector spaces over fields and division rings. The aim is to develop at least partial right analogues of results on left Engel elements and related nilpotent radicals in such S proved recently by Casolo & Puglisi, by Traustason and by the current author. Perhaps surprisingly, while there is an absolute bound on these central heights for descending series, for ascending series the central height can be any ordinal number.     

Monogenic functions and representations of nilpotent Lie groups in quantum mechanics

We describe several different representations of nilpotent step two Lie groups in spaces of monogenic Clifford‐valued functions. We are inspired by the classic representation of the Heisenberg group in the Segal–Bargmann space of holomorphic functions. Connections with quantum mechanics are described. Copyright © 1999 John Wiley & Sons, Ltd.