scholarly journals Structure of Finite-Dimensional Protori

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 93
Author(s):  
Wayne Lewis
Keyword(s):  
Structure Theorem ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Compact Groups ◽  
Real Vector ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Path Component ◽  
Free Abelian Groups ◽  
Torsion Free

A Structure Theorem for Protori is derived for the category of finite-dimensional protori (compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are parameterized in the structure theorem by the dual category of finite rank torsion-free abelian groups. A consequence is a universal resolution for a finite-dimensional protorus, independent of a choice of a particular subgroup. A resolution is also given strictly in terms of the path component of the identity and the union of all zero-dimensional subgroups. The structure theorem is applied to show that a morphism of finite-dimensional protori lifts to a product morphism between products of periodic locally compact groups and real vector spaces.

scholarly journals Locally compact products and coproducts in categories of topological groups

1977 ◽  
Vol 17 (3) ◽  
pp. 401-417 ◽  
Author(s):  
Karl Heinrich Hofmann ◽  
Sidney A. Morris
Keyword(s):  
Abelian Groups ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Torsion Free Group ◽  
Locally Compact Abelian Groups ◽  
Discrete Torsion ◽  
Compact Abelian Groups ◽  
Torsion Free

In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.

scholarly journals A Finiteness Condition for Locally Compact Abelian Groups

1971 ◽  
Vol 12 (1) ◽  
pp. 115-121 ◽  
Author(s):  
L. C. Grove ◽  
L. J. Lardy
Keyword(s):  
Finite Type ◽  
Abelian Groups ◽  
Finiteness Condition ◽  
Vector Spaces ◽  
Finite Sets ◽  
Locally Compact ◽  
Object A ◽  
Locally Compact Abelian Groups ◽  
Finite Dimensional ◽  
Compact Abelian Groups

A map f: A→B in category is called monic if fg = fh implies that g = h for all maps g, h: C → A; it is called epic if gf = hf implies that g = h for all maps g, h: B → C. An object A ∈ is called an S-object if every monic map f: A → A is also epic; it is called a Q-object if every epic map f: A → A is also monic. If A is both an S-object and a Q-object then A is called an SQ-object. In the category of sets the SQ-sets are the finite sets. In the category of vector spaces over a field F the SQ-spaces are precisely the finite dimensional spaces. In the light of these simple examples, it seems reasonable to view the SQ-objects of a category as being of ‘finite type’. We shall be chiefly concerned with investigating the SQ-objects in certain subcategories of the category of locally compact abelian groups.

From “Metabelian ℚ-Vector Spaces” to New ω-Stable Groups

1996 ◽  
Vol 2 (1) ◽  
pp. 84-93 ◽  
Author(s):  
Olivier Chapuis
Keyword(s):  
Group Theory ◽  
Model Theory ◽  
Stability Theory ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Universal Theory ◽  
Direct Power ◽  
Metabelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

The aim of this paper is to describe (without proofs) an analogue of the theory of nontrivial torsion-free divisible abelian groups for metabelian groups. We obtain illustrations for “old-fashioned” model theoretic algebra and “new” examples in the theory of stable groups. We begin this paper with general considerations about model theory. In the second section we present our results and we give the structure of the rest of the paper. Most parts of this paper use only basic concepts from model theory and group theory (see [14] and especially Chapters IV, V, VI and VIII for model theory, and see for example [23] and especially Chapters II and V for group theory). However, in Section 5, we need some somewhat elaborate notions from stability theory. One can find the beginnings of this theory in [14], and we refer the reader to [16] or [21] for stability theory and to [22] for stable groups.§1. Some model theoretic considerations. Denote by the theory of torsion-free abelian groups in the language of groups ℒgp. A finitely generated group G satisfies iff G is isomorphic to a finite direct power of ℤ. It follows that axiomatizes the universal theory of free abelian groups and that the theory of nontrivial torsion-free abelian groups is complete for the universal sentences. Denote by the theory of nontrivial divisible torsion-free abelian groups.

scholarly journals Topologies on locally compact groups

1988 ◽  
Vol 38 (1) ◽  
pp. 105-111 ◽  
Author(s):  
Joan Cleary ◽  
Sidney A. Morris
Keyword(s):  
Structure Theorem ◽  
Abelian Groups ◽  
Dual Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Local Weight ◽  
Cardinal Numbers ◽  
Hausdorff Group ◽  
The Relationship

Using the Iwasawa structure theorem for connected locally compact Hausdorff groups we show that every locally compact Hausdorff group G is homeomorphic to Rn × K × D, where n is a non-negative integer, K is a compact group and D is a discrete group. This makes recent results on cardinal numbers associated with the topology of locally compact groups more transparent. For abelian G, we note that the dual group, Ĝ, is homeomorphic to This leads us to the relationship card G = ω0(Ĝ) + 2ω0(G), where ω (respectively, ω0) denotes the weight (respectively local weight) of the topological group. From this classical results such as card G = 2 card Ĝ for compact Hausdorff abelian groups, and ω(G) = ω(Ĝ) for general locally compact Hausdorff abelian groups are easily derived.

scholarly journals Extending Topological Abelian Groups by the Unit Circle

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Hugo J. Bello ◽  
María Jesús Chasco ◽  
Xabier Domínguez
Keyword(s):  
Abelian Groups ◽  
Vector Spaces ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Topological Vector Spaces ◽  
Direct Limits ◽  
The Subject ◽  
Dense Subgroups ◽  
Three Space

A twisted sum in the category of topological Abelian groups is a short exact sequence0→Y→X→Z→0where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to0→Y→Y×Z→Z→0. We study the classSTG𝕋of topological groupsGfor which every twisted sum0→𝕋→X→G→0splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups withℒ∞topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups, and coproducts. As means to find further subclasses ofSTG𝕋, we use the connection between extensions of the form0→𝕋→X→G→0and quasi-characters onG, as well as three-space problems for topological groups. The subject is inspired on some concepts known in the framework of topological vector spaces such as the notion of𝒦-space, which were interpreted for topological groups by Cabello.

scholarly journals The main decomposition of finite-dimensional protori

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wayne Lewis ◽  
Peter Loth ◽  
Adolf Mader
Keyword(s):  
Finite Dimension ◽  
Abelian Groups ◽  
Direct Products ◽  
Pontryagin Duality ◽  
Finite Dimensional ◽  
Free Abelian Groups ◽  
Main Decomposition ◽  
Canonical Type ◽  
Fundamental Theorems ◽  
Torsion Free

AbstractA protorus is a compact connected abelian group of finite dimension. We use a result on finite-rank torsion-free abelian groups and Pontryagin duality to considerably generalize a well-known factorization of a finite-dimensional protorus into a product of a torus and a torus free complementary factor. We also classify direct products of protori of dimension 1 by means of canonical “type” subgroups. In addition, we produce the duals of some fundamental theorems of discrete abelian groups.

On the square subgroups of decomposable torsion-free abelian groups of rank three

2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Fysal Hasani ◽  
Fatemeh Karimi ◽  
Alireza Najafizadeh ◽  
Yousef Sadeghi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractThe square subgroup of an abelian group

scholarly journals EXAMPLES OF FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS IN CHARACTERISTIC 2

2020 ◽  
pp. 1-14
Author(s):  
NICOLÁS ANDRUSKIEWITSCH ◽  
DIRCEU BAGIO ◽  
SARADIA DELLA FLORA ◽  
DAIANA FLÔRES
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Group Algebras ◽  
Finite Abelian Groups ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Characteristic 2 ◽  
Nichols Algebras ◽  
Kirillov Dimension

Abstract We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

Sign in / Sign up

Export Citation Format

Share Document