scholarly journals Representations of infinite soluble groups

1983 ◽  
Vol 24 (1) ◽  
pp. 43-52 ◽  
Author(s):  
Ian M. Musson
Keyword(s):  
Group Algebra ◽  
Soluble Group ◽  
Polycyclic Group ◽  
Soluble Groups ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Torsion Free ◽  
Free Soluble Group

The purpose of this paper is to study the following two questions.(1) When does the group algebra of a soluble group have infinite dimensional irreducible modules?(2) When is the group algebra of a torsion free soluble group primitive?In relation to the first question, Roseblade [13] has proved that if G is a polycyclic group and k an absolute field then all irreducible kG-modules are finite dimensional. Here we prove a converse.

Irreducible Modules for Polycyclic Group Algebras

1981 ◽  
Vol 33 (4) ◽  
pp. 901-914 ◽  
Author(s):  
I. M. Musson
Keyword(s):  
Nilpotent Group ◽  
Irreducible Module ◽  
Polycyclic Group ◽  
Group Algebras ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules

If G is a polycyclic group and k an absolute field then every irreducible kG-module is finite dimensional [10], while if k is nonabsolute every irreducible module is finite dimensional if and only if G is abelian-by-finite [3]. However something more can be said about the infinite dimensional irreducible modules. For example P. Hall showed that if G is a finitely generated nilpotent group and V an irreducible kG-module, then the image of kZ in EndkGV is algebraic over k [3]. Here Z = Z(G) denotes the centre of G. It follows that the restriction Vz of V to Z is generated by finite dimensional kZ-modules. In this paper we prove a generalization of this result to polycyclic group algebras.We introduce some terminology.

scholarly journals Group algebras of some generalised soluble groups

1972 ◽  
Vol 18 (1) ◽  
pp. 1-5 ◽  
Author(s):  
R. P. Knott
Keyword(s):  
Free Group ◽  
Group Algebra ◽  
Soluble Group ◽  
Group Algebras ◽  
Soluble Groups ◽  
Torsion Free Group ◽  
Open Question ◽  
Torsion Free

In (8) Stonehewer referred to the following open question due to Amitsur: If G is a torsion-free group and F any field, is the group algebra, FG, of G over F semi-simple? Stonehewer showed the answer was in the affirmative if G is a soluble group. In this paper we show the answer is again in the affirmative if G belongs to a class of generalised soluble groups

On the Cohomological Dimension of Soluble Groups

1981 ◽  
Vol 24 (4) ◽  
pp. 385-392 ◽  
Author(s):  
D. Gildenhuys ◽  
R. Strebel
Keyword(s):  
Homomorphic Image ◽  
Soluble Group ◽  
Cohomological Dimension ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Image Position ◽  
Torsion Free ◽  
Free Soluble Group

AbstractIt is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and its homological and cohomological dimensions over the integers and rationals satisfy the inequalitiesWe prove that G must be finitely generated if the equality hG = cdQG holds. Moreover, we show that if G is a countable soluble group of finite Hirsch number, but not necessarily torsion-free, and if hG = cdQG, then hḠ = cdQḠ for every homomorphic image Ḡ of G.

scholarly journals Locally soluble groups with min-n.

1974 ◽  
Vol 17 (3) ◽  
pp. 305-318 ◽  
Author(s):  
H. Heineken ◽  
J. S. Wilson
Keyword(s):  
Soluble Group ◽  
Free Groups ◽  
Torsion Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Isomorphism Classes ◽  
Torsion Groups ◽  
Torsion Free ◽  
General Method

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.

Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras

1996 ◽  
Vol 39 (1) ◽  
pp. 111-114
Author(s):  
F. Okoh
Keyword(s):  
Rational Functions ◽  
Indecomposable Module ◽  
Torsion Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dynkin Diagrams ◽  
The Status ◽  
Hereditary Algebras ◽  
Divisible Modules ◽  
Torsion Free

AbstractIf R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

scholarly journals Groups which satisfy a weak form of Poincaré duality

1991 ◽  
Vol 34 (3) ◽  
pp. 463-486
Author(s):  
J. E. Roberts
Keyword(s):  
Finite Rank ◽  
Weak Form ◽  
Homological Dimension ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Soluble Groups ◽  
Finite Dimensional ◽  
Duality Property ◽  
The Given ◽  
Torsion Free

Our main result is that a “restricted Poincaré duality” property with respect to finite dimensional coefficient modules over a field holds for a certain class of groups which includes all soluble groups of finite Hirsch length. This relies on a generalisation to the given class of a module construction by Stammbach; an extension of his result on homological dimension to these groups is given. We also generalise the well-known result that torsion-free soluble groups of finite rank are countable.

Graded torsion-free 𝔰𝔩2(ℂ)-modules of rank 2

2021 ◽  
pp. 2250229
Author(s):  
Yuri Bahturin ◽  
Abdallah Shihadeh
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Rank 2 ◽  
Torsion Free

In this paper, we explore the possibility of endowing simple infinite-dimensional [Formula: see text]-modules by the structure of graded modules. The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in 7, 8.

scholarly journals Cyclotomic Nazarov-Wenzl Algebras

2006 ◽  
Vol 182 ◽  
pp. 47-134 ◽  
Author(s):  
Susumu Ariki ◽  
Andrew Mathas ◽  
Hebing Rui
Keyword(s):  
Irreducible Representations ◽  
Arbitrary Field ◽  
Dimensional Algebra ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Brauer Algebras

AbstractNazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain “cyclotomic quotients” of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank rn(2n−1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

Tensor Product Realizations of Simple Torsion Free Modules

2001 ◽  
Vol 53 (2) ◽  
pp. 225-243 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Complex Numbers ◽  
Infinite Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Elementary Construction ◽  
Free Modules ◽  
Torsion Free

AbstractLet be a finite dimensional simple Lie algebra over the complex numbers C. Fernando reduced the classification of infinite dimensional simple -modules with a finite dimensional weight space to determining the simple torsion free -modules for of type A or C. Thesemodules were determined by Mathieu and using his work we provide a more elementary construction realizing each one as a submodule of an easily constructed tensor product module.

The Torsion Free Pieri Formula

1998 ◽  
Vol 50 (2) ◽  
pp. 266-289 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Tensor Product ◽  
Arbitrary Degree ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional ◽  
Pieri Formula ◽  
Free Modules ◽  
Central Characters ◽  
Torsion Free

AbstractCentral to the study of simple infinite dimensional g𝓵(n, C)-modules having finite dimensional weight spaces are the torsion free modules. All degree 1 torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree 1 with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct.

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