Dimension Subgroups Research Articles

Augmentation powers and group homology

Generalizing the notion of polynomial 2-cocycles, a homological approach to the identification of normal subgroups determined by two-sided ideals in group rings is developed. In particular, transfinite dimension subgroups, i.e., the subgroups determined by the transfinite augmentation powers are investigated.

The maximal subgroups of positive dimension in exceptional algebraic groups

Introduction Preliminaries Maximal subgroups of type $A_1$ Maximal subgroups of type $A_2$ Maximal subgroups of type $B_2$ Maximal subgroups of type $G_2$ Maximal subgroups $X$ with rank$(X)\geq3$ Proofs of Corollaries 2 and 3 Restrictions of small $G$-modules to maximal subgroups The tables for Theorem 1 and Corollary 2 Appendix: $E_8$ structure constants References.

Kaluza-Klein supergravity on AdS3×S3

We construct a Chern-Simons type gauged N=8 supergravity in three spacetime dimensions with gauge group SO(4) x T_\infty over the infinite dimensional coset space SO(8,\infty)/(SO(8) x SO(\infty)), where T_\infty is an infinite dimensional translation subgroup of SO(8,\infty). This theory describes the effective interactions of the (infinitely many) supermultiplets contained in the two spin-1 Kaluza-Klein towers arising in the compactification of N=(2,0) supergravity in six dimensions on AdS_3 x S^3 with the massless supergravity multiplet. After the elimination of the gauge fields associated with T_\infty, one is left with a Yang Mills type gauged supergravity with gauge group SO(4), and in the vacuum the symmetry is broken to the (super-)isometry group of AdS_3 x S^3, with infinitely many fields acquiring masses by a variant of the Brout-Englert-Higgs effect.

Dimension subgroups andp-th powers inp-groups

We prove that if the nilpotence class of ap-group is strictly less thanp kthen every product ofp k-thpowers can be written as thep-th power of an element. Scoppola and Shalev have proven the same thing for groups of class strictly less thanp k−p k−1. They also provide an example which proves that ours is the best possible result. This is a generalization of the well known fact that in groups of class strictly less thanp every product ofp-powers is again ap-th power. Along the way we prove results of independent interest on dimension subgroups ofp-groups.

Uniform finite generation of compact Lie groups

Abstract Consider a compact connected Lie group G and the corresponding Lie algebra L . Let {X1,…,Xm} be a set of generators for the Lie algebra L . We prove that G is uniformly finitely generated by {X1,…,Xm}. This means that every element K∈G can be expressed as K=eXt1eXt2···eXtl, where the indeterminates X are in the set {X1,…,Xm}, t i ∈ R , i=1,…,l , and the number l is uniformly bounded. This extends a previous result by F. Lowenthal in that we do not require the connected one dimensional Lie subgroups corresponding to the X i , i=1,…,m , to be compact. We link the results to the existence of universal logic gates in quantum computing and discuss the impact on bang bang control algorithms for quantum mechanical systems.

The dimension subgroup conjecture holds for odd order groups

The dimension subgroup conjecture holds for odd order groups

Topology of symplectomorphism groups of rational ruled surfaces

LetMMbe eitherS2×S2S^2\times S^2or the one point blow-upCP2#CP¯2{\mathbb {C}}P^2\#\overline {{\mathbb {C}}P}^2ofCP2{\mathbb {C}}P^2. In both casesMMcarries a family of symplectic formsωλ\omega _{\lambda }, whereλ>−1\lambda > -1determines the cohomology class[ωλ][\omega _\lambda ]. This paper calculates the rational (co)homology of the groupGλG_\lambdaof symplectomorphisms of(M,ωλ)(M,\omega _\lambda )as well as the rational homotopy type of its classifying spaceBGλBG_\lambda. It turns out that each groupGλG_\lambdacontains a finite collectionKk,k=0,…,ℓ=ℓ(λ)K_k, k = 0,\dots ,\ell = \ell (\lambda ), of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute", i.e. all the higher Whitehead products that they generate vanish asλ→∞\lambda \to \infty. However, for each fixedλ\lambdathere is essentially one nonvanishing product that gives rise to a “jumping generator"wλw_\lambdainH∗(Gλ)H^*(G_\lambda )and to a single relation in the rational cohomology ringH∗(BGλ)H^*(BG_\lambda ). An analog of this generatorwλw_\lambdawas also seen by Kronheimer in his study of families of symplectic forms on44-manifolds using Seiberg–Witten theory. Our methods involve a close study of the space ofωλ\omega _\lambda-compatible almost complex structures onMM.

Subgroups Determined by Certain Products of Augmentation Ideals

Let G be a group, ZG the integral group ring of G, and I(G) its augmentation ideal. Let H be a subgroup of G. It is proved that the subgroup of G determined by the product I(H)I(G)I(H) equals γ3(H), i.e., the third term in the lower central series of H. Also, the subgroup determined by I(H)I(G)In(H) (resp., In(H)I(G)I(H)) for n > 1 equals Dn+2(H), the (n + 2)th dimension subgroup of H.

Cohomology of the Orlik–Solomon Algebras and Local Systems

The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.

Isomorphisms of Integral Group Rings of Infinite Groups

This paper deals with the isomorphism problem for integral group rings of infinite groups. In the first part we answer a question of Mazur by giving conditions for the isomorphism problem to be true for integral group rings of groups that are a direct product of a finite group and a finitely generated free abelian group. It is also shown that the isomorphism problem for infinite groups is strongly related to the normalizer conjecture. Next we show that the automorphism conjecture holds for infinite finitely generated abelian groups G if and only if Z G has only trivial units. In the second part we partially answer a problem of Sehgal. It is shown that the class of a finitely generated nilpotent group G is determined by its integral group ring provided G has only odd torsion. When G has nilpotency class two then the finitely generated restriction is not needed. This, together with a result of Ritter and Sehgal, settles the isomorphism problem for finitely generated nilpotency class two groups. A link is pointed out between this problem and the dimension subgroup problem.

Subgroups Determined by Certain Products of Augmentation Ideals

Let G be a group, ZG the integral group ring of G, and I(G) its augmentation ideal. Let H be a subgroup of G. It is proved that the subgroup of G determined by the product I(H)I(G)I(H) equals γ3(H), i.e., the third term in the lower central series of H. Also, the subgroup determined by I(H)I(G)I n (H) (resp., I n (H)I(G)I(H)) for n > 1 equals Dn+2(H), the (n + 2)th dimension subgroup of H.

Definable Compactness and Definable Subgroups of o-Minimal Groups

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

Finite dimensional representations and subgroup actions on homogeneous spaces

LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/Gℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/Gℤ; that ifH is epimorphic inG then the action ofH onG/Gℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/Gℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.

On Generalized Third Dimension Subgroups

AbstractLet G be any group, and H be a normal subgroup of G. Then M. Hartl identified the subgroup of G. In this note we give an independent proof of the result of Hartl, and we identify two subgroups of G for some subgroup K of G containing [H, G].

On the subgroup structure of exceptional groups of Lie type

We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X ( q ) X(q) of Lie type in the natural characteristic. Our approach is to show that for sufficiently large q q (usually q > 9 q>9 suffices), X ( q ) X(q) is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.

A Universal Coefficient Decomposition for Subgroups Induced by Submodules of Group Algebras

AbstractDimension subgroups and Lie dimension subgroups are known to satisfy a ‘universal coefficient decomposition’, i.e. their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the ‘universal’ coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary N-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods.

Dimension subgroups of extensions with an Abelian kernel

It is known that the dimension subgroup problem can be solved affirmatively in the class of groups whose lower central series quotients are torsion-free. In this paper it is proved that the same result is true for an arbitrary extension of a group belonging to this class by an Abelian group.

Polynomiality Properties of Group Extensions with a Torsion-Free Abelian Kernel

In studying torsion-free nilpotent groups of class 2 it is a key fact that each central group extension Zn↣E↠Zmis representable by abilinearcocycle. So for investigating nilpotent groups of higher class it is natural to ask for a generalization of this fact, namely when Zmis replaced by sometorsion-free nilpotentgroupGand Znby sometorsion-free abeliangroupBwithnilpotent G-action. In this paper we study a suitable notion of (bi)polynomialcocycles (in the strong sense of polynomiality introduced by Passi) and prove the desiredrepresentability theorem(4.3). This was known before only forcentralextensions withdivisiblekernel and with kernel Z ifGis abelian, nilpotent of class 2 or if the quotients of the lower central series ofGare torsion-free. Our representability result implies aconvergence theoremfor an approximation ofH2(G, B) bypolynomial cohomology groups(4.4). Since the latter ones are well accessible to computation one obtains a formula forH2(G, B) in terms of a presentation ofGwhich can be evaluated by integral matrix calculus. More precisely, a given presentation ofGamounts to a three-term cochain complex consisting offinitely generated free Z-modulesif the groupsGandBare finitely generated; the cohomology of this complex is identified withH2(G, B) in such a way that representing 2-cocycles are explicitly given in terms of integer valued rational polynomial functions (6.4). As an application, we establish an explicitbijectionbetween torsion-free nilpotentgroupsand Z-torsion-free nilpotentLie ringsboth finitely generated and nilpotent ofclass≤3 (7.2). In case a given group extension is representable by a polynomial cocycle we also determine theminimal degreeof polynomiality for which this holds. Indeed, dropping the assumption thatGis torsion-free nilpotent, we give anintrinsic characterizationof all group extensions with torsion-free abelian kernel which are representable by a polynomial cocycle of degree ≤n, provided that the cokernel is finitely generated and acts nilpotently on the kernel (4.2). Thus a polynomiality theory for group extensions as asked for by Passi is now achieved in case the kernel istorsion-free. A motivation for this—apart from calculatingH2(G, B) explicitly—comes from a question posed by J. Milnor in 1977, namely whether all finitely generated, torsion-free virtually polycyclic groups arise as fundamental groups of compact, complete affinely flat manifolds. Even the case of torsion-free nilpotent groups is interesting but far from being understood, notably since counterexamples of this type have recently been discovered. A connection of Milnor's question for nilpotent groups with polynomial constructions was first indicated in the work of P. Igodt and K. B. Lee. Recently a very close connection of this type was established, and anobstruction theorydeveloped for the problem in terms of polynomial cohomology. On the other hand, polynomiality properties of extensions with atorsionkernel are closely related todimension subgroups, as was observed by Passi in the case ofcentralextensions. An extension of this idea to thenoncentralcase is provided by Theorem 3.4 below. This might be of interest in connection with the recent discovery of Gupta and Kuz'min that the quotient groupDn(G)/γn(G) is an abelian normal but in general noncentral subgroup ofG/γn(G). Moreover, we obtain afunctorial equivalencebetween extensions oftorsion-free nilpotent groupsand of Z-torsion-free nilpotent modules(5.1). This improves a general result of Reiner and Roggenkamp in the special situation of torsion-free nilpotent groups. As applications, it yields an inductive cohomological description ofautomorphism groupsof torsion-free nilpotent groups [9] and a generalization of the classicalDold-Kan equivalence—between simplicial abelian groups and chain complexes of abelian groups—to simplicial groups of class 2. Applications in localization theory are also to be expected, namely to questions concerning the genus of torsion-free nilpotent groups. All the above-mentioned results are based on the more technical work of the first section. There, also, generalizations of theorems of Witt and Quillen are obtained concerning certain Lie algebras associated with groups; these results might be of independent interest.

Solutions in self-dual gravity constructed via chiral equations.

The chiral model for self-dual gravity given by Husain in the context of the chiral equations approach is discussed. A Lie algebra corresponding to a finite dimensional subgroup of the group of symplectic diffeomorphisms is found, and then use for expanding the Lie algebra valued connections associated with the chiral model. The self-dual metric can be explicitly given in terms of harmonic maps and in terms of a basis of this subalgebra.

Relativistic Gauge Invariant Potentials

A global method characterizing the invariant connections on an abelian principal bundle under a group of transformations is applied in order to get gauge invariant electromagnetic (elm.) potentials in a systematic way. So, we have classified all the elm. gauge invariant potentials under the Poincare subgroups of dimensions 4, 5, and 6, up to conjugation. It is paid attention in particular to the situation where these subgroups do not act transitively on the space-time manifold. We have used the same procedure for some galilean subgroups to get nonrelativistic potentials and study the way they are related to their relativistic partners by means of contractions. Some conformal gauge invariant potentials have also been derived and considered when they are seen as consequence of an enlargement of the Poincare symmetries.