Semidirect Product Group Research Articles

Geometric quantization of reduced contangent bundles

The method of geometric quantization is applied to the symplectic space obtained by Marsden-Weinstein reduction of a cotangent bundle T ∗N . Specifically it is assumed that the symmetry [gauge] group acts freely on N so that N is a principal H-bundle over the assumed Riemannian manifold Q = N/ H. The reduced phase space obtained is the same of that for a charged particle moving on Q in an external Yang-Mills gauge field which is given by a connection on N → Q. An explicit map is found from a subalgebra of the classical observables to the corresponding quantum operators. These operators are found to be the generators of a representation of the semi-direct product group, Aut N ⋉ C ∞ (Q) . A generalized Aharanov-Bohm effect is shown to be a natural consequence of the quantization procedure. In particular, the role of the connection in the quantum mechanical system is made clear. The quantization of the Hamiltonian is also considered. Additionally, our approach allows the related quantization procedures proposed by Mackey and by Isham to be fully understood.

Cohomology of Semidirect Product Groups

We give examples of semidirect product groups G⋉A such that the HochschildSerre spectral sequence H(G,HA) ⇒ H(G⋉A) for Z/p-cohomology has nonzero differentials. Until now, few such examples have been known, especially when the normal subgroup A is abelian. In particular, Benson and Feshbach [2] mentioned that in all known semidirect products with A abelian, the spectral sequence satisfies: (1) All differentials after d2 are 0. (2) All differentials are 0 if A = (S1)n. (To be consistent with the notation for discrete groups A, HA here means the cohomology of the classifying space of A.) (3) All differentials are 0 if A = (Z/2)n and we consider cohomology with Z/2 coefficients. We give examples to show that all three statements can fail. In fact, there can be nonzero differentials at dp or later in all of these cases. I expect that there can be nonzero differentials arbitrarily far along in the spectral sequence in all of these cases, but the problem remains open. (For semidirect products G ⋉ A with A not abelian, Benson and Feshbach [2] gave examples of nonzero differentials arbitrarily far along in the spectral sequence for Z/2-cohomology.) It turns out that there is a very general reason why there will be nonzero differentials in some examples. If X is a G-space, then H(G,C(X)) admits Steenrod operations compatible with those on HG because it is the cohomology of the space (X × EG)/G, whereas there is no reason for H(G,M) to have Steenrod operations for a general G-module M . Thus Steenrod operations provide a fundamental obstruction for a G-module to be the representation of G on the cohomology of a G-space, as G. Carlsson found [4]; there is a useful exposition by Benson and Habegger [3]. If a semidirect product G ⋉ A has the G-action on A given by the dual of such a G-module, we can show that there must be nonzero differentials in the Hochschild-Serre spectral sequence. It is interesting to contrast these examples with Nakaoka’s theorem that the Hochschild-Serre spectral sequence has no differentials for any wreath product G⋉ Hn ([6], p. 50). Here G and H are any finite groups and G acts on Hn through a permutation representation G → Sn. It would be good to characterize algebraically the class of G-modules M over Z/p, say for a p-group G, such that the semidirect product G ⋉M has no differentials in the spectral sequence: it seems to be fairly close to the class of permutation modules, but there are some other interesting examples. I would like to thank J. L. Alperin, Steve Siegel, and Peter Sin for many conversations on group cohomology. Steve Siegel read an early version of this paper and made useful suggestions. Also, this work was partially supported by the NSF.

Minimal Projections in L 1 -Algebras and Open Points in the Dual Spaces of Semi-Direct Product Groups

Minimal Projections in <i>L</i> ¹ -Algebras and Open Points in the Dual Spaces of Semi-Direct Product Groups

Induced cayley graphs and d0ouble covers

Motivated by a construction of highly expanding simple Cayley graphs of dihedral groups derived from-or induced by-highly expanding Cayley digraphs fo cyclic groups presented by F. R. K. chung, constructions of simple Cayley graphs on a semidirect product of groups and Cayley digraphs of one of its factors are suggested. In case of the non-normal factor being the cyclic group of order 2. a condition is given to derive spectral bounds of the Cayley graph of the product from those of the Cayley graph of the normal factor.

A Riemannian geometrical description for Lie-Poisson systems and its application to idealized magnetohydrodynamics

This paper presents a general theorem which enables description of Lie-Poisson systems for semi-direct product groups in terms of Riemannian geometry. The method employed is unique in that it changes the non-quadratic right- or left-invariant Hamiltonian to the quadratic form in a right- or left-invariant I-form on the corresponding group by removing the respective invariance, which obtains the Riemannian metric and its induced Riemannian (Levi-Civita) connection. The resultant geodesic equation proves to be equivalent to the equation of motion, while the corresponding Jacobi equation determines its instability. In addition, this method is applied to idealized magnetohydrodynamics having isentropic flow, with a simple example being provided that considers the motion of an isentropic gas with no magnetic field present.

Riemannian geometry of the motion of an ideal incompressible magnetohydrodynamical fluid

This paper presents a Riemannian geometrical formalism of the hydrodynamics of an ideal incompressible fluid, originally proposed by Arnold, and then extends it to include the magnetohydrodynamics (MHD) of such a fluid. For the motion of an ideal incompressible fluid, a right-invariant metric, the induced Riemannian (Levi-Civita) connection, and then the geodesics are firstly defined on the Lie group D ( M) of all diffeomorphisms of an N-manifold M, being the ambient space of the Lie group D ν( M) of volume-preserving diffeomorphisms of M. Next, the associated covariant derivatives on the tangent space T θ D ( M) to D ( M) at θ ϵ D ν( M) ⊂ D ( M) are orthogonally projected onto T θ D ν( M), and finally the geodesic equation on D ν( M) is obtained. In the same way, the equation of motion for an ideal incompressible MHD fluid is obtained, too. In this case, the appropriate configuration space is the semidirect product of D ν( M) with the space Λ 1( M) of 1-forms on M, denoted S ν( M) = D ν ( M) × Λ 1( M), and as its ambient space the semidirect product group S ( M) = D ( M) × Λ 1( M) is considered. In addition, as on a finite-dimensional manifold, classical tensor calculus is performed on the considered infinite-dimensional Lie groups when M is an N-/three-dimensional flat torus. The curvature tensors responsible for the instability of these systems are also obtained.

Loops which are semidirect products of groups

Loops which are semidirect products of groups

Ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups

Three-dimensional ideal magnetohydrodynamic (3d-iMHD) equations are shown to be a geodesic equation on an infinite-dimensional Lie group. The group is the semidirect product of a group of volume-preserving diffeomorphisms (particle motion) and a linear space of divergenceless vector fields (current density). The present theory includes the work of Zeitlin and Kambe (1993) for two-dimensional iMHD flow as a special case. Passive scalar motion in N-dimensional ideal hydrodynamic flow is also generally shown to be described by a geodesic equation.

Les groupes, les hypergroupes et l'énigme des Murngin

This is a contribution to the algebraic study of an Australian kinship system known as the ‘Murngin system’. It is shown how the introduction of hypergroups and semi-direct products of groups can shed new light on some aspects of this particular system and further enhance the abstract theory of kinship.

Is a Semidirect Product of Groups Necessarily a Group?

Is a Semidirect Product of Groups Necessarily a Group?

SKEW PRODUCT ACTIONS OF SEMI-DIRECT PRODUCT GROUPS

Introduction Let f be a homomorphism from a second countable locally compact group G into the continuous automorphisms of a second countable compact group K. Let g ∗ k = f(g)(k) and suppose further the mapping given by (g, k) → g ∗ k is continuous from G × K into K, g ∈ G, k ∈ K. Then the cartesian product K × G becomes a group with multiplication given by (k1, g1)(k2, g2) = (k1(g1 ∗ k2), g1g2), where (ki, gi) ∈ K × G, i = 1, 2. We denote this semi-direct product group by K ×s G. Let (S, μ), a standard Borel space, be an ergodic Borel K ×s G-space with a probability invariant measure μ. In this paper we state necessary and sufficient conditions so the quotient Borel K ×s G-mapping from S into the space of K-orbits in S has relative discrete spectrum. These conditions are in terms of the stabilizers and the natural action of G on the dual K of K. (We remind the reader K is the set of unitary equivalence classes of irreducible unitary representations of K). Choose a Borel subset S of S that meets each K-orbit in S exactly once. Let p : (S, μ) → (S, μ) be a mapping given by p(s) = s where s is the unique point in S that meets the K-orbit s · K of s and μ = μ ◦ p−1.

Cylindrical representations of some infinite dimensional nuclear Lie groups

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie groupGand for any strictly positive integerkthere exist non located and non trivially decomposable representations of orderkof the nuclear Lie group(M;G) of all theG-valued test-functions on a given paracompact manifoldM.

Compact open sets in duals and projections inL1-algebras of certain semi-direct product groups

The main purpose of this paper is to study projections, that is, self-adjoint idempotents, inL1-algebras of semi-direct productsG= ℝ ⋉ ℝd,d≥ 2. We establish necessary and sufficient conditions for the existence of non-zero projections in terms of the action of ℝ on ℝd. In the cases where such projections exist, we describe minimal ones in detail.

The diffeomorphism group and flat principal bundles

Representations of the semidirect product group D■Diff(M) in the context of fiber bundle theory are studied. Here, D is the group of compactly supported functions and Diff(M) is the group of compactly supported diffeomorphisms of a manifold M. The carrier space is taken as the space of equivariant functions on a flat principal bundle over M where M is multiply connected. Two principal bundles are taken as equivalent if they are related by a gauge transformation. For the U(1) case it is found that the representations are irreducible, and the equivalence classes of representations are in one to one correspondence with the equivalence classes of bundles. Simple examples are discussed.

A new quantum representation for canonical gravity and SU(2) Yang-Mills theory

Starting from Rovelli-Smolin's infinite-dimensional graded Poisson-bracket algebra of loop variables, we propose a new way of constructing a corresponding quantum representation. After eliminating certain quadratic constraints, we “integrate” an infinite-dimensional subalgebra of loop variables, using a formal group law expansion. With the help of techniques from the representation theory of semidirect-product groups, we find an exact quantum representation of the full classical Poisson-bracket algebra of loop variables, without any higher-order correction terms. This opens new ways of tackling the quantum dynamics for both canonical gravity and Yang-Mills theory.

Motions and pattern analysis: harmonic analysis on motion groups and their homogeneous spaces

Fourier-like transforms that are defined on several groups of motions on the plane and on the plane itself (viewed as the homogeneous space of these motion groups) are presented. These transforms should have many applications in the area of pattern recognition, detection, and the representation of motions in pattern analysis. The method is based upon group-representation theory and abstract harmonic analysis on semidirect product groups. However, most of the ideas are generalizations and interpretations of classical concepts and methods such as the Fourier descriptor method used for boundary recognition. Computational aspects are investigated, and examples are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Minimal and distal functions on semidirect products of groups

Minimal and distal functions on semidirect products of groups

Semidirect product groups with Abelian automorphism groups

AbstractMiller's group of order 64 is a smallest example of a nonabelian group with an abelian automorphism group, and is the first in an infinite family of such groups formed by taking the semidirect product of a cyclic group of order 2m (m ≥ 3) with a dihedral group of order 8. This paper gives a method for constructing further examples of non abelian 2-groups which have abelian automorphism groups. Such a 2-group is the semidirect product of a cyclic group and a special 2-group (satisfying certain conditions). The automorphism group of this semidirect product is shown to be isomorphic to the central automorphism group of the corresponding direct product. The conditions satisfied by the special 2-group are determined by establishing when this direct product has an abelian central automorphism group.

Minimal and distal functions on semidirect products of groups II.

Minimal and distal functions on semidirect products of groups II.

Cohomology of step functions under irrational rotations

This paper studies the solvability of the functional equationg(x+θ)=φ(x)g(x), given an irrationalθ and a step functionf mappingR/Z (with Lebesgue measure) to the unit circle. Results are applied to find parameterized families of representations of non-regular semi-direct product groups and to display irregularities in the uniform distribution of the sequenceZθ.