Semidirect Product Group Research Articles

Euler-Poincaré Formalism of Coupled KdV Type Systems and Diffeomorphism Group on S1

This paper describes a wide class of coupled KdV equations. The first set of equations directly follow from the geodesic flows on the Bott-Virasoro group with a complex field. But the set of 2-component systems of nonlinear evolution equations, which includes dispersive water waves, Ito's equation, many other known and unknown equations, follow from the geodesic flows of the right invariant L 2 metric on the semidirect product group , where Diff( S 1 ) is the group of orientation preserving diffeomorphisms on a circle. We compute the Lie-Poisson brackets of the Antonowicz-Fordy system, and the mode expansion of these beackets yield the twisted Heisenberg-Virasoro algebra. We also give an outline to study geodesic flows of a H 1 metric on .

The Group of Outer Automorphisms of the Semidirect Product of the Additive Group of a Ring by a Group of Units#

Let R be a ring with identity, and let U be a subgroup of the unit group of R. Denote by R + the underlying additive group of R. Consider the action of U on R + given by left multiplication, and the corresponding semidirect product group R + ⋊ U. We explicitly compute the automorphism group and the outer automorphism group of R + ⋊ U, under certain assumptions on R and U. The groups of outer automorphisms of the three euclidean triangle groups, which motivated our investigations, are obtained as particular cases. Many other examples to which our method applies are given.

On the Structure of Solutions to the Periodic Hunter--Saxton Equation

We prove the local existence of strong solutions of the periodic Hunter--Saxton equation, and we show that all strong solutions except space-independent solutions blow up in finite time.

Coherent state lattices and square integrability of representations

In this paper, we discretize the continuous theory of coherent states on a general semidirect product group G = V x S, where V is a vector space and S subset of GL(V) is a semisimple connected Lie group. We show that it is always possible to construct a discrete frame associated with a unitary irreducible representation of G, which is square integrable modulo an affine admissible section. We also prove the converse result that the existence of a discrete frame associated with a unitary representation U of G, indexed by a homogeneous space Gamma, implies the square integrability of U on Gamma.

Semidirect product constructions of directed strongly regular graphs

We construct a new infinite family of directed strongly regular graphs, as Cayley graphs of certain semidirect product groups. This generalizes an earlier construction of Klin, Munemasa, Muzychuk, and Zieschang on some dihedral groups.

Unified Theory of Fundamental Interactions

Based on local gauge invariance, four different kinds of fundamental interactions in nature are unified in a theory which has Gravitational Gauge Group gauge symmetry. In this approach, gravitational field, like electromagnetic field, intermediate gauge field, and gluon field, is represented by gauge potential. Four kinds of fundamental interactions are formulated in the similar manner, and therefore can be unified in a direct or semi-direct product group. The model discussed in this paper is a renormalizable quantum model and can be regarded as an extension of the standard model to gravitational interactions, so it can be used to study quantum effects of gravitational interactions.

Discretizing continuous wavelet transforms using integrated wavelets

We present integrated wavelets as a method for discretizing the continuous wavelet transform. Using the language of group theory, the results are presented for wavelet transforms over semidirect product groups. We obtain tight wavelet frames for these wavelet transforms. Further integrated wavelets yield tight families of convolution operators independent of the choice of discretization of scale and orientation parameters. Thus these families can be adapted to specific problems. The method is more flexible than the well-known dyadic wavelet transform. We state an exact algorithm for implementing this transform. As an application the enhancement of digital mammograms is presented.

Wigner functions for a class of semi-direct product groups

Following a general method proposed earlier, we construct here Wigner functions defined on coadjoint orbits of a class of semidirect product groups. The groups in question are such that their unitary duals consist purely of representations from the discrete series and each unitary irreducible representation is associated with a coadjoint orbit. The set of all coadjoint orbits (hence UIRs) is finite and their union is dense in the dual of the Lie algebra. The simple structure of the groups and the orbits enables us to compute the various quantities appearing in the definition of the Wigner function explicitly. Possible use of the Wigner functions so constructed, in image analysis and quantum optical measurements, is suggested.

Tensor operators and constructing indecomposable representations of semidirect product groups

Consider a semidirect product group G=H⋉V, where H is reductive and V is a vector group. Two irreps π1 and π2 of H can be “assembled” into a representation of G if it is possible to construct an indecomposable representation Π of G whose restriction to H is π1⊕π2. It is shown that this is equivalent to the existence of a tensor operator from π2 to π1 carrying a representation of H which is equivalent to a nontrivial quotient of the representation which defines the semidirect product. This provides a systematic method for deciding whether two irreps can be assembled, and, if so, in how many inequivalent ways. The method is applied in many of the standard examples that arise in physical questions.

Unification of Non-Abelian SU(N) Gauge Theory and Gravitational Gauge Theory

In this paper, a general theory on unification of non-Abelian SU(N) gauge interactions and gravitational interactions is discussed. SU(N) gauge interactions and gravitational interactions are formulated on the similar basis and are unified in a semi-direct product group GSU(N). Based on this model, we can discuss unification of fundamental interactions of Nature.

Unification of Electromagnetic Interactions and Gravitational Interactions

Unified theory of gravitational interactions and electromagnetic interactions is discussed in this paper. Based on gauge principle, electromagnetic interactions and gravitational interactions are formulated in the same manner and are unified in a semi-direct product group of U(1) Abelian gauge group and gravitational gauge group.

Continuous Wavelet Transforms from Semidirect Products: Cyclic Representations and Plancherel Measure

Continuous wavelet transforms arising from the quasiregular representation of a semidirect product group G = R k ⋊ H have been studied by various authors. Recently the attention has shifted from the irreducible case to include more general dilation groups H, for instance cyclic (more generally: discrete) or one-parameter groups. These groups do not give rise to irreducible square-integrable representations, yet it is possible (and quite simple) to give admissibility conditions for a large class of them. We put these results in a theoretical context by establishing a connection to the Plancherel theory of the semidirect products, and show how the admissibility conditions relate to abstract admissibility conditions which use Plancherel theory.

Hopf Algebra Extensions Arising from Semi-Direct Products of Groups

The aim of this paper is to study Hopf algebra extensions arising from semi-direct products of groups in terms of group cohomology. This enables us to compute and describe explicitly some groups of Hopf algebra extensions.

On the projective representations of the Bondi-Metzner-Sachs group

With the Bondi‐Metzner‐Sachs (BMS) group in general relativity as the main motivation and example, a theorem is proved which may be described as follows. Let G be a complex semisimple, connected and simply connected Lie group with compact real form K , and let A be a metrizable, complete and locally convex real topological vector space on which there is a continuous G action. Consider the semidirect product topological group Gx s A (which is, in general, infinite‐dimensional) constructed naturally out of G and A . If the set of equivalence classes of irreducible representations of K in A satisfies certain hypotheses, then the second cohomology group of G x s A in the sense of continuous group cohomology is trivial. When G = SL (2, C ) and A is an appropriate function space of real‐valued functions of the 2‐sphere endowed with a specific G action (e.g. A may consist of C k , k ⩾ 3, real‐valued functions defined on the 2‐sphere), the semidirect product group is the universal cover of the BMS group. The theorem implies the existence of lifting of the projective unitary representations of the BMS group to the linear unitary representations of its universal cover. In the quantum context when we consider massless quantum fields at null infinity of a non‐stationary, asymptotically Minkowskian space‐time, in place of the projective unitary representations of the BMS group, there is no loss of generality in considering the linear unitary representations of its universal cover instead.

Double Structures and Multiple Symmetry Groups for the Reduced 4-Dimensional String Background Equations

By using the extended double complex method proposed previously, the 1-loop string background equations with axion and dilaton fields in 4 dimensions with two commuting Killing symmetries and δc = 0 are reduced essentially to two double Ernst σ-models. Then the string Geroch group acting on the solution space is extended to a multi-fold version of a semidirect product of the string Geroch group and the Virasoro group. The usual string Geroch group is just one component of a subgroup in the multi-fold semidirect product group. It is also found that a dobule Z2 symmetry exists for every case of the 4-dimensional reduced string equations and for most of the cases here, this Z2 symmetry is new and cannot be obtained in the usual (non-double) scheme. These results show that the reduced string background equations considered possess more and richer symmetry structures than previously expected. Moreover, a double form of string soliton method is briefly described and, as an application, a multiple family of soliton solutions of the considered reduced string equations is given, which shows that the double method is more effective.

The use of group and algebraic properties of tensor expressions in an object-oriented CA system

An implementation of an object-oriented CA system for reducing tensor expression to the canonical form is described. The specific feature of this system is the use of operations on mathematical systems, such as a direct and semidirect product of groups, to increase the efficiency of computations. The system is implemented in Java.

AN ANALYSIS OF THE REPRESENTATIONS OF THE MAPPING CLASS GROUP OF A MULTIGEON THREE-MANIFOLD

It is known that the distinct unitary irreducible representations (UIR's) of the mapping class group G of a three-manifold ℳ give rise to distinct quantum sectors ("θ sectors") in quantum theories of gravity based on a product space–time of the form ℝ×ℳ. In this paper, we study the UIR's of G in an effort to understand the physical implications of these quantum sectors. The mapping class group of a three-manifold which is the connected sum of ℝ3 with a finite number of irreducible primes is a semidirect product group. Following Mackey's theory of induced representations, we provide an analysis of the structure of the general finite-dimensional UIR of such a group. In the picture of quantized primes as particles (topological geons), this general group-theoretic analysis enables one to draw several qualitative conclusions about the geons' behavior in different quantum sectors, without requiring an explicit knowledge of the UIR's corresponding to the individual primes. An important general result is that the classification of the UIR's of the so-called particle subgroup (equivalently, the UIR's of G in which the slide diffeomorphisms are represented trivially) is reduced to the problem of finding the UIR's of the internal diffeomorphism groups of the individual primes. Moreover, this reduction is entirely consistent with the geon picture, in which the UIR of the internal group of a prime determines the species of the corresponding quantum geon, and the remaining freedom in the overall UIR of G expresses the possibility of choosing an arbitrary statistics (Bose, Fermi or para) for the geons of each species. For UIR's which represent the slides nontrivially, we do not provide a complete classification, but we find some new types of efforts due to the slides, including quantum breaking of internal symmetry and of particle indistinguishability. In connection with the latter, a novel kind of statistics arises which is determined by representations of proper subgroups of the permutation group, rather than of the group as a whole. Finally, we observe that for a generic three-manifold there will be an infinity of inequivalent UIR's and hence an infinity of "consistent" theories, when topology change is neglected.

A unified approach to concrete Plancherel theory¶of homogeneous spaces

A new paradigm for concrete Plancherel analysis on homogeneous spaces is established wherein the distinction between finite and infinite multiplicity is de-emphasized. A unified treatment — incorporating the Penney-Fujiwara formulation for finite multiplicity and the Bonnet formulation for infinite multiplicity — is presented. The heretofore unrecognized similarities in the two theories are uncovered and emphasized. The unified theory is illustrated with explicit computations of the Plancherel formula for a certain class of homogeneous spaces of semidirect product groups.

The symmetry groups of non-rigid molecules: a Lie algebraic and group contraction approach

The description of the symmetry of non-rigid molecules is explored through a Lie algebraic approach. It is shown that the abstract process of contraction to an appropriate Lie algebra corresponds to restriction of molecular motion to regions located around the minima of a global potential. The contracted algebra may be determined by recasting the Hamiltonian in terms of variables which vanish at global potential minima, and the resultant direct or semidirect product group structure accords with that proposed by other workers. This approach is applied to two familiar problems of molecular symmetry, namely inversion through a potential barrier and hindered rotation; the results derived confirm and extend those obtained by other methods.

Algebraic solutions for point groups: The tetrahedral group for the group chain T⊃D2⊃C2

A factorization lemma is discovered for the irreducible symmetry operators (ISO’s) of a semidirect product group of two abelian groups. By means of it, simple algebraic expressions of the T⊃D2 single-valued irreducible matrices, ISO’s and symmetry adapted functions are derived. Using the double-induced technique, the algebraic solutions for the double-valued representations of group T in the group chain T⊃C2 are also obtained. The simplicity and elegance of the algebraic solution of the point group can be compared with that of the analytic solution of the rotation group.