Semidirect Product Group Research Articles

INDECOMPOSABLE REPRESENTATIONS OF THE EUCLIDEAN ALGEBRA 𝔢(3) FROM IRREDUCIBLE REPRESENTATIONS OF

AbstractThe Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra $\mathfrak {sl}(4,\mathbb {C})$ and show that the irreducible representations V (m,0,0) and V (0,0,m) of $\mathfrak {sl}(4,\mathbb {C})$ are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

Indecomposable representations of the Euclidean algebra (3) from irreducible representations of the symplectic algebra (4, ℂ)

The Euclidean group E(3) is the noncompact, semidirect product group E(3) ≊ SO(3) ℝ3. It is the Lie group of orientation-preserving isometries of 3-dimensional Euclidean space. The Euclidean algebra (3) is the complexification of the Lie algebra of E(3). We embed (3) into the 10-dimensional symplectic algebra (4, ℂ), the simple Lie algebra of type C2. We show that, up to conjugation by an element of Sp(4, ℂ), there is only one embedding of (3) into (4, ℂ), and then prove that the irreducible representations of (4, ℂ) remain indecomposable upon restriction to (3), thus creating a new class of indecomposable (3)-representations.

Semidirect product gauge group[SU(3)c×SU(2)L]⋊U(1)Yand quantization of hypercharge

In the Standard Model the hypercharges of quarks and leptons are not determined by the gauge group $SU(3)_{\rm c} \times SU(2)_{\rm L} \times U(1)_{\rm Y}$ alone. We show that, if we choose the semidirect product group $[SU(3)_{\rm c} \times SU(2)_{\rm L}] \rtimes U(1)_{\rm Y}$ as its gauge group, the hyperchages are settled to be $n/6 \mod {\mathbb{Z}}\;(n = 0,1,3,4) $. In addition, the conditions for gauge-anomaly cancellation give strong constraints. As a result, the ratios of the hypercharges are uniquely determined and the gravitational anomaly is automatically canceled. The standard charge assignment to quarks and leptons can be properly reproduced. For exotic matter fields their hypercharges are also discussed.

Embeddings of the Euclidean algebra $\mathfrak {e}(3)$e(3) into $\mathfrak {sl}(4,\mathbb {C})$sl(4,C) and restrictions of irreducible representations of $\mathfrak {sl}(4,\mathbb {C})$sl(4,C)

The Euclidean group E(3) is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. It is the noncompact, semidirect product group \documentclass[12pt]{minimal}\begin{document}$E(3) \cong SO(3) \ltimes \mathbb {R}^{3}$\end{document}E(3)≅SO(3)⋉R3. The Euclidean algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3) is the complexification of the Lie algebra of E(3). We classify the embeddings of the Euclidean algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3) into the simple Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C) and, as an application of this classification, discuss the restriction to various embeddings of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3) of certain irreducible representations of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(4,\mathbb {C})$\end{document}sl(4,C). In particular, we consider which of these restrictions are \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}(3)$\end{document}e(3)-indecomposable.

Semidirect product groups, vacuum alignment, and tribimaximal neutrino mixing

The neutrino oscillation data are in very good agreement with the tribimaximal mixing pattern: \sin^2\theta_{23}=1/2, \sin^2\theta_{12}=1/3, and \sin^2\theta_{13}=0. Attempts to generate this pattern based on finite family symmetry groups typically assume that the family symmetry is broken to different subgroups in the charged lepton and the neutrino mass matrices. This leads to a technical problem, where the cross-couplings between the Higgs fields responsible for the two symmetry breaking chains force their vacuum expectation values to align, upsetting the desired breaking pattern. Here, we present a class of models based on the semidirect product group (S_3)^4 \rtimes A_4, where the lepton families belong to representations which are not faithful. In effect, the Higgs sector knows about the full symmetry while the lepton sector knows only about the A_4 factor group. This can solve the alignment problem without altering the desired properties of the family symmetry. Inclusion of quarks into the framework is straightforward, and leads to small and arbitrary CKM mixing angles. Supersymmetry is not essential for our proposal, but the model presented is easily supersymmetrized, in which case the same family symmetry solves the SUSY flavor problem.

The Momentum Map Representation of Images

This paper discusses the mathematical framework for designing methods of large deformation matching (LDM) for image registration in computational anatomy. After reviewing the geometrical framework of LDM image registration methods, a theorem is proved showing that these methods may be designed by using the actions of diffeomorphisms on the image data structure to define their associated momentum representations as (cotangent lift) momentum maps. To illustrate its use, the momentum map theorem is shown to recover the known algorithms for matching landmarks, scalar images and vector fields. After briefly discussing the use of this approach for Diffusion Tensor (DT) images, we explain how to use momentum maps in the design of registration algorithms for more general data structures. For example, we extend our methods to determine the corresponding momentum map for registration using semidirect product groups, for the purpose of matching images at two different length scales. Finally, we discuss the use of momentum maps in the design of image registration algorithms when the image data is defined on manifolds instead of vector spaces.

Some nonunitary, indecomposable representations of the Euclidean algebra

The Euclidean group E(3) is the noncompact, semidirect product group . It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra is the complexification of the Lie algebra of E(3). We construct three distinct families of finite-dimensional, nonunitary representations of and show that each representation is indecomposable. The representations of the first family are explicitly realized as subspaces of the polynomial ring with the action of given by differential operators. The other families are constructed via duals and tensor products of the representations within the first family. We describe subrepresentations, quotients and duals of these indecomposable representations.

Structure, Classification, and Conformal Symmetry, of Elementary Particles over Non-Archimedean Space–Time

It is known that no length or time measurements are possible in sub-Planckian regions of spacetime. The Volovich hypothesis postulates that the micro-geometry of spacetime may therefore be assumed to be non-archimedean. In this letter, the consequences of this hypothesis for the structure, classification, and conformal symmetry of elementary particles, when spacetime is a flat space over a non-archimedean field such as the p-adic numbers, is explored. Both the Poincaré and Galilean groups are treated. The results are based on a new variant of the Mackey machine for projective unitary representations of semidirect product groups which are locally compact and second countable. Conformal spacetime is constructed over p-adic fields and the impossibility of conformal symmetry of massive and eventually massive particles is proved.

On the Square Integrability of Quasi Regular Representation on Semidirect Product Groups

Let H be a locally compact group and K be a locally compact abelian group. Also let G=H×τK denote the semidirect product group of H and K, respectively. Then the unitary representation (U,L2(K)) on G defined by \(U(h,x)f(y)=\delta(h)^{\frac{1}{2}}f(\tau_{h^{-1}}(yx^{-1}))\) is called the quasi regular representation. The properties of this representation in the case K=(ℝn,+), have been studied by many authors under some specific assumptions. In this paper we aim to consider a general case and extend some of these properties when K is an arbitrary locally compact abelian group. In particular we wish to show that the two conditions (i) \(\delta\Delta_{H}\not\equiv 1\) , and (ii) the stabilizers Hω are compact for a.e. \(\omega \in \widehat{K}\) ; both are necessary for square integrability of U. Furthermore, we shall consider some sufficient conditions for the square integrability of U. Also, for the square integrability of subrepresentations of U, we will introduce a concrete form of the Duflo-Moore operator.

GEODESIC FLOW ON EXTENDED BOTT–VIRASORO GROUP AND GENERALIZED TWO-COMPONENT PEAKON TYPE DUAL SYSTEMS

This paper discusses an algorithmic way of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two-component peakon type dual systems from their two-component soliton equations counter part. We study the essential aspects of Hamiltonian flows on coadjoint orbits of the centrally extended semidirect product group [Formula: see text] to give a systematic derivation of the dual counter parts of various two-component of integrable systems, viz., the dispersive water wave equation, the Kaup–Boussinesq system and the Broer–Kaup system, using moment of inertia operators method and the (frozen) Lie–Poisson structure. This paper essentially gives Lie algebraic explanation of Olver–Rosenau's paper [31].

ON CONSTRUCTION OF COHERENT STATES ASSOCIATED WITH SEMIDIRECT PRODUCTS

Most of the interesting groups encountered in the physics literature are semidirect product groups. These groups are of the general form G = H ×τ K, where H and K are locally compact groups. In this paper, we consider the quasi regular representation on such a group G and investigate when it is possible to construct a family of coherent states associated to this representation.

Semidirect products of lattices

The semidirect product of lattices is a lattice analogue of the semidirect product of groups. In this article we introduce this construction, show some basic facts and study a class of lattices closed under semidirect products. We also generalise this notion presenting the semidirect product of semilattices.

Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups

In this paper, we consider the hidden subgroup problem (HSP) over the class of semi-direct product groups $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_q$, for $p$ and $q$ prime. We first present a classification of these groups in five classes. Then, we describe a polynomial-time quantum algorithm solving the HSP over all the groups of one of these classes: the groups of the form $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_p$, where $p$ is an odd prime. Our algorithm works even in the most general case where the group is presented as a black-box group with not necessarily unique encoding. Finally, we extend this result and present an efficient algorithm solving the HSP over the groups $\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$.

CRITERION OF PROPER ACTIONS ON HOMOGENEOUS SPACES OF CARTAN MOTION GROUPS

The Cartan motion group associated to a Riemannian symmetric space X is a semidirect product group acting isometrically on its tangent space. For two subsets in a locally compact group G, Kobayashi introduced the concept of "properness" as a generalization of properly discontinuous actions of discrete subgroups on homogeneous spaces of G. In this paper, we give a criterion of properness for homogeneous spaces of Cartan motion groups. Our criterion has a similar feature to the case where G is a reductive Lie group.

ON THE AUTOMORPHISM GROUPS OF A FAMILY OF BINARY QUANTUM ERROR-CORRECTING CODES

Based on the classical binary simplex code [Formula: see text] and any fixed-point-free element f of [Formula: see text], Calderbank et al. constructed a binary quantum error-correcting code [Formula: see text]. They proved that [Formula: see text] has a normal subgroup H, which is a semidirect product group of the centralizer Z(f) of f in GLm(2) with [Formula: see text], and the index [Formula: see text] is the number of elements of Ff = {f, 1 - f, 1/f, 1 - 1/f, 1/(1 - f), f/(1 - f)} that are conjugate to f. In this paper, a theorem to describe the relationship between the quotient group [Formula: see text] and the set Ff is presented, and a way to find the elements of Ff that are conjugate to f is proposed. Then we prove that [Formula: see text] is isomorphic to S3 and H is a semidirect product group of [Formula: see text] with [Formula: see text] in the linear case. Finally, we generalize a result due to Calderbank et al.

Explicit [formula omitted] of some finite group rings

We compute K 2 of some finite group algebras of characteristic 2, giving explicit Steinberg or Dennis–Stein symbols as generators. The groups include finite abelian groups of 4-rank at most 1, some direct products involving semidirect product groups, and some small alternating groups.

Quantum algorithms for the hidden subgroup problem on some semi-direct product groups by reduction to Abelian cases

We consider the hidden subgroup problem on the semi-direct product of cyclic groups Z N ⋊ Z p , where p is a prime that does not divide p j − 1 for any of the prime factors p j of N, and show that the hidden subgroup problem can be reduced to other ones for which solutions are already known.

Wavelet Transforms for Semidirect Product Groups with Not Necessarily Commutative Normal Subgroups

Let G be the semidirect product group of a separable locally compact unimodular group N of type I with a closed subgroup H of Aut(N). The group N is not necessarily commutative. We consider irreducible subrepresentations of the unitary representation of G realized naturally on L2(N), and investigate the wavelet transforms associated to them. Furthermore, the irreducible subspaces are characterized by certain singular integrals on N analogous to the Cauchy-Szego integral.

V-admissibility, Poincaré group and Sturmian-based vector coherent states

We show that there exists a V-admissible-type subspace in the carrier space of the Wigner representation of the Poincar? group in 1+3 dimensions. This subspace is built up from an orthonormal set of Sturmian-type functions which verify naturally the assumption of rotational invariance under which relativistic coherent states frames were previously obtained. Then, we propose an extension of the concept of V-admissibility for the class of semi-direct product groups, which takes into account the relativity groups which were not covered by the existing approaches. In the light of this extension, the square-integrability modulo an appropriate subgroup of the Wigner representation of the group of Poincar? is revisited and some physical implications are discussed.

An invariant operator due to F Klein quantizes H Poincaré's dodecahedral 3-manifold

The eigenmodes of the Poincar\'e dodecahedral 3-manifold $M$ are constructed as eigenstates of a novel invariant operator. The topology of $M$ is characterized by the homotopy group $\pi_1(M)$, given by loop composition on $M$, and by the isomorphic group of deck transformations $deck(\tilde{M})$, acting on the universal cover $\tilde{M}$. ($\pi_1(M)$, $\tilde{M}$) are known to be the binary icosahedral group ${\cal H}_3$ and the sphere $S^3$ respectively. Taking $S^3$ as the group manifold $SU(2,C)$ it is shown that $deck(\tilde{M}) \sim {\cal H}^r_3$ acts on $SU(2,C)$ by right multiplication. A semidirect product group is constructed from ${\cal H}^r_3$ as normal subgroup and from a second group ${\cal H}^c_3$ which provides the icosahedral symmetries of $M$. Based on F. Klein's fundamental icosahedral ${\cal H}_3$-invariant, we construct a novel hermitian ${\cal H}_3$-invariant polynomial (generalized Casimir) operator ${\cal K}$. Its eigenstates with eigenvalues $\kappa$ quantize a complete orthogonal basis on Poincar\'{e}'s dodecahedral 3-manifold. The eigenstates of lowest degree $\lambda=12$ are 12 partners of Klein's invariant polynomial. The analysis has applications in cosmic topology \cite{LA},\cite{LE}. If the Poincar\'{e} 3-manifold $M$ is assumed to model the space part of a cosmos, the observed temperature fluctuations of the cosmic microwave background must admit an expansion in eigenstates of ${\cal K}$.