Semidirect Product Group Research Articles

The Three Faces of U(3)

U(n) is a semi-direct product group characterized by nontrivial homomorphisms mapping U(1) into the automorphism group of SU(n). For U(3), there are three nontrivial homomorphisms that induce three separate defining representations. In a toy model of U(3) Yang–Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. Employing a mod3 permutation induced by a large gauge transformation acting on the defining representation vector space, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same U(3) quantum numbers. One can interpret this as three species of matter fields in the defining representation.

On the structure and classification of Bernstein algebras

Linear algebra tools are used to give a new approach to the open problem of the classification of Bernstein algebras. We prove that any Bernstein algebra [Formula: see text] is isomorphic to a semidirect product [Formula: see text] associated to a commutative algebra [Formula: see text] such that [Formula: see text], for all [Formula: see text] and an idempotent endomorphism [Formula: see text] of [Formula: see text] satisfying two compatibility conditions. The set of types of [Formula: see text]-dimensional Bernstein algebras is parametrized by an explicitly constructed classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups [Formula: see text].

Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces

We consider algebras of polynomials and analytic functions that are invariant with respect to semidirect products of groups of bounded operators on Banach spaces with symmetric bases. In particular, we consider algebras of so-called block-symmetric and double-symmetric analytic functions on Banach spaces ℓp(Cn) and the homomorphisms of these algebras. In addition, we describe an algebraic basis in the algebra of double-symmetric polynomials and discuss a structure of the spectrum of the algebra of double-symmetric analytic functions on ℓp(Cn).

Poisson–Lie group structures on semidirect products

We look at the Poisson structure on the total space of the dual bundle to the Lie algebroid arising from a matched pair of Lie groups. This dual bundle, with the natural semidirect product group structure, becomes a Poisson–Lie group as suggested by a recent work of Stachura. Moreover, when we start from the matched pairs given by the Iwasawa decomposition of simple Lie groups, we find that the associated Lie bialgebra is coboundary.

Isomorphism classes of idempotent evolution algebras

We showed that isomorphism classes of idempotent evolution algebras are in bijection with the orbits of the semidirect product group of the symmetric group and the torus, considered the combinatoric problem of enumeration of isomorphism classes for these algebras over arbitrary finite fields, derived a general counting formula, and obtained explicit formulas for the numbers of isomorphism classes in dimensions 2, 3, and 4 over any finite field.

The Fubini's Theorem for the Even Index Lie Subgroup

In this paper we present a new proof to generalize the Fubini's Theorem for the even index subgroup of a compact Lie group, using a recent result on the semi-direct product of groups. We also present some applications of this result for the invariant theory.

Equivariant Observers for Second-Order Systems on Matrix Lie Groups

This paper develops an equivariant symmetry for second order kinematic systems on matrix Lie groups and uses this symmetry for observer design. The state of a second order kinematic system on a matrix Lie group is naturally posed on the tangent bundle of the group with the inputs lying in the tangent of the tangent bundle known as the double tangent bundle. We provide a simple parameterization of both the tangent bundle state space and the input space (the fiber space of the double tangent bundle) and then introduce a semi-direct product group and group actions onto both the state and input spaces. We show that with the proposed group actions the second order kinematics are equivariant. An equivariant lift of the kinematics onto the symmetry group is derived and used to design nonlinear observers on the lifted state space using nonlinear constructive design techniques. The observer design is specialised to kinematics on groups that themselves admit a semi-direct product structure and include applications in rigid-body motion amongst others. A simulation based on an ideal hovercraft model verifies the performance of the proposed observer architecture.

Banach convolution modules of group algebras on covariant functions of characters of normal subgroups

This paper investigates structure of Banach convolution modules induced by group algebras on covariant functions of characters of closed and normal subgroups. Suppose that G is a locally compact group with the group algebra L1(G). Let N be a closed and normal subgroup of G. Assume that T is the circle group and ξ:N→T is a character. Further, suppose that 1≤p<∞ and Lξp(G,N) is the Lp-space of all covariant functions of ξ on G. It is shown that Lξp(G,N) is a Banach L1(G)-module. We then study convolution module actions of group algebras on covariant functions of characters for the case of canonical normal subgroups in semi-direct product groups.

A criterion of proper action on some compact extensions of ℝn and applications

Let K be a compact subgroup of [Formula: see text] and [Formula: see text] the semi-direct product group. Let H be a closed subgroup of G and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. We establish a geometrical criterion of the proper action of [Formula: see text] on [Formula: see text], which requires an accurate description of the structure of closed connected subgroups of Euclidean motion groups. As a consequence, we establish a criterion for the existence of a compact Clifford–Klein form [Formula: see text] and study the Calabi–Markus phenomenon.

Struktur Grup pada Pocket Cube

The objective of this research is to examine the abstraction of a semi-direct product group that originates from a specific object, which is the pocket cube. The process involves constructing groups by identifying free groups that are produced from reduced rotation sequences with finite standard rotations within the pocket cube. Moreover, by taking the action group to the position and orientation of the cubino on the pocket cube, the set of member pairs in S_8 and Z_3^8 is derived. The investigation then focuses on various valid random patterns on the pocket cube to determine how they correspond to each element in the semi-direct product of S_8 and Z_3^8. Additionally, the study describes the randomness of the pocket cube pattern when two reduced rotation sequences are applied sequentially by analyzing the operational representation of the semi-direct product group members.Keywords: group, pocket cube, semi-direct product.

Singularity-Free Lie Group Integration and Geometrically Consistent Evaluation of Multibody System Models Described in Terms of Standard Absolute Coordinates

Abstract A classical approach to the multibody systems (MBS) modeling is to use absolute coordinates, i.e., a set of (possibly redundant) coordinates that describe the absolute position and orientation of the individual bodies with respect to an inertial frame (IFR). A well-known problem for the time integration of the equations of motion (EOM) is the lack of a singularity-free parameterization of spatial motions, which is usually tackled by using unit quaternions. Lie group integration methods were proposed as an alternative approach to the singularity-free time integration. At the same time, Lie group formulations of EOM naturally respect the geometry of spatial motions during integration. Lie group integration methods, operating directly on the configuration space Lie group, are incompatible with standard formulations of the EOM, and cannot be implemented in existing MBS simulation codes without a major restructuring. The contribution of this paper is twofold: (1) A framework for interfacing Lie group integrators to standard EOM formulations is presented. It allows describing MBS in terms of various absolute coordinates and at the same using Lie group integration schemes. (2) A method for consistently incorporating the geometry of rigid body motions into the evaluation of EOM in absolute coordinates integrated with standard vector space integration schemes. The direct product group SO(3)×ℝ3 and the semidirect product group SE(3) are used for representing rigid body motions. The key element is the local-global transitions (LGT) transition map, which facilitates the update of (global) absolute coordinates in terms of the (local) coordinates on the Lie group. This LGT map is specific to the absolute coordinates, the local coordinates on the Lie group, and the Lie group used to represent rigid body configurations.

Spontaneous CP violation and symplectic modular symmetry in Calabi-Yau compactifications

We explore the geometrical origin of CP and the spontaneous CP violation in Calabi-Yau compactifications. We find that the CP symmetry is identified with an outer automorphism of the symplectic modular group in the large complex structure regime of Calabi-Yau threefolds, thereby enlarging the symplectic modular group to their semidirect product group. The spontaneous CP violation is realized by the introduction of fluxes, whose effective action is invariant under CP as well as the discrete Z2 symmetry or Z4 R-symmetry. We explicitly demonstrate the spontaneous CP violation on a specific Calabi-Yau threefold.

Congruences on the partial automorphism monoid of a free group action

We study congruences on the partial automorphism monoid of a finite rank free group action. We determine a decomposition of a congruence on this monoid into a Rees congruence, a congruence on a Brandt semigroup and an idempotent separating congruence. The constituent parts are further described in terms of subgroups of direct and semidirect products of groups. We utilize this description to demonstrate how the number of congruences on the partial automorphism monoid depends on the group and the rank of the action.

Random walk on finite extensions of lattices

We obtain a precise formula for the probability that a random walker returns to the origin after n steps on some semi-direct product groups obtained by extending a Euclidean lattice by a finite group. Prior to this work, to the best of our knowledge, for the class of groups considered, only asymptotic estimates were available in the literature.

Coding Matrices for the Semi-Direct Product Groups

We shall determine the coding matrix of the semi-direct product group $ G = C_{n} \rtimes_{\phi} C_{m} $ ; $ \phi : C_{m} \longrightarrow Aut(C_{n}) $ of two cyclic groups in order to generalize the known result for the dihedral group $D_{2n}$, which is known to be a semi-direct of the two cyclic groups $C_{n} \ , \ C_{2}$.

On the Cortex of the Groups 𝕋n ⋉ ℍn

Let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group and let [Formula: see text] be the [Formula: see text]-dimensional torus acting on [Formula: see text] by automorphisms. We consider the semidirect product group [Formula: see text] The cortex, [Formula: see text] of [Formula: see text] is the set of all unitary irreducible representations [Formula: see text] in the unitary dual [Formula: see text] of [Formula: see text] that cannot be Hausdorff separated from the identity representation [Formula: see text] of [Formula: see text] In this paper, we describe explicitly the cortex ([Formula: see text]) of [Formula: see text] using the coadjoint orbits of the group.

Two kinds of constructions of directed strongly regular graphs

In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given. Based on the cyclotomic numbers of finite fields, we present two infinite families of directed strongly regular Cayley graphs from semi-direct products of groups.

Localization operators for generalized Weyl–Heisenberg group

In this paper, we consider the generalized Weyl–Heisenberg group $${\mathbb {H}}(G_\tau )$$ associated with the semi direct product group $$G_\tau =H \times _\tau K$$ in which H is a locally compact group, K is a locally compact abelian group and $$\tau :H \rightarrow \hbox {Aut}(K)$$ is a continuous homomorphism. We give a square integrable representation on $${\mathbb {H}}(G_\tau )$$ and then as a result, we obtain admissible wavelets on this group. Moreover, we define the localization operator on $${\mathbb {H}}(G_\tau )$$ and we show that it is a bounded and compact operator. Furthermore, we investigate the boundedness and compactness of localization operators with two admissible wavelets on $$L^p({\hat{K}})$$ , ( $${\hat{K}}$$ the dual group of K), for the generalized Weyl–Heisenberg group. Finally, some example are given.

The $$C^*$$-algebra of the semi-direct product $$K &lt; imes A$$

Let $$G=K < imes A$$ be the semi-direct product group of a compact group K acting on an abelian locally compact group A. We describe the $$C^*$$ -algebra $$C^*(G)$$ of G in terms of an algebra of operator fields defined over the spectrum of G, generalizing previous results obtained for some special classes of such groups.

Groups with some families of complemented subgroups

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that $$G=HK$$ and $$H \cap K=1$$ . We prove that, for a locally soluble group G, all cyclic subgroups are complemented if and only if it is the semidirect product of groups $$A= {{\,\mathrm{Dr}\,}}_{i \in I} A_i$$ by $$B={{\,\mathrm{Dr}\,}}_{j \in J} B_j$$ , where all factors $$A_i$$ and $$B_j$$ are finite of prime order, and A has a set of maximal subgroups normal in G with trivial intersection. An analysis of the structure of periodic locally soluble groups of infinite rank shows, in particular, that if G is a periodic locally soluble group whose infinite rank subgroups are complemented, then every subgroup of G is complemented.