Semidirect Product Research Articles

Some properties of relative Rota–Baxter operators on groups

We find connection between relative Rota–Baxter operators and usual Rota–Baxter operators. We prove that any relative Rota–Baxter operator on a group H with respect to ( G , Ψ ) defines a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G . On the other side, we give condition under which a Rota–Baxter operator on the semi-direct product H ⋊ Ψ G defines a relative Rota–Baxter operator on H with respect to ( G , Ψ ) . We introduce homomorphic post-groups and find their connection with λ-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family of post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group.

The common solution space of general relativity

We review the solution space for the field equations of Einstein's General Relativity for various static, spherically symmetric spacetimes. We consider the vacuum case, represented by the Schwarzschild black hole; the de Sitter-Schwarzschild geometry, which includes a cosmological constant; the Reissner-Nordström geometry, which accounts for the presence of charge. Additionally we consider the homogenenous and anisotropic locally rotational Bianchi II spacetime in the vacuum. Our analysis reveals that the field equations for these scenarios share a common three-dimensional group of point transformations, with the generators being the elements of the D⊗sT2 Lie algebra, known as the semidirect product of dilations and translations in the plane. Due to this algebraic property the field equations for the aforementioned gravitational models can be expressed in the equivalent form of the null geodesic equations for conformally flat geometries. Consequently, the solution space for the field equations is common, and it is the solution space for the free particle in a flat space. This approach open new directions on the construction of analytic solutions in gravitational physics and cosmology.

A unified-description of curvature, torsion, and non-metricity of the metric-affine geometry with the Möbius representation

We establish the mathematical fundamentals for a unified description of curvature, torsion, and non-metricity 2-forms in the way extending the so-called Möbius representation of the affine group, which is the method to convert the semi-direct product into the ordinary matrix product, to revive the fertility of gauge theories of gravity. First of all, we illustrate the basic concepts for constructing the metric-affine geometry. Then the curvature and torsion 2-forms are described in a unified manner by using the Cartan connection of the Möbius representation of the affine group. In this unified-description, the curvature and torsion are derived by Cartan’s structure equation with respect to a common connection 1-form. After that, extending the Möbius representation, the dilation and shear 2-forms, or equivalently, the non-metricity 2-form, are introduced in the same unified manner. Based on the unified-description established in this paper, introducing a new group parametrization and applying the Inönü–Wigner group contraction to the full theory, the relationships among symmetries, geometric quantities, and geometries are investigated with respect to the three gauge groups: the metric-affine group and its extension, and an extension of the (anti)-de Sitter group in which the non-metricity exists. Finally, possible applications to theories of gravity are briefly discussed.

Extending structures for dendriform algebras

In this paper, we devote to extending structures for dendriform algebras. First, we define extending datums and unified products of dendriform algebras, and theoretically solve the extending structure problem. As an application, we consider flag datums as a special case of extending structures, and give an example of the extending structure problem. Second, we introduce matched pairs and bicrossed products of dendriform algebras and theoretically solve the factorization problem for dendriform algebras. Moreover, we also introduce cocycle semidirect products and nonabelian semidirect products as special cases of unified products. Finally, we define the deformation map on a dendriform extending structure (more general case), not necessary a matched pair, which is more practical in the classifying complements problem.

Nielsen fixed point theory for split n-valued maps on the Klein bottle

In this work we begin the study of n-valued maps on the Klein bottle, denoted by K, where we focus on the split ones. We provide an explicit description of Pn(K) (the n-th pure braid group of K) as an iterated semi-direct product of the form Fn⋊θn−1(⋯⋊(F2⋊θ1π1(K))), where the Fi′s are free groups on i letters. Given a split n-valued map Φ={f1,…,fn} its pointed homotopy class is determined by a pair of braids in Pn(K). We also provide a formula for N(Φ), the Nielsen number of Φ, which is completely determined by two braids, which in turn also determine the homotopy classes of the functions fi′s. If Φ={f1,f2} is a 2-valued map with N(Φ)=0, we show that there exists at least one 2-valued map Φ′={f1′,f2′}, such that Φ′ is fixed point free and for i=1,2 it holds that [fi]=[fi′], where [] denotes a pointed homotopy class of maps. Finally, we display an infinite family of pointed homotopy classes in [K,F2(K)], such that N(ϕ)=0, for any ϕ in the family. Furthermore, the map from [K,F2(K)] to [K,K×K], induced by the inclusion F2(K)→K×K, takes this family to one single element in [K,K×K]. We do not know if these 2-valued maps of the family can be deformed to fixed point free maps.

The The Cayley Graph of Semi-Direct Product of finite Groups: Interrelationships and Construction

In this paper, we study Cayley graph of the semi-direct product of two finite groups where is an odd prime numbe. Specifically, we endeavor to establish a comprehensive understanding of the Cayley graph by investigating the interrelationships among the constituent elements of the group. Leveraging this acquired knowledge, we systematically construct the Cayley graph, there by contributing to the scholarly discourse on this subject matter.

Fuchs' problem for endomorphisms of nonabelian groups

In 1960, László Fuchs posed the problem of determining which groups G are realizable as the group of units in some ring R. In [4], we investigated the following variant of Fuchs' problem, for abelian groups: which groups G are realized by a ring R where every group endomorphism of G is induced by a ring endomorphism of R? Such groups are called fully realizable. In this paper, we answer the aforementioned question for several families of nonabelian groups: symmetric, dihedral, quaternion, alternating, and simple groups; almost cyclic p-groups; and groups whose Sylow 2-subgroup is either cyclic or normal and abelian. We construct three infinite families of fully realizable nonabelian groups using iterated semidirect products.

δ-(bi)derivations and transposed Poisson algebra structures on the n-th Schrödinger algebra

The [Formula: see text]-th Schrödinger algebra [Formula: see text] is the semidirect product of the Lie algebra [Formula: see text] with the [Formula: see text]-th Heisenberg Lie algebra [Formula: see text]. In this paper, we will show that [Formula: see text] has neither nontrivial [Formula: see text]-derivations nor nontrivial transposed Poisson algebra structures, and doesn’t have nonzero [Formula: see text]-biderivations. In addition, all [Formula: see text]-derivations and [Formula: see text]-biderivations on [Formula: see text] are all described.

Horizon phase spaces in general relativity

We derive a prescription for the phase space of general relativity on two intersecting null surfaces using the null initial value formulation. The phase space allows generic smooth initial data, and the corresponding boundary symmetry group is the semidirect product of the group of arbitrary diffeomorphisms of each null boundary which coincide at the corner, with a group of reparameterizations of the null generators. The phase space can be consistently extended by acting with half-sided boosts that generate Weyl shocks along the initial data surfaces. The extended phase space includes the relative boost angle between the null surfaces as part of the initial data.We then apply the Wald-Zoupas framework to compute gravitational charges and fluxes associated with the boundary symmetries. The non-uniqueness in the charges can be reduced to two free parameters by imposing covariance and invariance under rescalings of the null normals. We show that the Wald-Zoupas stationarity criterion cannot be used to eliminate the non-uniqueness. The different choices of parameters correspond to different choices of polarization on the phase space. We also derive the symmetry groups and charges for two subspaces of the phase space, the first obtained by fixing the direction of the normal vectors, and the second by fixing the direction and normalization of the normal vectors. The second symmetry group consists of Carrollian diffeomorphisms on the two boundaries.Finally we specialize to future event horizons by imposing the condition that the area element be non-decreasing and become constant at late times. For perturbations about stationary backgrounds we determine the independent dynamical degrees of freedom by solving the constraint equations along the horizons. We mod out by the degeneracy directions of the presymplectic form, and apply a similar procedure for weak non-degeneracies, to obtain the horizon edge modes and the Poisson structure. We show that the area operator of the black hole generates a shift in the relative boost angle under the Poisson bracket.

Representations of toroidal and full toroidal Lie algebras over polynomial algebras

Toroidal Lie algebras are n variable generalizations of affine Kac-Moody Lie algebras. Full toroidal Lie algebra is the semidirect product of derived Lie algebra of toroidal Lie algebra and Witt algebra, also it can be thought of n-variable generalization of Affine-Virasoro algebras. Let h̃ be a Cartan subalgebra of a toroidal Lie algebra as well as full toroidal Lie algebra without containing the zero-degree central elements. In this paper, we classify the module structure on U(h̃) for all toroidal Lie algebras as well as full toroidal Lie algebras which are free U(h̃)-modules of rank 1. These modules exist only for type Al(l ≥ 1), Cl(l ≥ 2) toroidal Lie algebras and the same is true for full toroidal Lie algebras. Also, we determined the irreducibility condition for these classes of modules for both the Lie algebras.

Weak [formula omitted]-structures for some combinatorial group constructions

Bestvina [1] introduced the notion of a (weak) Z-structure and (weak) Z-boundary for a torsion-free group, motivated by the notion of boundary for hyperbolic and CAT(0) groups. Since then, some classes of groups have been shown to admit a (weak) Z-structure (see [5,20,22] for example); in fact, in all cases these groups are semistable at infinity and happen to have a pro-(finitely generated free) fundamental pro-group. The question whether or not every type F group admits such a structure remains open. In [33] it was shown that the property of admitting such a structure is closed under direct products and free products. Our main results are as follows.THEOREM: Let G be a torsion-free and semistable at infinity finitely presented group with a pro-(finitely generated free) fundamental pro-group at each end. If G has a finite graph of groups decomposition in which all the groups involved are of type F and inward tame (in particular, if they all admit a weak Z-structure) then G admits a weak Z-structure.COROLLARY: The class of those 1-ended and semistable at infinity torsion-free finitely presented groups which admit a weak Z-structure and have a pro-(finitely generated free) fundamental pro-group is closed under amalgamated products (resp. HNN-extensions) over finitely generated free groups.On the other hand, given a finitely presented group G and a monomorphism φ:G⟶G, we may consider the ascending HNN-extension G⁎φ=〈G,t;t−1gt=φ(g),g∈G〉. The results in [26] together with the Theorem above yield the following:PROPOSITION: If a finitely presented torsion-free group G is of type F and inward tame, then any (1-ended) ascending HNN-extension G⁎φ admits a weak Z-structure.In the particular case φ∈Aut(G), this ascending HNN-extension corresponds to a semidirect product G⋊φZ, and it has been shown in [18] that if G admits a Z-structure then so does G⋊φZ.

Metarouting with automatic tunneling in multilayer networks

Metarouting allows for the modeling of routing protocols using an algebraic structure called routing algebra. Routing protocols requiring design or validation can easily be modeled using this approach. To date, however, existing research on routing algebras has mainly focused on applying this approach to routing protocols that are generally used in networks which have a single addressing and forwarding protocol. The basic algebraic structures used in such contexts are semirings, Sobrinho’s algebras and algebras of endomorphisms. In this paper, we propose the modification of these existing routing algebras to deal with networks that contain multiple forwarding protocols where tunnels are omnipresent. To achieve this, we define new algebraic structures derived from the three aforementioned ones, in order to model the generalized routing problem with automatic tunneling entitled valid paths algebra. All of our routing algebras are defined as semi-direct products of two structures, the well-known shortest paths algebra and the proposed valid paths algebra. These new algebras are isotonic and non-monotonic with a partial order. We propose a fixed point for those new algebras and we prove the iterative convergence to the optimal solution of the valid shortest paths problem.

On normalizers of maximal tori in classical Lie groups

In general, the normalizer N G ( H G ) N_G(H_G) of a maximal torus H G H_G in a semisimple complex Lie group G G does not admit a presentation as a semidirect product of H G H_G and the corresponding Weyl group W G W_G . Meanwhile, such splitting exists for classical groups corresponding to the root systems A ℓ A_\ell , B ℓ B_\ell , D ℓ D_\ell . For the remaining classical groups corresponding to the root systems C ℓ C_\ell , there still exists an embedding of the Tits extension W G T W^T_G of W G W_G into the normalizer N G ( H G ) N_G(H_G) . Here an explicit unified construction is proposed for the lifts of the Weyl groups into normalizers of maximal tori for general linear and orthogonal Lie groups via embedding into the general linear groups. For the symplectic group S p 2 ℓ \mathrm {Sp}_{2\ell } , an explanation of the impossibility of embedding W S p 2 ℓ W_{\mathrm {Sp}_{2\ell }} into the normalizer N S p 2 ℓ ( H S p 2 ℓ ) N_{\mathrm {Sp}_{2\ell }}(H_{\mathrm {Sp}_{2\ell }}) is suggested. Also, explicit formulas are computed for the adjoint action of the lifts of the Weyl groups on g = L i e ( G ) \mathfrak {g}=\mathrm {Lie}(G) . Finally, examples of Weyl groups lifts are given for the groups closely associated with classical Lie groups.

Structure and Representations for Electrical Lie Algebra of Type D5

In this paper, we prove that the electrical Lie algebra [Formula: see text] is isomorphic to the semidirect product of [Formula: see text] and a 2-step nilpotent Lie algebra. Furthermore, we classify the irreducible highest weight modules for [Formula: see text].

Efficient quantum algorithms for some instances of the semidirect discrete logarithm problem

The semidirect discrete logarithm problem (SDLP) is the following analogue of the standard discrete logarithm problem in the semidirect product semigroup G⋊End(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\\rtimes {{\\,\ extrm{End}\\,}}(G)$$\\end{document} for a finite semigroup G. Given g∈G,σ∈End(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\in G, \\sigma \\in {{\\,\ extrm{End}\\,}}(G)$$\\end{document}, and h=∏i=0t-1σi(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h=\\prod _{i=0}^{t-1}\\sigma ^i(g)$$\\end{document} for some integer t, the SDLP(G,σ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G,\\sigma )$$\\end{document}, for g and h, asks to determine t. As Shor’s algorithm crucially depends on commutativity, it is believed not to be applicable to the SDLP. For generic semigroups, the best known algorithm for the SDLP is based on Kuperberg’s subexponential time quantum algorithm. Still, the problem plays a central role in the security of certain proposed cryptosystems in the family of semidirect product key exchange. This includes a recently proposed signature protocol called SPDH-Sign. In this paper, we show that the SDLP is even easier in some important special cases. Specifically, for a finite group G, we describe quantum algorithms for the SDLP in G⋊Aut(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\\rtimes {\ extrm{Aut}}(G)$$\\end{document} for the following two classes of instances: the first one is when G is solvable and the second is when G is a matrix group and a power of σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} with a polynomially small exponent is an inner automorphism of G. We further extend the results to groups composed of factors from these classes. A consequence is that SPDH-Sign and similar cryptosystems whose security assumption is based on the presumed hardness of the SDLP in the cases described above are insecure against quantum attacks. The quantum ingredients we rely on are not new: these are Shor’s factoring and discrete logarithm algorithms and well-known generalizations.

Krylov complexity for Jacobi coherent states

We develop computational tools necessary to extend the application of Krylov complexity beyond the simple Hamiltonian systems considered thus far in the literature. As a first step toward this broader goal, we show how the Lanczos algorithm that iteratively generates the Krylov basis can be augmented to treat coherent states associated with the Jacobi group, the semi-direct product of the 3-dimensional real Heisenberg-Weyl group H1, and the symplectic group, Sp(2, ℝ) ≃ SU(1, 1). Such coherent states are physically realized as squeezed states in, for example, quantum optics [1]. With the Krylov basis for both the SU(1, 1) and Heisenberg-Weyl groups being well understood, their semi-direct product is also partially analytically tractable. We exploit this to benchmark a scheme to numerically compute the Lanczos coefficients which, in principle, generalizes to the more general Jacobi group Hn ⋊ Sp(2n, ℝ).

Semidirect products of digroups and skew braces

Semidirect products of digroups and skew braces

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.

Classification of flat Lorentzian nilpotent Lie algebras

AbstractWe give a complete classification of flat Lorentzian nilpotent Lie algebras, this is to say of pseudo‐Euclidean Lie algebras associated to nilpotent Lie groups endowed with a left‐invariant Lorentzian metric of vanishing curvature. We prove that every such a Lie algebra is a direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand and show that, if denotes the ‐dimensional Heisenberg Lie algebra, then the only non‐abelian Lie algebras admitting flat Lorentzian metrics which are indecomposable are and the semidirect products and , defined by some particular derivations . In all those cases we also find the equivalence classes of flat Lorentzian products.

Biderivations of the mirror Heisenberg–Virasoro algebra

The mirror Heisenberg–Virasoro algebra [Formula: see text] is the semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra. In this paper, all biderivations of the mirror Heisenberg–Virasoro algebra are determined by using the gradation of biderivations. As an application, we show that any linear commuting map [Formula: see text] on the mirror Heisenberg–Virasoro algebra is the form [Formula: see text], where [Formula: see text] is an element from the basic field and [Formula: see text] is a linear map having its range in the center of [Formula: see text].