Maximal Symmetry Group Research Articles

Classification of Lie point symmetries for quadratic Liénard type equation $\ddot{x}+f(x)\dot{x}^2+g(x)=0$ẍ+f(x)ẋ2+g(x)=0

In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Li\documentclass[12pt]{minimal}\begin{document}$\acute{e}$\end{document}énard type equation, \documentclass[12pt]{minimal}\begin{document}$\ddot{x} + f(x){\dot{x}}^{2} + g(x)\break = 0$\end{document}ẍ+f(x)ẋ2+g(x)=0, where f(x) and g(x) are arbitrary functions of x. The symmetry analysis gets divided into two cases, (i) the maximal (eight parameter) symmetry group and (ii) non-maximal (three, two, and one parameter) symmetry groups. We identify the most general form of the quadratic Li\documentclass[12pt]{minimal}\begin{document}$\acute{e}$\end{document}énard equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable as well as isochronic. We present specific examples of physical interest. For the non-maximal cases, the identified equations are all integrable and include several physically interesting examples such as the Mathews-Lakshmanan oscillator, particle on a rotating parabolic well, etc. We also analyse the underlying equivalence transformations.

QED2+1 in Graphene: Symmetries of Dirac Equation in 2+1 Dimensions

It is well-known that the tight-binding Hamiltonian of graphene describes the low-energy excitations that appear to be massless chiral Dirac fermions. Thus, in the continuum limit one can analyze the crystal properties using the formalism of quantum electrodynamics in 2+1 dimensions (QED2+1) which provides the opportunity to verify the high energy physics phenomena in the condensed matter system. We study the symmetry properties of 2+1-dimensional Dirac equation, both in the non-interacting case and in the case with constant uniform magnetic field included in the model. The maximal symmetry group of the massless Dirac equation is considered by putting it in the Jordan block form and determining the algebra of operators leaving invariant the subspace of solutions. It is shown that the resulting symmetry operators expressed in terms of Dirac matrices cannot be described exclusively in terms of gamma matrices (and their products) entering the corresponding Dirac equation. It is a consequence of the reducibility of the considered representation in contrast to the 3+1-dimensional case. Symmetry algebra is demonstrated to be a direct sum of two gl(2,C) algebras plus an eight-dimensional abelian ideal. Since the matrix structure which determines the rotational symmetry has all required properties of the spin algebra, the pseudospin related to the sublattices (M. Mecklenburg and B. C. Regan, Phys. Rev. Lett. 106, 116803 (2011)) gains the character of the real angular momentum, although the degrees of freedom connected with the electron's spin are not included in the model. This seems to be graphene's analogue of the phenomenon called "spin from isospin" in high energy physics.

On maximal Lie point symmetry groups admitted by scalar stochastic differential equations

It is proved that the Lie point symmetry group admitted by a scalar stochastic differential equation (SDE) of order n ⩾ 3 is at most (n + 2) dimensional. This result supplements those for first- and second-order SDEs. Maximal dimensionality can be used to carry out Lie group classifications. Lie point group classification of the third-order SDEs is provided.

FJRW-Rings and Mirror Symmetry

We verify the Landau-Ginzburg Mirror Symmetry Conjecture for Arnol'd's list of unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group, and include a discussion of eight axioms which facilitate the computation of FJRW-rings.

Duality approach to one-dimensional degenerate electronic systems

We investigate the possible classification of zero-temperature spin-gapped phases of multicomponent electronic systems in one spatial dimension. At the heart of our analysis is the existence of non-perturbative duality symmetries which emerge within a low-energy description. These dualities fall into a finite number of classes that can be listed and depend only on the algebraic properties of the symmetries of the system: its physical symmetry group and the maximal continuous symmetry group of the interaction. We further characterize possible competing orders associated to the dualities and discuss the nature of the quantum phase transitions between them. Finally, as an illustration, the duality approach is applied to the description of the phases of two-leg electronic ladders at incommensurate filling.

UNIQUENESS OF THE McVITTIE SOLUTION AS PLANE SYMMETRIC SOURCELESS ELECTROMAGNETIC FIELD SPACETIMES

It is proved that plane symmetric Lorentzian manifolds representing a sourceless electromagnetic field admit only a four-dimensional maximal symmetry group and consist of only the McVittie5 spacetime and its non-static analogue.4

PROJECTIVE LINEAR GROUPS AS MAXIMAL SYMMETRY GROUPS

AbstractA maximal symmetry group is a group of isomorphisms of a three-dimensional hyperbolic manifold of maximal order in relation to the volume of the manifold. In this paper we determine all maximal symmetry groups of the typesPSL(2,q) andPGL(2,q). Depending on the primepthere are one or two such groups withq=pkandkalways equals 1, 2 or 4.

Projective geometry of systems of second-order differential equations

It is proved that every projective connection on an -dimensional manifold is locally defined by a system of second-order ordinary differential equations resolved with respect to the second derivatives and with right-hand sides cubic in the first derivatives, and that every differential system defines a projective connection on . The notion of equivalent differential systems is introduced and necessary and sufficient conditions are found for a system to be reducible by a change of variables to a system whose integral curves are straight lines. It is proved that the symmetry group of a differential system is a group of projective transformations in -dimensional space with the associated projective connection and has dimension . Necessary and sufficient conditions are found for a system to admit the maximal symmetry group; basis vector fields and structure equations of the maximal symmetry Lie algebra are produced. As an application a classification is given of the systems of two second-order differential equations admitting three-dimensional soluble symmetry groups.

Chern–Simons vs. Yang–Mills gaugings in three dimensions

Recently, gauged supergravities in three dimensions with Yang–Mills and Chern–Simons type interactions have been constructed. In this article, we demonstrate that any gauging of Yang–Mills type with semisimple gauge group G 0, possibly including extra couplings to massive Chern–Simons vectors, is equivalent on-shell to a pure Chern–Simons type gauging with non-semisimple gauge group G 0⋉ T⊂ G , where T is a certain translation group, and where G is the maximal global symmetry group of the ungauged theory. We discuss several examples.

Symmetriegruppen und fundamentale Pflasterungen der Fläche vom Geschlecht -3. I. Maximale Gruppen mit Sechseckbereichen

For the non-orientable compact surface of genus 3 all the possible (discontinuous) groups of its symmetries will be given. In this first part of the paper the 8 hexagonal fundamental domains have been discussed for determining the maximal symmetry groups in a combinatorial way. The computer classification of the 65 unfoldings of the -3-surface in the former work [15] by the author will be the basic tool.

Comment: Towards a Complete Classification of Spherically Symmetric Lorentzian Manifolds According to Their Ricci Collineations

General expressions for the components of the Ricci collineation vector are derived and the related constraints are obtained. These constraints are then solved to obtain Ricci collineations and the related constraints on the Ricci tensor components for all spacetime manifolds (degenerate or non-degenerate, diagonal or non-diagonal) admitting symmetries larger than so(3) and already known results are recovered. A complete solution is achieved for the spacetime manifolds admitting so(3) as the maximal symmetry group with non-degenerate and non diagonal Ricci tensor components. It is interesting to point out that there appear cases with finite number of Ricci collineations although the Ricci tensor is degenerate and also the cases with infinitely many Ricci collineations even in the case of non-degenerate Ricci tensor. Interestingly, it is found that the spacetime manifolds with so(3) as maximal symmetry group may admit two extra proper Ricci collineations, although they do not admit a G5 as the maximal symmetry group. Examples are provided which show and clarify some comments made by Camci et al. [Camci, U., and Branes, A. (2002). Class. Quantum Grav.19, 393–404]. Theorems are proved which correct the earlier claims made in [Carot, J., Nunez, L. A., and Percoco, U. (1997). Gen. Relativ. Gravit.29, 1223–1237; Contreras, G., Nunez, L. A., and Percolo, U. (2000). Gen. Relativ. Gravit.32, 285–294].

Invariance properties of the Dirac equation with external electro-magnetic field

In this paper, we attempt to obtain the nature of the external field such that the Dirac equation with external electro-magnetic field is invariant. The Poincare group, which is the maximal symmetry group for field free case, is constrained by the presence of the external field. Introducing infinitesimal transformation ofx and ψ, we apply Lie’s extended group method to obtain the class of external field which admit of the invariance of the equation. It is important to note that the constraints for the existence of invariance are explicity on the electric and magnetic field, though only potentials explicity appears in the equation.

Lorentz and CPT violations from Chern-Simons modifications of QED

The possibility of a small modification of spinor Quantum Electro-Dynamics is reconsidered, in which Lorentz and CPT non-covariant kinetic terms for photons and fermions are present. The corresponding free field theory is carefully discussed. The finite one-loop parity-odd induced effective action is unambiguously calculated using the physical cutoff method, which manifestly encodes the maximal residual symmetry group allowed by the presence of the Lorentz and CPT breaking axial-vector. This very same induced effective action, which is different from those ones so far quoted in the Literature, is also re-derived by means of the dimensional regularization, provided the maximal residual symmetry is maintained in the enlarged $D$-dimensional space-time. As a consequence, it turns out that the requirement of keeping the maximal residual symmetry at the quantum level just corresponds to the physical renormalization prescription which naturally fixes the one-loop parity-odd induced effective action.

Solving Third Order Differential Equations with Maximal Symmetry Group

The largest group of Lie symmetries that a third-order ordinary differential equation (ode) may allow has seven parameters. Equations sharing this property belong to a single equivalence class with a canonical representative v′′′(u)=0. Due to this simple canonical form, any equation belonging to this equivalence class may be identified in terms of certain constraints for its coefficients. Furthermore a set of equations for the transformation functions to canonical form may be set up for which large classes of solutions may be determined algorithmically. Based on these steps a solution algorithm is described for any equation with this symmetry type which resembles a similar scheme for second order equations with projective symmetry group.

Set Ideals with Maximal Symmetry Group and Minimal Dynamical Systems

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <∣X∣, the stabilizer S{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory. 1991 Mathematics Subject Classification 20B35, 20B27.

Solving Second Order Ordinary Differential Equations with Maximal Symmetry Group

Second order ordinary differential equations of the form $y''+A{y'}^3+B{y'}^2+Cy'+D$ with $y\equiv y(x)$ and A, B, C and D functions of x and y are of special interest because they may allow the largest possible group of point symmetries if its coefficients satisfy certain constraints. For large classes of these equations a solution algorithm is described that determines its general solution in closed form by reducing it to a linear third-order equation. If the results obtained by Sophus Lie in the last century are supplemented by more recent concepts like Janet bases and Loewy decompositions, a systematic solution procedure is obtained that is easily implemented in a computer algebra system.

On 3rd order ordinary differential equations with maximal symmetry group

The following theorem is proved by investigating the Janet bases of determining systems: In order that a 3rd order quasilinear ordinary differential equation has a seven-parameter symmetry group it must have a certain structure, and a set of necessary and sufficient conditions for its coefficients must be satisfied. This theorem generalizes similar results for linear equations and for quasilinear equations of 2nd order. It is shown how this theorem facilitates the computation of closed form solutions.

Gauging kinematical and internal symmetry groups for extended systems: the Galilean one-time and two-times harmonic oscillators

The possible external couplings of an extended non-relativistic classical system are characterized by gauging its maximal dynamical symmetry group at the centre-of-mass. The Galilean one-time and two-times harmonic oscillators are exploited as models. The following remarkable results are then obtained: 1), a peculiar form of interaction of the system as a whole with the external gauge fields; 2), a modification of the dynamical part of the symmetry transformations, which is needed to take into account the alteration of the dynamics itself, induced by the gauge fields. In particular, the Yang - Mills fields associated to the internal rotations have the effect of modifying the time derivative of the internal variables in a scheme of minimal coupling (introduction of an internal covariant derivative); 3), given their dynamical effect, the Yang - Mills fields associated to the internal rotations apparently define a sort of Galilean spin connection, while the Yang - Mills fields associated to the quadrupole momentum and to the internal energy have the effect of introducing a sort of dynamically induced internal metric in the relative space.

Projective transformations and symmetries of differential equation

The group properties of the equations of geodesics on a pseudo-Riemannian manifold are considered, in particular, when these are written as a system of second-order differential equations (resolved with respect to the second derivatives) with third-degree polynomials in the derivatives of the unknown function on the right-hand sides. Each point symmetry of such systems is proved to be a projective transformation. A connection between projective transformation in and symmetries of Hamiltonian systems and Lie-Bäcklund transformations of Hamilton-Jacobi equation with quadratic Hamiltonians is discovered. This provides tools for developing a systematic geometric approach to defining and investigating point and non-point symmetries of large classes of differential equations and partial differential equations and to obtaining their solutions. The dimension of the maximal symmetry group for system of second-order ordinary differential equations resolved with respect to the higher derivatives is found, and this group is proved to be the projective group. As a consequence, the dimension of the maximal symmetry group of the Newton equations is found. In case of three spatial dimensions this group (which is a 24-dimensional projective group) is proved to have as a subgroup the Poincaré group, which is fundamental for special relativity theory.

On soluble groups of automorphisms of nonorientable Klein surfaces

We classify up to topological type nonorientable bordered Klein surfaces with maximal symmetry and soluble automorphism group provided its solubility degree does not exceed 4. Using this classification we show that a soluble group of automorphisms of a nonorientable Riemann surface of algebraic genus q 2 has at most 24(q 1) elements and that this bound is sharp for infinitely many values of q.