Classical Lie Groups Research Articles

Residual symmetry and interactions between solitons and periodic waves of the Drinfeld-Sokolov-Satsuma-Hirota system

Abstract In this paper, the Drinfeld-Sokolov-Satsuma-Hirota (DSSH) system is studied by using residual symmetry and consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the DSSH system is localized to a Lie point symmetry in a properly prolonged system, based on which we get a new Bäcklund transformation for this system. New symmetry reduction solutions of the DSSH system are obtained by applying the classical Lie group approach on the prolonged system. Moreover, the DSSH system is proved to be CRE integrable and new interesting interaction solutions between solitons and periodic waves are generated and analyzed.

Logarithmic Sobolev-Type Inequalities on Lie Groups

In this paper we show a number of logarithmic inequalities on several classes of Lie groups: log-Sobolev inequalities on general Lie groups, log-Sobolev (weighted and unweighted), log-Gagliardo–Nirenberg and log-Caffarelli–Kohn–Nirenberg inequalities on graded Lie groups. Furthermore, on stratified groups, we show that one of the obtained inequalities is equivalent to a Gross-type log-Sobolev inequality with the horizontal gradient. As a result, we obtain the Gross log-Sobolev inequality on general stratified groups but, very interestingly, with the Gaussian measure on the first stratum of the group. Moreover, our methods also yield weighted versions of the Gross log-Sobolev inequality. In particular, we also obtain new weighted Gross-type log-Sobolev inequalities on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document} for arbitrary choices of homogeneous quasi-norms. As another consequence we derive the Nash inequalities on graded groups and an example application to the decay rate for the heat equations for sub-Laplacians on stratified groups. We also obtain weighted versions of log-Sobolev and Nash inequalities for general Lie groups.

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.

Taut almost cosymplectic hyperbolas and almost bi-contact metric structures on three-manifolds

Cappelletti-Montano et al. [6] introduced and studied taut cosymplectic circles/spheres, which are the analogues in the cosymplectic setting of taut contact circles/spheres introduced and studied by Geiges-Gonzalo [11], [13], [14]. The main purpose of this paper is to introduce and study, in dimension three, the hyperbolic analogue in the cosymplectic setting. We give a characterization of a taut almost cosymplectic hyperbola, classify compact three-manifolds which admit a cosymplectic hyperbola, and give a complete classification of 3D Lie groups which admit a left invariant taut cosymplectic hyperbola/circle/hyperboloid/sphere. Besides, we introduce the notion of almost bi-contact metric structure. In particular, in dimension three, an almost bi-contact structure defines a taut almost cosymplectic hyperbolas or a taut almost cosymplectic circle. Then we characterize, in dimension three, the existence of an almost bi-contact metric structure, in particular with the associated 1-forms closed, and thus a complete classification of almost bi-α-coKähler three-manifolds with constant extrinsic curvature is established.

Conformal vector fields on Lie groups: The trans-Lorentzian signature

A pseudo-Riemannian Lie group is a connected Lie group endowed with a left-invariant pseudo-Riemannian metric of signature (p,q). In this paper, we study pseudo-Riemannian Lie groups (G,〈⋅,⋅〉) with non-Killing left-invariant conformal vector fields. Firstly, we prove that if h is a Cartan subalgebra for a semisimple Levi factor of the Lie algebra g, then dim⁡h≤max⁡{0,min⁡{p,q}−2}. Secondly, we classify trans-Lorentzian Lie groups (i.e., min⁡{p,q}=2) with non-Killing left-invariant conformal vector fields, and prove that [g,g] is at most three-step nilpotent. Thirdly, based on the classification of the trans-Lorentzian Lie groups, we show that the corresponding Ricci operators are nilpotent, and consequently the scalar curvatures vanish. As a byproduct, we prove that four-dimensional trans-Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are necessarily conformally flat, and construct a family of five-dimensional ones which are not conformally flat.

Manifestly covariant worldline actions from coadjoint orbits. Part I. Generalities and vectorial descriptions

We derive manifestly covariant actions of spinning particles starting from coadjoint orbits of isometry groups, by using Hamiltonian reductions. We show that the defining conditions of a classical Lie group can be treated as Hamiltonian constraints which generate the coadjoint orbits of another, dual, Lie group. In case of (inhomogeneous) orthogonal groups, the dual groups are (centrally-extended inhomogeneous) symplectic groups. This defines a symplectic dual pair correspondence between the coadjoint orbits of the isometry group and those of the dual Lie group, whose quantum version is the reductive dual pair correspondence à la Howe. We show explicitly how various particle species arise from the classification of coadjoint orbits of Poincaré and (A)dS symmetry. In the Poincaré case, we recover the data of the Wigner classification, which includes continuous spin particles, (spinning) tachyons and null particles with vanishing momenta, besides the usual massive and massless spinning particles. In (A)dS case, our classification results are not only consistent with the pattern of the corresponding unitary irreducible representations observed in the literature, but also contain novel information. In dS, we find the presence of partially massless spinning particles, but continuous spin particles, spinning tachyons and null particles are absent. The AdS case shows the largest diversity of particle species. It has all particles species of Poincaré symmetry except for the null particle, but allows in addition various exotic entities such as one parameter extension of continuous particles and conformal particles living on the boundary of AdS. Notably, we also find a large class of particles living in “bitemporal” AdS space, including ones where mass and spin play an interchanged role. We also discuss the relative inclusion structure of the corresponding orbits.

Global Coordinated Stabilization of Multiple Simple Mechanical Control Systems on a Class of Lie Groups

Global Coordinated Stabilization of Multiple Simple Mechanical Control Systems on a Class of Lie Groups

Ninth Variation of Classical Group Characters of Type A-D and Littlewood Identities

We introduce certain generalisations of the characters of the classical Lie groups, extending the recently defined factorial characters of Foley and King. In this extension, the factorial powers are replaced with an arbitrary sequence of polynomials, as in Sergeev–Veselov's generalised Schur functions and Okada's generalised Schur P- and Q-functions. We also offer a similar generalisation for the rational Schur functions. We derive Littlewood-type identities for our generalisations. These identities allow us to give new (unflagged) Jacobi–Trudi identities for the Foley–King factorial characters and for rational versions of the factorial Schur functions. We also propose an extension of the original Macdonald's ninth variation of Schur functions to the case of symplectic and orthogonal characters, which helps us prove Nägelsbach–Kostka identities.

Correction to: Emergent Dynamics of a Generalized Lohe Model on Some Class of Lie Groups

Correction to: Emergent Dynamics of a Generalized Lohe Model on Some Class of Lie Groups

Half-integrality of line bundles on partial flag schemes of classical Lie groups

In this paper, we develop a theory of Galois descent for equivariant line bundles on partial flag schemes. In particular, we study computational aspects of the classification of descent data of equivariant line bundles attached to characters of parabolic subgroups. As an application, we classify equivariant line bundles on partial flag schemes of the standard Z[1/2]-forms of classical Lie groups.

Reality of unipotent elements in classical Lie groups

In this paper, we classify the real and strongly real unipotent elements in classical simple Lie groups. We introduce an infinitesimal version of the classical reality in Lie groups. This helps us to achieve the desired classification. The reversers of such unipotent elements are also constructed.

Residue representations - The rank one case

With each resonance of the Laplacian acting on the compactly supported sections of a homogeneous vector bundle over a Riemannian symmetric space of the non-compact type, one can associate a residue representation. The purpose of this paper is to study them. The symmetric space is assumed to have rank-one but the irreducible representation τ of K defining the vector bundle is arbitrary. We give an algorithm that aims at determining if these representations are irreducible, finding their Langlands parameters, their Gelfand-Kirillov dimensions and wave front sets. As an example, we apply this algorithm to the Laplacian of the p-forms in the cases of all the classical real rank-one Lie groups.

Optimal subalgebra, conservations laws and symmetry reductions with analytical solutions using Lie symmetry analysis and geometric approach for the (1+1)-dimensional Manakov model

In this article, the (1+1)-dimensional Manakov model has been examined for finding its exact closed form solitonic solutions with the help of symmetry generators. These symmetry generators are explored using the Lie symmetry analysis, commonly known as the classical Lie group approach and the geometric approach. In a geometric approach, the extended Harrison and Estabrook’s differential forms have been used for obtaining the infinitesimal generators of the Manakov model. As there are infinite possibilities for the linear combination of infinitesimal generators, so by using Olver’s standard approach a one-dimensional optimal system of subalgebra has been established. Additionally, the ‘new conservation theorem’ put forth by Ibragimov has been utilized in order to devise the conservation laws for the (1+1)-dimensional Manakov model. Finally, the exact closed form solutions are obtained with the help of Lie symmetries corresponding to the defined model.

Concentration of measure for classical Lie groups

We study the concentration of measure in metric-measurable (mm)-spaces. We define the notion of concentration locus of a flag sequence of metric-measurable (mm)-spaces. Some examples of infinite group action on an infinite dimensional compact and non-compact manifold show the role played by the trajectory of concentration locus. We also provide some applications in physics, which emphasize the role of concentration of measure in gravitational effects.

Lie group method for unsteady heat flow of a third-grade fluid between two parallel heated plates

In this paper, we use the classical Lie group method, to investigate the symmetries of the heat transfer flow of a third-grade fluid. This approach allows one to reduce the coupled partial differential equations governing the problem, to a system of nonlinear ordinary differential equations. Point symmetries of such systems are used to construct some classes of solutions. By using travelling wave solutions, we studied the influence of third-grade fluid parameters on the flow.

Combinatorics of Vogan diagrams for almost-Kähler manifolds

Let [Formula: see text] be a non-compact classical semisimple Lie group and let [Formula: see text] be the adjoint orbit with respect to a fixed element in [Formula: see text]. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on [Formula: see text] purely in terms of the combinatorics of the associated Vogan diagram. The formulae are given separately for Lie groups whose Lie algebras are of type [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] denotes the rank of the Lie algebra.

Lie group symmetry analysis and invariant difference schemes of the two-dimensional shallow water equations in Lagrangian coordinates

The two-dimensional shallow water equations in Lagrangian coordinates are considered. Lie group classification for the class of the elliptic paraboloid bottom topography is performed.The transformations mapping the two-dimensional shallow water equations with a plane or rotation symmetric bottom into the gas dynamics equations of a polytropic gas with polytropic exponent γ=2 are represented. The group foliation of the two-dimensional shallow water equations in Lagrangian coordinates is discussed.New invariant conservative finite-difference schemes for the equations and their one-dimensional reductions are constructed. The schemes are derived either by extending the known one-dimensional schemes or by direct algebraic construction based on some assumptions on the form of the energy conservation law. Among the proposed schemes there are schemes possessing conservation laws of mass and energy.

Newell–Littlewood numbers II: extended Horn inequalities

The Newell–Littlewood numbers N μ,ν,λ are tensor product multiplicities of Weyl modules for classical Lie groups, in the stable limit. For which triples of partitions (μ,ν,λ) does N μ,ν,λ >0 hold? The Littlewood–Richardson coefficient case is solved by the Horn inequalities (in work of A. Klyachko and A. Knutson-T. Tao). We extend these celebrated linear inequalities to a much larger family, suggesting a general solution.

On the form of Lie symmetries of systems with three pdes: The examples of two variable coefficient Hirota Satsuma systems

We consider a general class of systems of three partial differential equations and we provide restrictions on the form of Lie symmetry operators admitted by such systems. When these restrictions are known in advance, the symmetry analysis becomes simpler. Special cases of this class are two generalizations of Hirota–Satsuma systems with variable coefficients. We derive the equivalence groups for these two systems. With the aid of the equivalence groups and the restricted form of the Lie symmetry operators, we present an enhanced Lie group classification for the two Hirota–Satsuma systems. For each class a specific system is single out which has the property to be mapped into a constant coefficient system.

Toda equations for surface defects in SYM and instanton counting for classical Lie groups

The partition function of super Yang-Mills theories with arbitrary simple gauge group coupled to a self-dual Ω background is shown to be fully determined by studying the renormalization group equations relevant to the surface operators generating its one-form symmetries. The corresponding system of equations results in a non-autonomous Toda chain on the root system of the Langlands dual, the evolution parameter being the RG scale. A systematic algorithm computing the full multi-instanton corrections is derived in terms of recursion relations whose gauge theoretical solution is obtained just by fixing the perturbative part of the IR prepotential as its asymptotic boundary condition for the RGE. We analyze the explicit solutions of the τ-system for all the classical groups at the diverse levels, extend our analysis to affine twisted Lie algebras and provide conjectural bilinear relations for the τ-functions of linear quiver gauge theory.