Classical Lie Groups Research Articles

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.

One-dimensional MHD flows with cylindrical symmetry: Lie symmetries and conservation laws

A recent paper considered symmetries and conservation laws of the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. This paper analyzes the one-dimensional magnetohydrodynamics flows with cylindrical symmetry in the mass Lagrangian coordinates. The medium is assumed inviscid and thermally non-conducting. It is modeled by a polytropic gas. Symmetries and conservation laws are found. The cases of finite and infinite electric conductivity need to be analyzed separately. For finite electric conductivity σ(ρ,p) we perform Lie group classification, which identifies σ(ρ,p) cases with additional symmetries. The conservation laws are found by direct computation. For cases with infinite electric conductivity variational formulations of the equations are considered. Lie group classifications are obtained with the entropy treated as an arbitrary element. A variational formulation allows to use the Noether theorem for computation of conservation laws. The conservation laws obtained for the variational equations are also presented in the original (physical) variables.

Symmetry methods for a hyperbolic model for a class of populations

We investigate a hyperbolic system that describe the dispersal dynamics of a population, introduced by Méndez and Camacho (1997)[1], in Lie symmetry classification perspective. A Lie group classification is provided for different forms of two constitutive functions: the propagation coefficient D(u) and the reaction term r(u). We establish the Lie symmetry determining equations by utilizing an equivalence generator and the projection theorem. Several extensions of principal Lie algebra are found for different forms of D(u) and r(u). By performing classical reductions we obtain several exact solutions.

On the Symmetric Einstein Equation for Three-Dimensional Lie Groups with Left-Invariant Riemannian Metric and Semi-Symmetric Connection

Riemannian manifolds with a Levi-Civita connection and constant Ricci curvature, or Einstein manifolds, were studied in the works of many mathematicians. This question has been most studied in the homogeneous Riemannian case. In this direction, the most famous ones are the results of works by D.V. Alekseevsky, M. Wang, V. Ziller, G. Jensen, H.Laure, Y.G. Nikonorov, E.D. Rodionov and other mathematicians. At the same time, the question of studying Einstein manifolds has been little studied for the case of an arbitrary metric connection. This is primarily due to the fact that the Ricci tensor of metric connection is not, generally speaking, symmetric.
 In this paper, we consider semisymmetric connections on 3-dimensional Lie groups with a left-invariant Riemannian metric. For the symmetric part of the Ricci tensor, the Einstein equation is studied. As a result of the research carried out, a classification of 3-dimensional metric Lie groups and the corresponding semisymmetric connections in the case of the Einstein symmetric equation has been obtained.
 Earlier in the works of P.N. Klepikov, E.D. Rodionov and O.P. Khromova, the classical Einstein equation was studied, and it was proved that if the classical.
 The Einstein equation holds for a 3-dimensional Lie group with left-invariant (pseudo) Riemannian metric and semi-symmetric connection. Then, either the connection is a Levi-Civita connection, or the curvature tensor of the connection is equal to zero.

Symplectic groups over noncommutative algebras

We introduce the symplectic group \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\) over a noncommutative algebra A with an anti-involution \(\sigma \). We realize several classical Lie groups as \({{\,\mathrm{Sp}\,}}_2\) over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\) on isotropic A-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic A-lines as invariants of this action. When the algebra A is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space \(X_{{{\,\mathrm{Sp}\,}}_2(A,\sigma )}\), and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as \({{\,\mathrm{Sp}\,}}_2(A,\sigma )\)) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A.

Classification of Lorentzian Lie Groups Based on Codazzi Tensors Associated with Yano Connections

In this paper, we derive the expressions of Codazzi tensors associated with Yano connections in seven Lorentzian Lie groups. Furthermore, we complete the classification of three-dimensional Lorentzian Lie groups in which Ricci tensors associated with Yano connections are Codazzi tensors. The main results are listed in a table, and indicate that G1 and G7 do not have Codazzi tensors associated with Yano connections, G2, G3, G4, G5 and G6 have Codazzi tensors associated with Yano connections.

Lie group classification of the nonlinear transmission line model and exact traveling wave solutions

A nonlinear transmission line (NLTL) model is very essential tools in understanding of propagation of electrical solitons which can propagate in the form of voltage waves in nonlinear dispersive media. These models are often formulated using nonlinear partial differential equations. One of the basic tools available to study these equations are numerical methods such as finite difference method, finite element method, etc, have been developed for nonlinear partial differential equations. These methods require a great amount of time and memory due to the discretization and usually the effect of round-off error causes loss of accuracy in the results. So in this paper, we use one of the most famous analytical methods the Lie group analysis due to Sophus Lie. One of the advantages of this approach is that requires only algebraic calculations. The main aim of this study is to explore the nonlinear transmission line model with arbitrary capacitor's voltage dependence, through the use of Lie group classification, we show that the specifying form of arbitrary capacitor's voltage are power law nonlinearity, exponential law nonlinearity and constant capacitance. The exact solutions and similarity reductions generated from the symmetries are also provided. Furthermore, translational symmetries were utilized to find a family of traveling wave solutions via the \(\tanh\)-method of the governing nonlinear problem.

Integrable property, Lie symmetry analysis and explicit solutions to the generalized [formula omitted] model

In this paper, the combination of Painlevé analysis and Lie group method is performed, the Painlevé properties (PPs) and Bäcklund transformations (BTs) of the nonlinear ϕ4 types of equations are obtained under some conditions. Then, all of the point symmetries of the generalized ϕ4 model are given by Lie group classification method. Furthermore, the exact solutions to the equations are investigated by the symmetry reductions and analytic method.

Conservation laws of the complex short pulse equation and coupled complex short pulse equations

In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated. The two complex nonlinear models are turned into multi-component real models by proper transformations. Lie symmetries are obtained via the classical Lie group method, and the results for the coupled complex short pulse equations contain the existing results as particular cases. Based on the linearizing operator and adjoint linearizing operator for the two real systems, adjoint symmetries can be obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair (SA) method. Relationships between the nonlinear self-adjointness method and the SA method are investigated.

First integrals, solutions and conservation laws of the derivative nonlinear Schrödinger equation

We study the derivative nonlinear Schrödinger equation which has several applications, such as the propagation of circular polarized nonlinear Alfvén waves in plasmas. We present general and special solutions of this equation using first integrals. Classical Lie group theory along with power series method is also applied to obtain exact analytical solutions of this equation. Finally, conservation laws of the underlying equation are constructed through the use of the multiplier method.

Non‐classical Lie symmetries for nonlinear time‐fractional Heisenberg equations

The current work is based on performing the non‐classical and classical Lie group analysis for a system of nonlinear time‐fractional Heisenberg equation. Here, we apply analysis of the Lie group symmetry as one of the powerful tools dealing with the large class of fractional order differential equations in the Riemann–Liouville (RL) sense. Indeed, classical and non‐classical Lie symmetries group is used to similarity reduction of nonlinear time‐fractional Heisenberg equation. Especially, based on the obtained Lie symmetry generators, we construct conservation laws for the classical and non‐classical vector fields related to the Heisenberg time‐fraction equation.

Dirac series for complex classical Lie groups: A multiplicity-one theorem

This paper computes the Dirac cohomology HD(π) of irreducible unitary Harish-Chandra modules π of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are shown to be unitarily induced from unipotent representations. When nonzero, there is a unique, multiplicity free K-type in π contributing to HD(π). This confirms conjectures formulated by the first named author and Pandžić in 2011.

Exotic group C * -algebras of simple Lie groups with real rank one

Exotic group C * -algebras are C * -algebras that lie between the universal and the reduced group C * -algebra of a locally compact group. We consider simple Lie groups G with real rank one and investigate their exotic group C * -algebras C L p+ * (G), which are defined through L p -integrability properties of matrix coefficients of unitary representations. First, we show that the subset of equivalence classes of irreducible unitary L p+ -representations forms a closed ideal of the unitary dual of these groups. This result holds more generally for groups with the Kunze–Stein property. Second, for every classical simple Lie group G with real rank one and every 2≤q<p≤∞, we determine whether the canonical quotient map C L p+ * (G)↠C L q+ * (G) has non-trivial kernel. Our results generalize, with different methods, recent results of Samei and Wiersma on exotic group C * -algebras of SO 0 (n,1) and SU(n,1). In particular, our approach also works for groups with property (T).

The direct image of generalized divisors and the Norm map between compactified Jacobians

The Norm map between Jacobians and the Prym scheme are widely used in the description of spectral data of G-Higgs pairs, for G varying among classical Lie group. The aim of this paper is to generalize the Norm map to compactified Jacobians and provide a compactification for the Prym scheme. Given a finite, flat morphism between curves (e.g. embeddable noetherian schemes of pure dimension 1), we first define the notion of direct and inverse image for generalized divisors and generalized line bundles. In the case when we deal with (possibly reducible, non-reduced) projective curves over a field and the codomain curve is smooth, we introduce the compactified Jacobians parametrizing torsion-free rank-1 sheaves and then we study the Norm and inverse image maps between compactified Jacobians. Finally, we introduce and study the Prym stack defined as the kernel of the Norm map.

Lie Group Classification of Generalized Variable Coefficient Korteweg-de Vries Equation with Dual Power-Law Nonlinearities with Linear Damping and Dispersion in Quantum Field Theory

Many physical phenomena in fields of studies such as optical fibre, solid-state physics, quantum field theory and so on are represented using nonlinear evolution equations with variable coefficients due to the fact that the majority of nonlinear conditions involve variable coefficients. In consequence, this article presents a complete Lie group analysis of a generalized variable coefficient damped wave equation in quantum field theory with time-dependent coefficients having dual power-law nonlinearities. Lie group classification of two distinct cases of the equation was performed to obtain its kernel algebra. Thereafter, symmetry reductions and invariant solutions of the equation were obtained. We also investigate various soliton solutions and their dynamical wave behaviours. Further, each class of general solutions found is invoked to construct conserved quantities for the equation with damping term via direct technique and homotopy formula. In addition, Noether’s theorem is engaged to furnish more conserved currents of the equation under some classifications.

Plane one-dimensional MHD flows: Symmetries and conservation laws

The paper considers the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. The inviscid, thermally non-conducting medium is modeled by a polytropic gas. The equations are examined for symmetries and conservation laws. For the case of finite electric conductivity we establish Lie group classification, i.e. we describe all cases of the conductivity σ(ρ,p) for which there are symmetry extensions. The conservation laws are derived by direct computation. For the case of infinite electrical conductivity the equations can be brought into a variational form in the Lagrangian coordinates. Lie group classification is performed for the entropy function as an arbitrary element. Using the variational structure, we employ the Noether theorem for obtaining conservation laws. The conservation laws are also given in the physical variables.

C-minimal topological groups

Abstract In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235–239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211–223], we prove that a c-minimal locally solvable Lie group is compact. It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that G / N G/N is c-(totally) minimal. Applying this result, we prove that a locally compact group G is c-totally minimal if and only if its connected component c ⁢ ( G ) c(G) is compact and G / c ⁢ ( G ) G/c(G) is c-totally minimal. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229–327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces \((M=G/H,g)\) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (G/H, g), such that G is one of the compact classical Lie groups \({{\,\mathrm{SO}\,}}(n)\), \(\mathrm{U}(n)\), and H is a diagonally embedded product \(H_1\times \cdots \times H_s\), where \(H_j\) is of the same type as G. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H, g) with H semisimple.

Newell-Littlewood numbers

The Newell-Littlewood numbers are defined in terms of their celebrated cousins, the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. They are the structure coefficients of the K. Koike-I. Terada basis of the ring of symmetric functions. Recent work of H. Hahn studies them, motivated by R. Langlands’ beyond endoscopy proposal; we address her work with a simple characterization of detection of Weyl modules. This motivates further study of the combinatorics of the numbers. We consider analogues of ideas of J. De Loera-T. McAllister, H. Derksen-J. Weyman, S. Fomin–W. Fulton-C.-K. Li–Y.-T. Poon, W. Fulton, R. King-C. Tollu-F. Toumazet, M. Kleber, A. Klyachko, A. Knutson-T. Tao, T. Lam-A. Postnikov-P. Pylyavskyy, K. Mulmuley-H. Narayanan-M. Sohoni, H. Narayanan, A. Okounkov, J. Stembridge, and H. Weyl.