Showing papers on "Lie group published in 1992"

k‐symplectic structures

Abstract: The basic properties of k‐symplectic structures are introduced and developed in analogy with the well‐known symplectic differential geometry. Examples of such structures related to k‐symplectic Lie algebras are given.

Reduction of some classical non-holonomic systems with symmetry

Abstract: Two types of nonholonomic systems with symmetry are treated: (i) the configuration space is a total space of a G-principal bundle and the constraints are given by a connection; (ii) the configuration space is G itself and the constraints are given by left-invariant forms. The proofs are based on the method of quasicoordinates. In passing, a derivation of the Maurer-Cartan equations for Lie groups is obtained. Simple examples are given to illustrate the algorithmical character of the main results.

Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems

Abstract: Background of the theory of Lie algebras and Lie groups and their representations.- 1.1 Lie algebras and Lie groups.- 1.1.1 Basic definitions.- 1.1.2 Contractions and deformations.- 1.1.3 Functional algebras.- 1.2 ?-graded Lie algebras and their classification.- 1.2.1 Definitions.- 1.2.2 Semisimple, nilpotent and solvable Lie algebras. The Levi-Malcev theorem.- 1.2.3 Simple Lie algebras of finite growth: Classification and Dynkin-Coxeter diagrams.- 1.2.4 Root systems and the Weylgroup.- 1.2.5 A parametrization and ordering of roots of simple finite-dimensional Lie algebras.- 1.2.6 The real forms of complex simple Lie algebras.- 1.3 sl(2)-subalgebras of Lie algebras.- 1.3.1 Embeddings of sl(2) into Lie algebras.- 1.3.2 Infinite-dimensional graded Lie algebras corresponding to embeddings of sl(2) into simple finite-dimensional Lie algebras.- 1.3.3 Explicit realization of simple finite-dimensional Lie algebras for the principal embedding of sl(2).- 1.4 The structure of representations.- 1.4.1 Terminology.- 1.4.2 The adjoint representation.- 1.4.3 The regular representation and Casimir operators.- 1.4.4 Bases in the space of representation.- 1.4.5 Fundamental representations.- 1.5 A parametrization of simple Lie groups.- 1.6 The highest vectors of irreducible representations of semisimple Lie groups.- 1.6.1 Generalities.- 1.6.2 Expression for the highest matrix elements in terms of the adjoint representation.- 1.6.3 A formal expression for the highest matrix elements of the fundamental representations.- 1.6.4 Recurrence relations for the highest matrix elements of the fundamental representations.- 1.6.5 The highest matrix elements of irreducible representations expressed via generalized Euler angles.- 1.7 Superalgebras and superspaces.- 1.7.1 Superspaces.- 1.7.2 Classical Lie superalgebras.- Representations of complex semisimple Lie groups and their real forms.- 2.1 Infinitesimal shift operators on semisimple Lie groups.- 2.1.1 General expression of infinitesimal operators.- 2.1.2 The asymptotic domain.- 2.2 Casimir operators and the spectrum of their eigenvalues.- 2.2.1 General formulation of the problem.- 2.2.2 Quadratic Casimir operators.- 2.2.3 Construction of Casimir operators for semisimple Lie groups.- 2.3 Representations of semisimple Lie groups.- 2.3.1 Integral form of realization of operator-irreducible representations.- 2.3.2 The matrix elements of finite transformations.- 2.4 Intertwining operators and the invariant bilinear form.- 2.4.1 Intertwining operators and problems of reducibility, equivalence and unitarity of representations.- 2.4.2 Construction of intertwining operators.- 2.4.3 The invariant Hermitian form.- 2.5 Harmonic analysis on semisimple Lie groups.- 2.5.1 General method.- 2.5.2 Characters of operator-irreducible representations.- 2.5.3 Plancherel measure of the principal continuous series of unitary representations.- 2.6 Whittaker vectors.- A general method of integrating two-dimensional nonlinear systems.- 3.1 General method.- 3.1.1 Lax-type representation.- 3.1.2 Examples.- 3.1.3 Construction of solutions.- 3.2 Systems generated by the local part of an arbitrary graded Lie algebra.- 3.2.1 Exactly integrable systems.- 3.2.2 Systems associated with infinite-dimensional Lie algebras.- 3.2.3 Hamiltonian formalism.- 3.2.4 Solutions of exactly integrable systems (Goursat problem).- 3.3 Generalization for systems with fermionic fields.- 3.4 Lax-type representation as a realization of self-duality of cylindrically-symmetric gauge fields.- Integration of nonlinear dynamical systems associated with finite-dimensional Lie algebras.- 4.1 The generalized (finite nonperiodic) Toda lattice.- 4.1.1 Preliminaries.- 4.1.2 Construction of exact solutions on the base of the general scheme of Chapter 3.- 4.1.3 Examples.- 4.1.4 Construction of solutions without appealing to the Lax-type representation.- 4.1.4.1 Symmetry properties of the Toda lattice for the series A, B, C and the reduction procedure.- 4.1.4.2 Direct solution of the system (3.1.10) for the series A.- 4.1.4.3 Invariant generalization of the reduction scheme for arbitrary simple Lie algebras.- 4.1.5 The one-dimensional generalized Toda lattice.- 4.1.6 Boundary value problem (instantons and monopoles).- 4.2 Complete integration of the two-dimensionalized system of Lotka-Volterra-type equations (difference KdV) as the Backlund transformation of the Toda lattice.- 4.3 String-type systems (nonabelian versions of the Toda system).- 4.4 The case of a generic Lie algebra.- 4.5 Supersymmetric equations.- 4.6 The formulation of the one-dimensional system (3.2.13) based on the notion of functional algebra.- Internal symmetries of integrable dynamical systems.- 5.1 Lie-Backlund transformations. The characteristic algebra and defining equations of exponential systems.- 5.2 Systems of type (3.2.8), their characteristic algebra and local integrals.- 5.3 A complete description of Lie-Backlund algebras for the diagonal exponential systems of rank 2.- 5.4 The Lax-type representation of systems (3.2.8) and explicit solution of the corresponding initial value (Cauchy) problem.- 5.5 The Backlund transformation of the exactly integrable systems as a corollary of a contraction of the algebra of their internal symmetry.- 5.6 Application of the methods of perturbation theory in the search for explicit solutions of exactly integrable systems (the canonical formalism).- 5.7 Perturbation theory in the Yang-Feldmann formalism.- 5.8 Methods of perturbation theory in the one-dimensional problem.- 5.9 Integration of nonlinear systems associated with infinite-dimensional Lie algebras.- Scalar Lax-pairs and soliton solutions of the generalized periodic Toda lattice.- 6.1 A group-theoretical meaning of the spectral parameter and the equations for the scalar LA-pair.- 6.2 Soliton solutions of the sine-Gordon equation.- 6.3 Generalized Bargmann potentials.- 6.4 Soliton solutions for the vector representation of Ar.- Exactly integrable quantum dynamical systems.- 7.1 The Hamiltonian (canonical) formalism and the Yang-Feldmann method.- 7.2 Basics from perturbation theory.- 7.3 One-dimensional generalized Toda lattice with fixed end-points.- 7.3.1 Schroedinger's picture.- 7.3.2 Heisenberg's picture (the canonical formalism).- 7.3.3 Heisenberg's picture (Yang-Feldmann's formalism).- 7.4 The Liouville equation.- 7.5 Multicomponent 2-dimensional models. 1.- 7.6 Multicomponent 2-dimensional models. 2.- Afterword.

Twin buildings and groups of Kac-Moody type

Abstract: Introduction The combinatorial objects called buildings were first introduced (cf.) to provide a geometric approach to complex simple Lie groups – in particular the exceptional ones – and later on, more generally, to isotropic algebraic simple groups (cf. eg.). The buildings which do arise in that way are spherical, that is, have finite Weyl groups. This assertion has a partial converse, proved in: roughly speaking , there is a one-to-one correspondence between the (isomorphism classes of) buildings of irreducible spherical type and rank r ≥ 3 and the algebraic absolutely simple groups of relative rank r , where the notion of algebraic simple groups must be suitably extended so as to include, for instance, classical groups over division rings of infinite dimension over their centre. In order to have a similar statement in the rank 2 case, one must impose an extra condition, the Moufang condition, on the buildings under consideration. The correspondence in question is established via the classification of buildings of irreducible, spherical types and rank ≥ 3. For non-spherical buildings, a full classification is known only for the affine types, in rank ≥ 4 (cf.): buildings of such a type have a “spherical building at infinity”, which is the essential tool for classification. A construction procedure for buildings of more general types given in shows that there cannot be any hope for a complete classification of buildings of arbitrary types. On the other hand, there is another wide class of groups, the Kac-Moody groups, which give rise to buildings, usually of non-spherical and non-affine types; these are “concrete” objects, which one also wishes to characterize geometrically.

Lie-Poisson structure on some Poisson Lie groups

Abstract: Poisson Lie groups appeared in the work of Drinfel'd (see, e.g., [Drl, Dr2]) as classical objects corresponding to quantum groups. Going in the other direction, we may say that a Poisson Lie group is a group of symmetries of a phase space that are allowed to "twist," in a certain sense, the symplectic or Poisson structure. The Poisson structure on the group controls this twisting in a precise way. Quantizing both the phase space and the symmetries, one may obtain a quantum group acting on a quantum phase space. In recent work, Lu and Ratiu [LR] used so-called standard Poisson structures on a compact semisimple Lie group K and on its Poisson dual K* in order to give a new proof of the nonlinear convexity theorem of Kostant [Ko]. Their method is analogous to the famous symplectic proof of the linear convexity theorem given by Atiyah [A] and Guillemin and Stemnberg [GS]. The nonlinear convexity theorem, like the linear one, follows from a very general result on convexity of the image of the momentum map [A, GS]; however, in [LR] it is applied not to a coadjoint orbit in t*, but to a symplectic leaf in K* . The main result of this paper is that the standard Poisson structure on the Poisson dual K* to a compact semisimple Poisson Lie group K is actually isomorphic to the linear one on t* . This theorem seems to be related to some facts in the theory of quantum groups. Namely, (for generic q) the universal enveloping algebra U(t) and its quantum deformation Uq(t) are isomorphic as algebras (though not as coalgebras, of course, since the quantum version is not cocommutative). In particular, there is a bijective correspondence between their representations. A direct connection between our work and its quantum analogues, though, is still to be found. The present work supplies a positive answer to Question 5.1 in [LR] and strongly depends on that paper. To simplify reading, we keep the notations of [LR] wherever possible, on one hand, but give all necessary definitions, on the other. Our work is also related to [Du], in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in ?5. The paper is organized as follows. In ?2 we define Poisson Lie groups, discuss

Completely Integrable Gradient Flows

Abstract: In this paper we exhibit the Toda lattice equations in a double bracket form which shows they are gradient flow equations (on their isospectral set) on an adjoint orbit of a compact Lie group. Representations for the flows are given and a convexity result associated with a momentum map is proved. Some general properties of the double bracket equations are demonstrated, including a discussion of their invariant subspaces, and their function as a Lie algebraic sorter.

Non-compact WZW conformal field theories

Abstract: We discuss non-compact WZW sigma models, especially the ones with symmetric space H C /H as the target, for H a compact Lie group. They offer examples of non-rational conformal field theories. We remind their relation to the compact WZW models but stress their distinctive features like the continuous spectrum of conformal weights, diverging partition functions and the presence of two types of operators analogous to the local and non-local insertions recently discussed in the Liouville theory. Gauging non-compact abelian subgroups of H C leads to non-rational coset theories. In particular, gauging one-parameter boosts in the SL(2, C)/SU(2) model gives an alternative, explicitly stable construction of a conformal sigma model with the euclidean 2D black hole target. We compute the (regularized) toroidal partition function and discuss the spectrum of the theory. A comparison is made with more standard approach based on the U(1) coset of the SU(1, 1) WZW theory where stability is not evident but where unitarity becomes more transparent.

Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie)

Abstract: CONTENTSPrefaceChapter I. Definitions and elementary applications §1.1. One-parameter transformation groups §1.2. Prolongation formulae §1.3. Groups admissible by differential equations §1.4. Integration and reduction of order using one-parameter groups1.4.1. Integrating factor1.4.2. Method of canonical variables1.4.3. Invariant differentiation §1.5. Defining equations §1.6. Lie algebras §1.7. Contact transformationsChapter II. Integration of second-order equations admitting a two-dimensional algebra §2.1. Consecutive reduction of order2.1.1. An instructive example2.1.2. Solvable Lie algebras §2.2. The method of canonical variables2.2.1. Changes of variables and basis in an algebra2.2.2. Canonical form of two-dimensional algebras2.2.3. An integration algorithm2.2.4. An example of implementation of the algorithmChapter III. Group-theoretical classification of second-order equations §3.1. Equations admitting a three-dimensional algebra3.1.1. Classification in the complex domain3.1.2. Classification over the reals. Isomorphism and similarity §3.2. The general classification result §3.3. Two remarkable classes of equations3.3.1. The equation Linearizability criteria3.3.2. Equations Chapter IV. Ordinary differential equations with a fundamental system of solutions (following Vessiot-Guldberg-Lie) §4.1. The main theorem §4.2. Examples §4.3. Projective interpretation of the Riccati equation §4.4. Linearizable Riccati equationsChapter V. The invariance principle in problems of mathematical physics §5.1. Spherical functions §5.2. A group-theoretical touch to Riemann's method §5.3. Symmetry of fundamental solutions, or the first steps in group analysis in the space of distributions5.3.1. Something about distributions5.3.2. Laplace's equation5.3.3. The heat equation5.3.4. The wave equationChapter VI. Summary of resultsReferences

An Application of Homogenization Theory to Harmonic Analysis: Harnack Inequalities And Riesz Transforms on Lie Groups of Polynomial Growth

Abstract: We prove a homogenization formula for a sub-Laplacian are left invariant Hormander vector fields) on a connected Lie group Gof polynomial growth. Then using a rescaling argument inspired from M. Avellanedaand F. H. Lin [2], we prove Harnack inequalities for the positive solutions of the equation (∂/∂t+ L)u= 0. Using these inequalities and further exploiting the algebraic structure of Gwe prove that the Riesz transforms , are bounded on Lq,1 < q <+∞ and from L1 to weak-L1.

Rigidity for Anosov Actions of Higher Rank Lattices

Abstract: This is the simplest example of a large class of analytic "standard" actions of lattices in semisimple Lie groups on locally homogeneous spaces. A basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics (cf. [13], [50], [51]). A Cr-action So: IF > X -x X of a group F on a compact manifold X is said to be Anosov if there exists at least one element, Yh E F, such that 'p(Yh) is an Anosov diffeomorphism of X. We begin in this paper to study the Anosov differentiable actions of lattices, including many standard algebraic examples, and especially to study their stability properties. Our main theme is that either the Cr-rigidity or the Cr-deformation rigidity of an Anosov action (for 1 < r < cr. or even for the real analytic case) can be shown just by studying the behavior of the periodic orbits for the action. There are two notions of "structural stability" that appear in this paper, rigidity and deformation rigidity. A Cl-perturbation of a Cr-action So is simply another Cr-action Spl such that for a finite set of generators {81, . . , ad of F, the Cr-diffeomorphisms 'P(8i) and 'Pl(8i) are Cl-close for all i. An action So is said to be Cr-rigid (or topologically rigid if r = 0) if every sufficiently small C'perturbation of So is Cr-conjugate to So, for 0 < r < oo, or r = w in the case of real analytic actions. A Cl-deformation of an action So is a continuous path of Cr-actions (Pt defined for some 0 < t < e with Spo = SD* An action SD is said to be Cr-deform-

Symmetries of variable coefficient Korteweg–de Vries equations

Abstract: The Lie point symmetries of the equation ut+f(x, t)uux+g(x,t)uxxx=0 are studied. The symmetry group is shown to be, at most, four dimensional, and this occurs if and only if the equation is equivalent, under local point transformations, to the KdV equation with f=g=1. For nine different classes of functions f and g, the symmetry group turns out to be three dimensional. Two‐dimensional and one‐dimensional symmetry groups occur for 11 and 15 classes of equations, respectively.

Analysis of swept volume via Lie groups and differential equations

Abstract: The development of useful mathematical techniques for an alyzing swept volumes, together with efficient means of im plementing these methods to produce serviceable models, has important applications to numerically controlled (NC) machin ing, robotics, and motion planning, as well as other areas of automation. In this article a novel approach to swept volumes is delineated—one that fully exploits the intrinsic geometric and group theoretical structure of Euclidean motions in or der to formulate the problem in the context of Lie groups and differential equations.Precise definitions of sweep and swept volume are given that lead naturally to an associated ordinary differential equation. This sweep differential equation is then shown to be related to the Lie group structure of Euclidean motions and to generate trajectories that completely determine the geometry of swept volumes.It is demonstrated that the notion of a sweep differential equation leads to criteria that provide useful insights concern ing the geo...

On the structure of twisted group C*-algebras

Abstract: We first give general structural results for the twisted group algebras C*(G, σ) of a locally compact group G with large abelian subgroups. In particular, we use a theorem of Williams to realise C*(G, σ) as the sections of a C*-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when Γ is a discrete subgroup of a solvable Lie group G, the K-groups K * (C*(Γ, σ)) are isomorphic to certain twisted K-groups K*(G/Γ, δ(σ)) of the homogeneous space G/Γ, and we discuss how the twisting class δ(σ) ∈ H 3 (G/Γ, Z) depends on the cocycle σ

Introduction to the Differential Geometry of Quantum Groups

Abstract: An introduction to the noncommutative differential calculus on quantum groups. The invariant group average is also discussed.

Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications

Abstract: Basics of Representation Theory. Commutative Harmonic Analysis. Representations of Compact and Finite Groups. Lie Groups SU(2) and SO(3). Classical Compact Lie Groups and Algebras. The Heisenberg Group and Semidirect Products. Representations of SL 2 . Lie Groups and Hamiltonian Mechanics. Appendices: Spectral Decomposition of Selfadjoint Operators. Integral Operators. A Primer on Riemannian Geometry: Geodesics, Connection, Curvature. References. List of Frequently Used Notations. Index.

Cohomology of biquotients

Abstract: Biquotients are non-homogeneous quotient spaces of Lie groups. Using the Serre spectral sequence and the method of Borel, we compute the cohomology algebra of these spaces in cases where the Lie group cohomology is not too complicated. Among these are the biquotients which are known to carry a metric of positive curvature.

Ideas and methods in mathematical analysis, stochastics, and applications

Abstract: Preface Picture of Raphael Hoegh-Krohn Bibliography of Raphael Hoegh-Krohn 1. On the scientific work of Raphael Hoegh-Krohn Part I. Stochastic Analysis 2. Some Euclidean integer-valued random fields with Markov properties 3. A Beurling-Deny type structure theorem for Dirichlet forms on general state spaces 4. Log-concavity of radial Schroedinger wave functions and convergence of planetesimal diffusions 5. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals 6. Large deviations for weak solutions of stochastic differential equations 7. On a class of probabilistic integrodifferential equations 8. Operator integrals and martingale integrals with general parameter 9. Wick multiplication and Ito-Skorohod stochastic differential equations 10. Multiscale analysis in random dynamics 11. A limiting distribution connected with fractional parts of linear forms 12. Rapport sur les representations chaotiques 13. Markov properties of solutions of stochastic partial differential equations in a finite volume 14. On stochastic evolution equations with non-homogeneous boundary conditions 15. Grey noise 16. Existence of invariant measures for diffusion process with infinite dimensional state space Part II. Infinite Dimensional Groups 17. On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds 18. Fields of left-invariant standard Brownian motion processes on a smooth bundle of compact-simple Lie groups 19. Unitary highest weight representations of gauge groups Part III. Operator Algebras 20. Some non-commutative orbifolds 21. Ergodic actions of non-abelian compact groups 22. Positive projections onto Jordan algebras and their enveloping von Neumann algebras Part IV. Nonlinear Analysis and Applications 23. How many singularities can there be in an energy minimizing map from the ball to the sphere? 24. Front tracking for petroleum reservoirs 25. Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vries equations 26. A new representation of soliton solutions of the Kadomtsev-Petviashvili equation 27. On scalar conservation laws in one dimension.

Semiclassical maslov asymptotics with complex phases. I. General approach

Abstract: A method of constructing semiclassical asymptotics with complex phases is presented for multidimensional spectral problems (scalar, vector, and with operator-valued symbol) corresponding to both classically integrable and classically nonintegrable Hamiltonian systems. In the first case, the systems admit families of invariant Lagrangian tori (of complete dimension equal to the dimensionn of the configuration space) whose quantization in accordance with the Bohr—Sommerfeld rule with allowance for the Maslov index gives the semiclassical series in the region of large quantum numbers. In the nonintegrable case, families of Lagrangian tori with complete dimension do not exist. However, in the region of regular (nonchaotic) motion, such systems do have invariant Lagrangian tori of dimensionk

Generic bifurcation of Hamiltonian vector fields with symmetry

Abstract: One of the goals of this paper is to describe explicitly the generic movement of eigenvalues through a one-to-one resonance in a linear Hamiltonian system which is equivariant with respect to a symplectic representation of a compact Lie group. We classify this movement, and hence answer the question of when the collisions are 'dangerous' in the sense of Krein by using a combination of group theory and definiteness properties of the associated quadratic Hamiltonian. For example, for systems with no symmetry or O(2) symmetry generically the eigenvalues split, whereas for systems with S1 symmetry, generically the eigenvalues may split or pass. It is in this last case that one has to use both group theory and energetics to determine the generic eigenvalue movement. The way energetics and group theory are combined is summarized in table 1. The result is to be contrasted with the bifurcation of steady states (zero eigenvalue) where one can use either group theory alone (Golubitsky and Stewart) or definiteness properties of the Hamiltonian (Cartan-Oh) to determine whether the eigenvalues split or pass on the imaginary axis.

SU(1,1) Lie algebraic approach to linear dissipative processes in quantum optics

Abstract: An SU(1,1) Lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems. The Liouville space formulation, thermo field dynamics, and the disentanglement theorem of SU(1,1) Lie algebra play essential roles in this formulation. In the Liouville space, the time‐evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of SU(1,1) Lie algebra and the thermal state condition of thermo field dynamics. The presented formulation is used for investigating a dissipative nonlinear oscillator, the quantum mechanical model of phase modulation, and the photon echo in the localized electron–phonon system. This algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems.

Deformed conformal and supersymmetric quantum mechanics

Abstract: Within the standard quantum mechanics a q-deformation of the simplest N=2 supersymmetry algebra is suggested. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are q-isospectral, i.e., the spectrum of one can be obtained from another (with possible exception of the lowest level) by the q2-factor scaling. A special class of the self-similar potentials is shown to obey the dynamical conformal symmetry algebra suq(1,1). These potentials exhibit exponential spectra and corresponding raising and lowering operators satisfy the q-deformed harmonic oscillator algebra of Biedenharn and Macfarlane.

Hidden symmetries associated with the projective group of nonlinear first-order ordinary differential equations

Abstract: Hidden symmetries, those not found by the classical Lie group method for point symmetries, are reported for nonlinear first-order ordinary differential equations (ODEs) which arise frequently in physical problems. These are for the special class of the eight nonAbelian, two-parameter subgroups of the eight-parameter projective group. The first-order ODEs can be transformed by non-local transformations to new separable first-order ODEs which then can be reduced to quadratures. The first-order ODEs include Riccati equations and equations which in particular cases are of the form of Abel's equation. The procedure demonstrates the feasibility of integrating nonlinear ODEs that do not show any apparent Lie group point symmetry. Applications to the Vlasov characteristic equation and the reaction-diffusion equation are given.

Novel symmetry of non‐Einsteinian gravity in two dimensions

Abstract: The integrability of R2‐gravity with torsion in two dimensions is traced to an ultralocal dynamical symmetry of constraints and momenta in Hamiltonian phase space. It may be interpreted as a quadratically deformed iso(2,1)‐algebra with the deformation consisting of the Casimir operators of the undeformed algebra. The locally conserved quantity encountered in the explicit solution is identified as an element of the center of this algebra. Specific contractions of the algebra are related to specific limits of the explicit solutions of this model.

A simple method for group analysis and its application to a model of detonation

Abstract: A simple procedure is suggested to obtain extensions of the principal Lie algebra, for invariance transformations of a given family of equations, by using an equivalence algebra. An application to a qualitative model of detonation is performed.