All invariant functions of the group generators (generalized Casimir operators) are found for all real algebras dimension up to five and nilpotent six.

A symmetry of an equation will leave the set all solutions invariant. 'conditional symmetry' only a subset solutions, defined by some differential condition, The authors show how specific class conditional symmetries can be used to reduce partial ordinary one. In particular, for Boussinesq equation, these symmetries, together with ones, provide possible reductions equations. group theoretical explanation recently obtained new is provided.

2014 Nous présentons un exposé détaillé du formalisme de la diffusion élastique nucléon- nucléon en ajoutant nouveaux résultats à ceux déjà connus.Nous passons revue plusieurs représentations matrice tenant compte des principes symétrie, notamment conservation parité, l'invariance par renversement temps, principe Pauli et isotopique.Les quantités expérimentales système centre masse (c.m.s.) laboratoire (l.s.) sont exprimées fonction amplitudes diffusion.Les relations entre ces quantités,...

A superintegrable system is, roughly speaking, a that allows more integrals of motion than degrees freedom. This review is devoted to finite dimensional classical and quantum systems with scalar potentials are polynomials in the momenta. We present classification second-order two-dimensional Riemannian pseudo-Riemannian spaces. It based on study quadratic algebras equivalence different under coupling constant metamorphosis. The determining equations for existence arbitrary order real...

The Lie subalgebras of all real algebras dimension d?4 are classified into equivalence classes under their groups inner automorphisms. Tables representatives each conjugacy class given.

We present a general method for reducing the problem of finding all continuous subgroups given Lie group G with nontrivial invariant subgroup N, to that classifying N and factor G/N. The is applied classify Poincaré (PG) Lorentz extended by dilatations [the homogeneous similitude (HSG)]. Lists representatives each conjugacy class subalgebras algebras groups PG HSG are in form tables.

The Kadomtsev–Petviashvili (KP) equation (ut+3uux/2+ 1/4 uxxx)x +3σuyy/4=0 allows an infinite-dimensional Lie group of symmetries, i.e., a transforming solutions amongst each other. algebra this symmetry depends on three arbitrary functions time ‘‘t’’ and is shown to be related subalgebra the loop A(1)4. Low-dimensional subalgebras are identified, specifically all those dimension n≤3, also physically important six-dimensional containing translations, dilations, Galilei transformations,...

The Kadomtsev–Petviashvili (KP) hierarchy is an infinite set of nonlinear partial differential equations in which the number independent variables increases indefinitely as one proceeds down hierarchy. Since these were obtained part a group theoretical approach to soliton it would appear that KP provides integrable scalar with arbitrary variables. It shown, by investigating specific equation 3+1 dimensions, higher are only conditional sense. under study, taken isolation, does not pass...

We consider the ``missing label'' problem for basis vectors of SU(3) representations in a corresponding to group reduction SU(3)⊃O(3)⊃O(2). prove that only two independent O (3) scalars exist enveloping algebra S U (3), addition obvious ones, namely angular momentum L2 and Casimir operators C(2) C(3). Any one these (of third fourth order generators) can be added C(2), C(3), L2, L3 form complete set commuting operators. The eigenvalues X(3) X(4) are calculated analytically or numerically many...

The representation theory of the rotation group O(3) is developed in a new basis, consisting eigenfunctions operator E = −4(L12 + rL22), where 0 < r 1 and Li are generators. This basis |Jλ〉 shown to be unique nonequivalent alternative canonical (eigenfunctions L3). functions constructed as linear combinations fall into four symmetry classes, distinguished by their behavior under reflections inidividual space axes. Algebraic equations for eigenvalues λ derived. When realized terms on...

A new class of ‘‘solvable’’ nonlinear dynamical systems has been recently identified by the requirement that ordinary differential equations (ODE’s) describing each member this possess superposition principles. These ODE’s are generally not derived from a Hamiltonian and classified associated pairs Lie algebras vector fields. In paper, all such n≤3 integrated in unified way finding explicit integrals for them relating to ‘‘pivotal’’ their class: projective Riccati equations. Moreover,...

The symmetry group of the generalised non-linear Schrodinger equation i psi t+ Delta =a0 +a1 mod 2 +a2 4 in three space dimensions is shown to be extended Galilei G(3), for a1a2 not=0, and Galilei-similitude Gd(3) (including a dilation) a1=oor a2=0. All Lie subgroups G(3) are found. They will used subsequent paper obtain invariant solutions equation.

An infinite family of exactly solvable and integrable potentials on a plane is introduced. It shown that all already known rational with the above properties allowing separation variables in polar coordinates are particular cases this family. The underlying algebraic structure new revealed as well its hidden algebra. We conjecture members also superintegrable demonstrate for first few cases. A quasi-exactly-solvable generalization found.

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of much recent program: use groups to study difference We show that mismatch between continuous symmetries discrete equations can be resolved in at least two manners. One is generalized acting on solutions equations, but leaving lattice invariant. The other restrict them point symmetries, allow also transform lattice.

The representation theory of the group $O(4)$ is considered systematically in different bases, corresponding to reduction various continuous or discrete subgroups. results are applied hydrogen atom and we investigate six bases separation variables Schrödinger equation momentum space four coordinate space. It shown that a classification (and complete sets commuting operators determining bases) corresponds interactions, breaking original symmetry, while preserving certain aspects it. Vector...

The Lie algebra of the group point transformations, leaving Davey–Stewartson equations (DSE’s) invariant, is obtained. general element this depends on four arbitrary functions time. shown to have a loop structure, property shared by symmetry algebras all known (2+1)-dimensional integrable nonlinear equations. Subalgebras are classified and used reduce DSE’s various involving only two independent variables.