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The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid which are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose that several realizations of pre-rackoid structures. One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the integration of the Courant algebroid when the strong constraint of DFT is imposed. Finally, for a physical application, we exhibit an implementation of the (pre-)rackoid in a three-dimensional topological sigma model., 31 pages, 1 figure, version appeared in JMP

Nonlocally related partial differential equation (PDE) systems can play an important role in the analysis of a given PDE system. In this paper, a new systematic method for obtaining nonlocally related PDE systems is developed. In particular, it is shown that if a PDE system admits q point symmetries whose infinitesimal generators form a q-dimensional solvable Lie algebra, then, for each resulting q-dimensional solvable algebra chain, one can obtain systematically q nonlocally related PDE systems. Such nonlocally related systems are obtained for a general class of nonlinear reaction–diffusion equations admitting two- to four-dimensional solvable algebras.

We explicitly give new group invariant solutions to a class of Riemann-Liouville time fractional evolution systems with variable coefficients. These solutions are derived from every element in an optimal system of Lie algebras generated by infinitesimal symmetries of evolution systems in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions and show that these solutions solve diffusion-wave equations with variable coefficients. These solutions contain previously known solutions as particular cases. Some plots of solutions subject to the order of the fractional derivative are illustrated.

We study a class of nonlinear Schrodinger equations driven by the infinitesimal generators of some general rotationally invariant Levy processes and with general nonlinearity. Regularity, sign-preserving property, and radial symmetry are shown for the solutions to the Schrodinger equations.

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented, and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result, the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance.

We use finite field-dependent BRS transformations (FFBRS) to connect the Green functions in a set of two otherwise unrelated gauge choices. We choose the Lorentz and the axial gauges as examples. We show how the Green functions in axial gauge can be written as a series in terms of those in Lorentz gauges. Our method also applies to operator Green's functions. We show that this process involves another set of related FFBRS transfomations that is derivable from infinitesimal FBRS. We suggest possible applications., 20 pages, LaTex, Section 4 expanded, typos corrected; last 2 references modified; (this) revised version to appear in J. Math. Phys

The complete classification of Lie symmetries for the Kramers equation is described. Group invariance under infinitesimal transformations is used to generate a wide class of exact solutions. Some of the known solutions are obtained as particular cases. The general results can also be used to solve some types of moving-boundary problems. We researched some Q-conditional symmetry properties of this equation.

The self-dual Yang–Mills equation in the (2+1)-dimensional space–time is considered. Binary Darboux transformation of a new kind is applied to obtain the infinite hierarchy of solutions expressed through ones of corresponding Lax pairs on an initial solution of the equation studied. Sufficient conditions are given for the solutions built to take values in the class of unimodular positive-definite matrices. A new infinitesimal Darboux transformation is introduced and a particular solution of the linearization of the self-dual Yang–Mills equation is given. Two hierarchies of infinitesimal symmetries depending on a solution of the nonlinear equation are extracted.

We obtain a classification of infinitesimal symmetries of an autonomous higher‐order Lagrangian system by using the theory of lifts of vector fields to tangent bundles. Also, a classification of infinitesimal symmetries of a time‐dependent higher‐order Lagrangian system is obtained. The relationship between infinitesimal symmetries and constants of the motion is investigated.

The paper focuses on studying the Noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales. First, the Hamilton equations of nonholonomic nonconservative systems on time scales are established, which is based on the Lagrange equations for nonholonomic systems on time scales. Then, based upon the quasi-invariance of Hamilton action of systems under the infinitesimal transformations with respect to the time and generalized coordinate on time scale, the Noether identity and the conserved quantity of nonholonomic nonconservative systems on time scales are obtained. Finally, an example is presented to illustrate the application of the results.

Deformations of nilpotent parts of the Neveu–Schwarz superalgebra and of the Ramond superalgebra are studied herein. This study consists of the computation of the second cohomology space with coefficients in the adjoint representation by means of spectral sequences, the explicit construction of infinitesimal deformations, the extension of these infinitesimal deformations to one‐parameter deformations, the realization of these deformations, and the construction of multiparameter deformations.

The nonstandard (infinitesimal) analysis developed by Robinson is applied to certain foundational aspects of superanalysis. The construction of nonstandard hull due to Luxemburg is used to study ‘‘limit phenomena’’ arising in a Grassmann algebra Λ(q) as the number of generators q tends to infinity. An outcome of such an approach is that under transition t→∞ a superspace of purely odd dimension (0,q) transforms into a superspace of dimension (∞,∞) and not (0,∞) as one would expect. It is suggested that this construction may have something to do with the problem of adequate mathematical description of collective bosonic effects arising within infinite fermionic dimensions.

Crucial to a knowledge of the perturbations of Robertson–Walker cosmological models are complete sets of functions with which to expand such perturbations. For the open Robertson–Walker cosmology an answer to this question is given. In addition some observations concerning explicit solution by separation of variables of wave equations for spin 1 in a Riemannian space having an infinitesimal line element of which the Robertson–Walker models are a special case are made.

Infinitesimal Backlund transformations are found for the problem of classical general relativity in d dimensions, which by dimensional reduction may be interpreted as gravitation interacting with several Maxwell and scalar fields. The transformations are constructed in terms of the solution of a linear system whose integrability condition yields the original field equations. The Kac–Moody algebra, which generates those transformations, is analyzed.

A nonstandard path space is constructed that gives the mathematically rigorous formulation for the path integral representation of the fundamental solution to the Cauchy problem for the Dirac equation in (1+1)‐dimensional space‐time. Nonstandard analysis makes the mathematical concepts elementary, consequently, the procedures to prove theorems are considerably simplified. A difference scheme with infinitesimal spacing is available in determining the probability distribution over the path‐space and this method is also available for the heat equation and for the free Schrodinger equation.

A general relativistic time‐relaxation model is introduced and compared with the relativistic Boltzmann equation. It is proved that the model has the same features (formal properties of linear operator, positive transport coefficients, etc.) of the Boltzmann equation. It is also shown that the phase speed of infinitesimal disturbances, propagating in a gas described by our model, is less than the speed of light.

Infinitesimal unitarity of representations of the simple real Lie algebras su(2), su(1,1), and so(4) is discussed with respect to scalar products induced by sesquilinear forms on their universal enveloping algebras. Sesquilinear forms are explicitly calculated. The Verma modules, their irreducible quotients, their irreducible submodules, and their infinitesimal unitarizability are discussed.

Birkhoff equations on time scales and Noether theorem for Birkhoffian system on time scales are studied. First, some necessary knowledge of calculus on time scales are reviewed. Second, Birkhoff equations on time scales are obtained. Third, the conditions for invariance of Pfaff action and conserved quantities are presented under the special infinitesimal transformations and general infinitesimal transformations, respectively. Fourth, some special cases are given. And finally, an example is given to illustrate the method and results.

A.A. Kirillov introduced the family algebras in 2000. In this paper we study the noncommutative Poisson bracket P on the classical family algebra. We show that P is the first-order deformation from the classical family algebra to the quantum family algebra. We will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary therefore the deformation is infinitesimally trivial. In the last part of this paper we also talk about Mackey's analogue and the quantization problem of the family algebras., Comment: 19 pages, minor changes of the previous version

An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the $su(2)$ algebra. This has been computed for both the discrete, as well as for the Perelemov's $SU(2)$ coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by $n\in\mathbb{Z}/2$.

p.1-19 Submitted by Edileide Reis (leyde-landy@hotmail.com) on 2015-03-25T14:59:15Z No. of bitstreams: 1 M. C. Bertin.pdf: 543250 bytes, checksum: aa27522ae8bbcfc40230d1be9ab82f56 (MD5) Approved for entry into archive by Alda Lima da Silva (sivalda@ufba.br) on 2015-03-27T20:18:47Z (GMT) No. of bitstreams: 1 M. C. Bertin.pdf: 543250 bytes, checksum: aa27522ae8bbcfc40230d1be9ab82f56 (MD5) Made available in DSpace on 2015-03-27T20:18:47Z (GMT). No. of bitstreams: 1 M. C. Bertin.pdf: 543250 bytes, checksum: aa27522ae8bbcfc40230d1be9ab82f56 (MD5) Previous issue date: 2014 In this paper, we study singular systems with complete sets of involutive constraints. The aim is to establish, within the Hamilton-Jacobi theory, the relationship between the Frobenius’ theorem, the infinitesimal canonical transformations generated by constraints in involution with the Poisson brackets, and the lagrangian point (gauge) transformations of physical systems.

The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algebres de Lie nilpotentes. Application a l’etude de la variete des algebres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.

Let uk(2) be the infinitesimal quantum gl2 over k, where k is a field containing an lth primitive root e of 1 with l ⩾ 3 odd. We will determine the basic algebra for End uk(2)(Ωk⊗r), where Ωk is the natural module for uk(2).

We study the problem of the infinitesimal deformations of all real, simple, finite-dimensional Filippov (or n-Lie) algebras, considered as a class of n-Leibniz algebras characterized by having an n-bracket skewsymmetric in its n-1 first arguments. We prove that all n>3 simple finite-dimensional Filippov algebras are rigid as n-Leibniz algebras of this class. This rigidity also holds for the Leibniz deformations of the semisimple n=2 Filippov (i.e., Lie) algebras. The n=3 simple FAs, however, admit a non-trivial one-parameter infinitesimal 3-Leibniz algebra deformation. We also show that the $n\geq 3$ simple Filippov algebras do not admit non-trivial central extensions as n-Leibniz algebras of the above class., 19 pages, 30 refs., no figures. Some text rearrangements for better clarity, misprints corrected. To appear in J. Math. Phys

It is shown that the method of infinitesimal generators ("Racah's method") can be broadly and systematically formulated as a method applicable to the calculation of reduced coupling coefficients for a generic subalgebra chain G>H, provided the reduced matrix elements of the generators of G and the recoupling coefficients of H are known. The calculation of SO(5)>SO(4) reduced coupling coefficients is considered as an example, and a procedure for transformation of reduced coupling coefficients between canonical and physical subalegebra chains is presented. The problem of calculating coupling coefficients for generic irreps of SO(5), reduced with respect to any of its subalgebra chains, is completely resolved by this approach., 30 pages, 6 figures; to be published in J. Math. Phys

In the framework of general relativistic classical mechanics on a spacetime with absolute time, we classify the infinitesimal symmetries of the classical structure by means of distinguished Lie subalgebras of the Lie algebra of “special phase function.” These subalgebras are crucial also for the classification of infinitesimal quantum symmetries, which will be analyzed in a forthcoming paper.

The result of classical invariant theory commonly referred to as Cayley’s lemma is reviewed. Its analog in the invariant theory of Killing tensors defined in pseudo-Riemannian spaces of constant curvature is formulated and proven. Illustrative examples are provided.

A new method of constructing structure equations of Lie symmetry pseudogroups of differential equations, dispensing with explicit solutions of the (infinitesimal) determining systems of the pseudogroups, is presented, and illustrated by the examples of the Kadomtsev–Petviashvili and Korteweg–de Vries equations.

In this article explicit expressions are obtained for the action of the infinitesimal operators of the principal nonunitary series representations of the groups U(p,q) in a U(p) ×U(q) basis. It is moreover shown how the finite dimensional irreducible representations of the group U(p,q) and the group U(p+q) with respect to a U(p) ×U(q) basis are obtained from the principal nonunitary series representations of the group U(p,q).

In a recent paper [J. Math. Phys. 26, 3080 (1985)], the first of this series, it was shown that velocity‐dependent symmetry mappings of second‐order dynamical systems have a characteristic functional structure which is the same for all dynamical systems. The present paper continues the investigation of this characteristic functional structure. A shortened, improved, conceptionally simpler proof of the existence of the characteristic structure is given. It is shown how this characteristic functional structure may be formally incorporated into a new procedure for the determination of classical (velocity‐independent) symmetries of dynamical systems. In addition to providing insight into symmetry analysis this new procedure is also a practical alternative to existing symmetry methods for those dynamical systems for which there exist sufficient knowledge of the form of the dynamical solution, so that it is feasible to determine constants of motion by inversion (or the converse). Application of the procedure to...

We investigate a structure of continuous invariance transformations connected to the identity transformation. The transformations considered do not necessarily form a group. We clarify the relationship between the infinitesimal invariance transformation and the finite invariance transformation by showing explicitly how the infinitesimal transformations are woven into the finite one. The analysis leads to a new method of finding generators of the invariance group transformation. The results are useful in the study of symmetry properties, or group theoretic structure, of differential equations. We use the results in studying the group properties of the sine‐Gordon equation uxt=sinu, and indicate that the equation is invariant under an infinite number of one‐parameter groups; the groups obtained are of a more general type than that dealt with by Lie. These findings are used to prove the group theoretic origin of the well‐known conservation laws associated with the sine‐Gordon equation.

This paper is a continuation of a previous paper with a similar title [J. Math. Phys. 17, 1345 (1976)]. In this paper we develop further properties of time‐dependent symmetries of dynamical systems expressible in the form (a) Ei(χ,χ,x,t) ≡ Ei(χ1,...,χn; χ1,...,χn; x1,...,xn;t) = 0. Such dynamical symmetries are based upon infinitesimal transformations of the form (b) χi=xi +δxi, δxi≡ξi(x,t) δa, (c) t=t +δt, δt≡ξ0(x,t) δa, which satisfy the condition (d) δEi=0 whenever Ej=0. It is shown that if (ξiA, ξ0A), A=1,...,ρ, is a complete set of solutions of the symmetry equations as determined by (d), then these solutions generate a ρ‐parameter complete group of symmetry mappings, and the group structure implies linear dependency relations between first and second derived time‐dependent constants of motion as obtained by a related integral theorem. The complete groups of time‐dependent symmetry mappings are obtained for all conservative systems (n≳1) with spherically symmetric potentials. These groups are...

Infinitesimal transformations about n‐soliton solutions of sine‐Gordon and modified Korteweg–de Vries equations are obtained using respective Backlund transformations. We also obtain the eigenfunctions of corresponding generalized Zakharov–Shabat systems.

An explicit basis is constructed for the symmetric irreducible representation (irrep) of SU(9)⊇SO(9)⊇SO(5)×SU1(2)×SU2(2). It is also indicated how good angular momentum states can be constructed. The techniques used are based on the well‐known tensor algebra for the infinitesimal generators of simple Lie groups.

The infinitesimal operators of the most degenerate representations of the groups SO*(2n) and SU*(2n) are found in a discrete basis. The structure (composition series) of these representations is studied. The classification of unitary irreducible representations of these groups which belong to most degenerate series is given. The infinitesimal operators of irredicuble unitary representations of SO(2n) in a U(n) basis and of SU(2n) in an Sp(n) basis are found for the cases of highest weights (M, M, 0, ..., 0), M≥0.

It is shown that whenever fields governed by the equations ∂/∂tpα=−δH/δqα, ∂/∂tqα =δH/δpα allow a conservation law of the form ∂ρ/∂t+divJ=0, there exists a corresponding Lie–Backlund infinitesimal contact transformation which leaves the Hamiltonian equations invariant. A condition that an invariant Lie–Backlund infinitesimal contact transformation gives rise to a conservation law is established. Each such transformation, which may involve derivatives of arbitrary order, yields a one‐parameter local Lie group of invariance transformations. The results are established with the aid of a Lie bracket formalism for Hamiltonian fields. They account for a number of recently discovered conservation laws associated with nonlinear time evolution equations.

Using software developed for symbolic integration, infinitesimal symmetries, conserved currents, and first‐ and second‐order Lie–Backlund transformations for the Federbush model are established. Moreover four (x,t)‐dependent Lie–Backlund transformations are constructed leading to infinite hierarchies of Lie–Backlund transformations.

Nonstandard analysis is used to apply the usual concepts of finite dimensional vector spaces in order to define various Gaussian functional measures without using a limiting process. The approach is to define matrices on the infinite dimensional space and use straightforward techniques to determine the properties of the functions the measures are concentrated on. The power of nonstandard analysis allows one to work directly with infinite and infinitesimal quantities and ’’visualize’’ certain sets upon which the basic Gaussian form is or is not infinitesimal. The relation of the choice of Gaussian to the function (process) properties remains heuristic since a proof of the Holder continuity of the Weiner paths is not complete, but remains at a local (infinitesimal) level. However, given the Holder continuity the nonbounded variation property easily follows. Higher derivative Gaussian measures are also easily developed and their analytic properties displayed along with their covariances. By using Fourier analysis on finite abelian groups transferred to the nonstandard universe and applied to hyperfinite abelian groups a rigorous transformation from the discrete form of the Weiner measure to a Fourier series form is accomplished. It is shown here that the functions (processes) are infinitesimally close to those of the discrete version of the measure. The Fourier approach is also extended to more general measures. Finally, some speculations show directions in which this direct approach to Gaussian functional measures can be extended and generalized.

A purely algebraic infinitesimal method for obtaining multiplicity‐free Wigner coefficients is presented. The method is applied to obtain analytic expressions for the complete matrix elements of all O(n) generators. Moreover the structure of these matrix elements in terms of reduced matrix elements, Wigner coefficients, and reduced Wigner coefficients is made explicit. By comparison with the Wigner–Eckart theorem explicit analytic expressions are obtained for the fundamental Wigner coefficients of O(n). Finally the results are presented in a form which is directly analogous with the corresponding results for U(n).

Expressions for the remainder function of a time dependent infinitesimally generated canonical transformation have recently been found by Dewar, who considered the action of the transformation operators on Liouville’s equation. Here an alternate proof of the remainder function expression is given, based on the transformations of particle trajectories. Then, using this expression, a proof of the symmetry‐equals‐invariant theorem is given.

The concept of infinitesimal null isotropy is defined for a Lorentz manifold, in terms of null sectional curvature (as defined by Harris). It is shown that infinitesimal null isotropy is equivalent to infinitesimal spatial isotropy (as defined by Karcher), and that a null‐isotropic space for which null sectional curvature is infinitesimally spatially constant must have a Robertson–Walker metric.

Generalized Holstein–Primakoff realizations deduced by Deenen, Quesne, and Papanicolaou are obtained directly from the algebraic identities satisfied in collective subspaces by the infinitesimal generators of the corresponding dynamical groups.

The converse problem of similarity analysis is discussed for the infinitesimal symmetry transformations of ordinary second‐order differential equations which are nonlinear in x (and may be linear or nonlinear in x). A natural classification of the problem arises, according to the highest order N of nonlinearity in x. The completely general maximal Lie algebra is obtained for the case N≤3. In the case N≥4 one has, besides the system of differential equations for the infinitesimal generators, an extra set of anholonomic constraints, which operates as a symmetry‐breaking mechanism producing a strong reduction in the number of surviving parameters. Miscellaneous examples are given, which illustrate some features of similarity analysis of nonlinear systems. The infinitesimal point transformation symmetries of the Van der Pol oscillator are also briefly discussed.

The unitary continuous representations of U (n,1) and O (n,1) are discussed from the point of view of deformation of IU (n) and IO (n). It is shown that there are two general ways of writing the matrix elements of the infinitesimal generators of the groups U (n,1) and O (n,1). The first one is to write them as either pure real or pure imaginary. The second one is to write them as complex. We show how these different ways are related to each other.

The importance of the non‐pointlike transformations of symmetry is vindicated in relation with the first integrals. A new first integral of a broad class of systems of second order differential equations is obtained out of a symmetry of them without having to impose the restriction that the system be equivalent to a Lagrangian system. The existence of a reciprocal relationship among the local infinitesimal symmetries (l.i.s.) of a Newtonian system of differential equations and the pseudosymmetries of the associated dynamical system is proved. Several applications are developed, and some open problems concerning the dynamical systems of constant divergence are proposed.

A study is made of time−dependent dynamical symmetry mappings of Hamilton’s equations for classical particle systems. The conditions that an infinitesimal mapping (δxA, δt) ≡ (ξA(x,t) δa, ξ0(x,t) δa), A = 1, ..., 2n, be a symmetry mapping are expressed in terms of a ’’symmetry vector’’ ZA(x,t) ≡ ξA − ξ0 ηAB∂BH (x,t) (where ηAB defines the symplectic matrix of phase space). These conditions imply that ξ0 is arbitrary. It is shown that the symmetry deformation of a constant of motion M (x,t) will also be a (’’derived’’) constant of motion (time−dependent related integral theorem). It follows for the case H = H (x) that every time−dependent symmetry deformation of H (x) is a constant of motion, and it is shown conversely that every constant of motion M (x,t) can be expressed as a symmetry deformation of the Hamiltonian, that is, there exists a symmetry vector ZA(x,t) such that M = ZA∂AH. It is found that if ZA(≠0) is a symmetry vector, then M (x,t) ZA will be a (scaled) symmetry vector if and only if M is a ...

An axiomatic framework for relativistic direct‐interaction single channel scattering theories is formulated in terms of two representations U’ and U of the Poincare group. The infinitesimal generators P (momentum), J (angular momentum), E (energy), N (’’boost’’) of U, and P’, J’, E’ N’ of U’ are assumed to be related by the formulas of Bakamjian, Thomas, and Foldy: P’=P, J’=J, E’= (M’2+RP2)1/2, and N’= (E’X+XE’)/2+(J−X×P)(M’+E’)−1, where X is the center‐of‐mass position operator of U given by X=T−P× (J−T×P) M−1(M+E)−1 with T= (E−1N+NE−1)/2 and M’ is a positive operator that commutes with P, J, and X. Then, it is proved that, within the above‐mentioned framework, the Moller operators W±=lim exp(−itE’)exp(itE) for t→±∞ cannot satisfy the cluster decomposition property (also known as separability) except for interactions that vanish if any one of the particles is removed to infinity.

We study the distribution of 2‐plane elements on which infinitesimal parallel displacement of isovectors yields identity. This yields to an algebraic and differential classification and, in the generic case, to a quasimetric naturally associated with the field.

The primary purpose of this paper is to show that infinitesimal velocity‐dependent symmetry mappings [(a) xi =xi +δxi, δxi ≡ ξi(x,x,t)δa with associated change in path parameter (b) t=t+δt, δt ≡ ξ0(x,x,t)] of classical (including relativistic) particle systems (c) Ei(x,x,x,t) =0 are expressible in a form with a characteristic functional structure which is the same for all dynamical systems (c) and is manifestly dependent upon constants of motion of the system. In this characteristic form the symmetry mappings are determined by (d) ξi =Zi(x,x,t) +xiξ0,ξ0 arbitrary; the functions Zi appearing in (d) have the form (e) Zi =BAgiA(C1,...,Cr; t), 0≤r≤2n, A=1,...,2n, where the BA are arbitrary constants of motion and the C’s appearing in the functions giA are specified constants of motion.A procedure is given to determine the giA. For Lagrangian systems it follows that velocity‐dependent Noether mappings are a subclass of the above‐mentioned general symmetry mappings of the form (a)–(e). An analysis of v...

A general formulation is given for deriving a Kac–Moody algebra in the spectrum space from the infinitesimal regular Riemann–Hilbert transform. Such a Kac–Moody algebra can be obtained for many nonlinear systems, e.g., the (super) principal chiral model with or without a Wess–Zumino–Witten term, the sine‐Gordon, Liouville, and KdV equations; reduced two‐dimensional gravity (Belinski–Zakharov gravity), and self‐dual supersymmetric Yang–Mills theories.