On Integrable roots in split Lie triple systems

We focus on the notion of an integrable root in the framework of split Lie triple systems T with a coherent 0-root space. As a main result, it is shown that if T has all its nonzero roots integrable, then its standard embedding is a split Lie algebra having all its nonzero roots integrable. As a consequence, a local finiteness theorem for split Lie triple systems, saying that whenever all nonzero roots of T are integrable then T is locally finite, is stated. Finally, a classification theorem for split simple Lie triple systems having all its nonzero roots integrable is given.     

Outer automorphisms ofsl(2), integrable systems, and mappings

Outer automorphisms of infinite-dimensional representations of the Lie algebra sl(2) are used to construct Lax matrices for integrable Hamiltonian systems and discrete integrable mappings. The known results are reproduced, and new integrable systems are constructed. Classical r-matrices, corresponding to the Lax representation with the spectral parameter are dynamic. This scheme is advantageous because quantum systems naturally arise in the framework of the classical r-matrix Lax representation and the corresponding quantum mechanical problem admits a variable separation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 2, pp. 205–216, February, 1999.     

Nonlinear continuous integrable Hamiltonian couplings

Based on a kind of special non-semisimple Lie algebras, a scheme is presented for constructing nonlinear continuous integrable couplings. Variational identities over the corresponding loop algebras are used to furnish Hamiltonian structures for the resulting continuous integrable couplings. The application of the scheme is illustrated by an example of nonlinear continuous integrable Hamiltonian couplings of the AKNS hierarchy of soliton equations.     

Conservation laws and self-consistent sources for a super integrable equation hierarchy

In this paper, a super integrable equation hierarchy is considered based on a Lie superalgebra and supertrace identity. Then, a super integrable equation hierarchy with self-consistent sources is established. Furthermore, we introduce two variables F and G to construct conservation laws of the super integrable equation hierarchy and the first two conserved densities and fluxes are listed. It would be specially mentioned that the Fermi variables play an important role in super integrable systems which is different from the ordinary integrable systems.     

Factorization of the Loop Algebra and Integrable Toplike Systems

With any Lie algebra of Laurent series with coefficients in a semisimple Lie algebra and its decomposition into a sum of the subalgebra consisting of the Taylor series and a complementary subalgebra, we associate a hierarchy of integrable Hamiltonian nonlinear ODEs. In the case of the so(3) Lie algebra, our scheme covers all classical integrable cases in the Kirchhoff problem of the motion of a rigid body in an ideal fluid. Moreover, the construction allows generating integrable deformations for known integrable models.     

IntegrableN-dimensional systems on the Hopf algebra andq-deformations

We construct the class of integrable classical and quantum systems on the Hopf algebras describing n interacting particles. We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra A(g) of a simple Lie algebra g and prove that the integrals of motion depend only on linear combinations of k coordinates of the phase space, 2·ind g≤k≤g·ind g, whereind g andg are the respective index and Coxeter number of the Lie algebra g. The standard procedure of q-deformation results in the quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3, 1). An exact solution for the quantum analogue of the N-dimensional Hamiltonian system on the Hopf algebra A(sl(2)) is constructed using the method of noncommutative integration of linear differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 373–390, September, 2000     

A family of hyperbolic spin Calogero‐Moser systems and the spin Toda lattices

In this paper, we continue to develop a general scheme to study a broad class of integrable systems naturally associated with the coboundary dynamical Lie algebroids. In particular, we present a factorization method for solving the Hamiltonian flows. We also present two important classes of new examples, a family of hyperbolic spin Calogero‐Moser systems and the spin Toda lattices. To illustrate our factorization theory, we show how to solve these Hamiltonian systems explicitly. © 2004 Wiley Periodicals, Inc.     

Applications of a few Lie algebras

Two isomorphic groups R ² andM are firstly constructed. Then we extend them into the differential manifold R ²ⁿ and n products of the group M for which four kinds of Lie algebras are obtained. By using these Lie algebras and the Tu scheme, integrable hierarchies of evolution equations along with multi-component potential functions can be generated, whose Hamiltonian structures can be worked out by the variational identity. As application illustrations, two integrable Hamiltonian hierarchies with 4 component potential functions are obtained, respectively, some new reduced equations are followed to present. Specially remark that the integrable hierarchies obtained by taking use of the approach presented in the paper are not integrable couplings. Finally, we generalize an equation obtained in the paper to introduce a general nonlinear integrable equation with variable coefficients whose bilinear form, B¨acklund transformation, Lax pair and infinite conserved laws are worked out, respectively, by taking use of the Bell polynomials.     

Lie algebras and Lie super algebra for the integrable couplings of NLS–MKdV hierarchy

Lie algebras and Lie super algebra are constructed and integrable couplings of NLS–MKdV hierarchy are obtained. Furthermore, its Hamiltonian and Super-Hamiltonian are presented by using of quadric-form identity and super-trace identity. The method can be used to produce the Hamiltonian structures of the other integrable and super-integrable systems.     

Five constructions of integrable dynamical systems connected with the Korteweg-de Vries equation

Two countable sets of integrable dynamical systems which turn into the Korteweg-de Vries equation in a continous limit are constructed. The integrability of the dynamics of the scattering matrix entries for these systems is proved and an integrable reduction in the finitedimensional case is pointed out. A construction of the integrable dynamical systems connected with the simple Lie algebras and generalizing the discrete kdV equation is presented. Two general constructions of differential and integro-differential equations (with respect to time t) possessing a countable set of first integrals are found. These equations admit the Lax representation in some infinite-dimensional subalgebras of the Lie algebra of integral operators on an arbitrary manifold M ⁿ with measure . A construction of matrix equations having a set of attractors in the space of all matrix entries is given.     

一个新的非等谱可积族和一些相关对称

利用李群$M_nC$的一个子群我们引入一个线性非等谱问题,该问题的相容性条件可导出演化方程的一个非等谱可积族,该可积族可约化成一个广义非等谱可积族.这个广义非等谱可积族可进一步约化成在物理学中具有重要应用的标准非线性薛定谔方程和KdV方程.基于此,我们讨论在广义非等谱可积族等谱条件下的一个广义AKNS族$u_t=K_m(u)$的$K$对称和$\tau$对称.此外,我们还考虑非等谱AKNS族$u_t=\tau_{N+1}^l$的$K$对称和$\tau$对称.最后,我们得到这两个可积族的对称李代数,并给出这些对称和李代数的一些应用,即生成了一些变换李群和约化方程的无穷小算子.     

Yang族一个新的非线性可积耦合及其哈密尔顿结构(英文)

Based on a kind of non-semisimple Lie algebras, we establish a way to construct nonlinear continuous integrable couplings. Variational identities over the associated loop algebras are used to furnish Hamiltonian structures of the resulting continuous couplings.As an illustrative example of the scheme is given nonlinear continuous integrable couplings of the Yang hierarchy.     

LIE GROUPS ADMITTED BY AUTONOMOUSSYSTEM WITH GIVEN STRUCTURECONSTANTS

1IlltroductionandNotationsInthesecondhalfofthepreviouscentury,SophousLieintroducedthetheoryofLiegroups,inordertocreateatheoryofintegratingordinarydifferentialequations.Sofar,Liegrouptheoryhadplayedimportantimpactinintegrabilitytheoryofdifferentialequations(ref.[l,2,3]).Considerthefollowingfirstorderautonomoussystemofnordinarydifferentialequations'wherex=(x',x',''3x")ED,andDisasub-domainofR"(orC"),XifD-- R(orC),XfECoo(D),andtER(orC).Astheclassicalresult,theorderofthesystemcanbereduced…     

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure μ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Lie algebra” of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the L ²(ν)-closure of holomorphic polynomials by their values on the Cameron–Martin subgroup.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras II: A nonlinear system arising from the group G 6.1 and its exact integration

In the first paper of this series a correspondence was established between coupled systems of two-dimensional nonlinear wave equations and the six-dimensional simply transitive Lie algebras. In the present paper we make use of this result to construct a Darboux integrable and exactly integrable nonlinear system associated with the six-parameter nilpotent Lie group G _6,1 and we give its exact general solution in terms of four arbitrary functions. The procedure is shown to be an exact linearization of the nonlinear problem.     

Integrable decomposition of a hierarchy of soliton equations and integrable coupling system by semidirect sums of Lie algebras

Staring from a new spectral problem, a hierarchy of the soliton equations is derived. It is shown that the associated hierarchies are infinite-dimensional integrable Hamiltonian systems. By the procedure of nonlinearization of the Lax pairs, the integrable decomposition of the whole soliton hierarchy is given. Further, we construct two integrable coupling systems for the hierarchy by the conception of semidirect sums of Lie algebras.     

Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type

We consider two classes of integrable nonlinear hyperbolic systems on Lie algebras. These systems generalize the principal chiral model. Each system is related to a pair of compatible Lie brackets and has a Lax representation, which is determined by the direct sum decomposition of the Lie algebra of Laurent series into the subalgebra of Taylor series and the complementary subalgebra corresponding to the pair. New examples of compatible Lie brackets are given.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras I

We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle — the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in correspondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles.     

An algebraic method for classifying S-integrable discrete models

We discuss a method for classifying integrable equations on quad-graphs based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators of the ring generators. This function grows exponentially in the general case. Examples show that it grows more slowly for integrable equations. We propose a classification scheme based on this observation.     

A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups

Let G denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on G that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite-dimensional projections are smooth measures. We prove a unitary equivalence between a subclass of these square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Cameron–Martin” Lie subalgebra. The isomorphism defining the equivalence is given as a composition of restriction and Taylor maps.