raising and lowering operators for uq gl n a set of raising and lowering operators are constructed for uq gl n being hermitian conjugates of one another they are easily normalized and allow us to express arbitrary gelfand tzetlin basis states as a product of lowering operators acting on the maximal weight state

new liouville integrable noncanonical hamiltonian systems from the akns spectral problem liouville integrable noncanonical hamiltonian systems with variable coefficient symplectic forms are generated from the akns 3 3 matrix lax pairs by binary symmetry constraints of the akns hierarchy these integrable systems provide new integrable factorization for every akns system in the hierarchy

periodic and partially periodic representations of su n q the gelfand zetlin basis is adapted tosu n q forq a root of unit extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center a crucial identity is derived and proved which guarantees the periodicity of the action of the generators full periodicity is relaxed by stages some raising and lowering operators remaining injective while others become nilpotent with corresponding changes in the dimension of the representation in the extreme case of highest weight representations all the raising and lowering operators are nilpotent as an alternative approach an auxiliary algebra giving all the periodic representations is presented an explicit solution of this system forn 3 while fully equivalent to the g z basis turns out to be much simpler

extension of hereditary symmetry operators two models of candidates for hereditary symmetry operators are proposed and thus many nonlinear systems of evolution equations possessing infinitely many commuting symmetries may be generated some concrete structures of hereditary symmetry operators are carefully analysed on the basis of the resulting general conditions and several corresponding nonlinear systems are explicitly given as illustrative examples

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

soliton fission and fusion of a new two component korteweg de vries kdv equation abstract in this paper the painleve test is performed for a new two component korteweg de vries kdv equation proposed by foursov it is shown that this equation passes the integrability test and is p integrable by means of the truncated singular expansion some explicit solutions from the trivial zero solution are derived the phenomena of soliton fission and fusion are studied in detail

matrix elements and wigner coefficients for uq gl n in this series of papers the fundamental wigner coefficients and the matrix elements of the generators for the quantum group uq gl n are derived the uq gl n uq gl n 1 reduced wigner coefficients and reduced matrix elements are determined algebraically as eigenvalues of certain uq gl n 1 invariants in the quantum group uq gl n the matrix elements of the elementary quantum group generators in the gel fand tsetlin basis are derived below while all fundamental wigner coefficients are derived in the second paper of the series

exact evaluation of the collapse phase boundary for two dimensional directed polymers the phase boundary of the collapsed phase for a directed polymer on a square lattice in the presence of an attractive wall and monomer monomer interactions is calculated exactly using transfer matrix techniques the position of the multicritical point is also identified

biorthogonal coupling coefficients of uq n the coupling wigner clebsch gordan coefficients and their isofactors for the unitary quantum algebras uq n with the repeating irreducible representations in the coproduct decomposition are considered generalizing the u n case the biorthogonal systems of the uq n isofactors with the dual multiplicity labels are constructed by means of the recoupling technique in terms of the isofactors with simpler multiplicity structure a first construction correlated with the inverted littlewood richardson rules gives the bilinear combinations of isofactors after applying the proportionality of the q recoupling racah coefficients to the boundary q isofactors an alternative recursive construction gives the nonorthogonal q isofactors satisfying the most elementary boundary conditions and proportional to the uq n 1 recoupling coefficients for some less restricted values of parameters some multiplicity free and more general uq n recoupling coefficients are found the blocks bilinear combinations of which equal to the resubducing coefficients of the complementary chains of q algebras are proposed to use for the orthonormalization of some uq n biorthogonal isofactors including the general uq 3 case

multiplicity free uq n coupling coefficients the coupling wigner clebsch gordan coefficients of the unitary quantum algebras uq n are considered the tensorial properties of the generator powers and their ordered products used in the explicit projectors and weight lowering procedures are established different expressions for the multiplicity free isofactors of uq n coupling coefficients those coupling an arbitrary and symmetric representations are derived explicit expression of the arbitrary isofactors in terms of their boundary values are proposed proportionality of the semistretched isofactors to the stretched q recoupling coefficients of uq n 1 q analogues of 9j symbols is demonstrated the stretched isofactors of uq 3 are expressed in terms of clebsch gordan coefficients of uq 2

on a nonlinear hyperbolic variational equation i global existence of weak solutions we study the nonlinear hyperbolic partial differential equation u t uux x 1 2u x 2 this partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles in particular it describes waves in a massive director field of a nematic liquid crystal global smooth solutions of the partial differential equation do not exist since their derivatives blow up in finite time while weak solutions are not unique we therefore define two distinct classes of admissible weak solutions which we call dissipative and conservative solutions we prove the global existence of each type of admissible weak solution provided that the derivative of the initial data has bounded variation and compact support these solutions remain continuous despite the fact that their derivatives blow up there are no a priori estimates on the second derivatives in any l p space so the existence of weak solutions cannot be deduced by using sobolev type arguments instead we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation we also describe the qualitative properties of the partial differential equation including a comparison with the burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions we show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics and we prove that the large time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink wave

two integrable hamiltonian hierarchies in sl 2 r and so 3 r with three potentials by introducing two specific matrix spectral problems associated with sl 2 r and so 3 r matrix lie algebras we generate two integrable hamiltonian hierarchies with three potentials the computation and analysis on their hamiltonian structures by means of the trace identity show that the resulting hierarchies are liouville integrable namely that each hierarchy consists of commuting hamiltonian soliton equations

generalized gel fand invariants and characteristic identities for quantum groups the generalized gel fand invariants for an arbitrary quantum group are explicitly constructed and their eigenvalues in any irreducible representation are computed these invariants enable one to develop characteristic identities for the quantum group and as a natural application these identities are used to construct projection operators for tensor products of representations to illustrate the general theory the quantum group uq gl m is studied in detail

variational identities and applications to hamiltonian structures of soliton equations abstract this is an introductory report concerning our recent research on hamiltonian structures we will discuss variational identities associated with continuous and discrete spectral problems and their applications to hamiltonian structures of soliton equations our illustrative examples are the akns hierarchy and the volterra lattice hierarchy associated with semisimple lie algebras and two hierarchies of their integrable couplings associated with non semisimple lie algebras the resulting hamiltonian structures generate infinitely many commuting symmetries and conservation laws for the four soliton hierarchies the presented variational identities can be applied to hamiltonian structures of other soliton hierarchies

the string equation and the τ function witt constraints for the discrete kadomtsev petviashvili hierarchy using the tool of pseudo difference operators the string equation for the discrete kadomtsev petviashvili kp hierarchy is introduced and a general expression for the string equation is formulated furthermore the operators of the constraints that the string equation imposes on the function of the p reduced dkp hierarchy are presented the algebra which part of the constraints spans and eigenvalues of part of the constraints corresponding to the function are calculated it is also shown that this algebra is exactly a witt algebra not only for p 2 but also for a general p n and that the function of the p reduced dkp hierarchy constrained by the string equation is a vacuum vector of a witt algebra

lax representations and lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations a method of constructing lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed it is shown that under some simple but essential conditions the corresponding lax operators may constitute infinite dimensional lie algebras with respect to the binary operation given in this paper furthermore the detailed analyses for the kdv couple the akns couple and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples

integrable couplings of vector akns soliton equations by enlarging the associated matrix spectral problems two specific classes of multicomponent integrable couplings of the physically important vector akns soliton equations are constructed which can be linked to each other by a backlund transformation the resulting two hierarchies of integrable couplings possess the enlarged zero curvature representations and recursion structures and thus each system in the two hierarchies has infinitely many commuting symmetries

a new fractional order soliton equation hierarchy and its integrable coupling system a fractional order soliton equation hierarchy is presented which is a new liouville integrable hierarchy and possesses bi hamiltonian structure finally we obtain a integrable coupling system of this fractional order soliton equation hierarchy by applying a upper triangular matrix of lie algebras

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

an integrable coupling hierarchy of the mkdv_integrable systems its hamiltonian structure and corresponding nonisospectral integrable hierarchy a four by four matrix spectral problem is introduced locality of solution of the related stationary zero curvature equation is proved an integrable coupling hierarchy of the mkdv integrable systems is presented the hamiltonian structure of the resulting integrable coupling hierarchy is established by means of the variational identity it is shown that the resulting integrable couplings are all liouville integrable hamiltonian systems ultimately through the nonisospectral zero curvature representation a nonisospectral integrable hierarchy associated with the resulting integrable couplings is constructed

symmetry constraint of mkdv equations by binary nonlinearization abstract a symmetry constraint for mkdv integrable hierarchy is presented by binary nonlinearization the spatial part and the temporal parts of the lax pairs and the adjoint lax pairs of mkdv equations are all constrained as finite dimensional liouville integrable hamiltonian systems whose integrals of motion are explicitly given in terms of the proposed symmetry constraint mkdv equations are decomposed into two finite dimensional liouville integrable constrained systems and thus a kind of separation of variables for mkdv equations is established

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

a supertrace identity and its applications to superintegrable systems a supertrace identity on lie superalgebras is established it provides a tool for constructing super hamiltonian structures of zero curvature equations associated with lie superalgebras applications in the case of the lie superalgebra b 0 1 present super hamiltonian structures of a super akns soliton hierarchy and a super dirac soliton hierarchy

reduced wigner coefficients for uq gl n this is the second paper in a series concerned with the racah wigner calculus for the quantum group uq gl n in this approach all reduced wigner coefficients are determined algebraically by the eigenvalues of certain generalized casimir invariants here the method is applied to evaluate all fundamental wigner coefficients and reduced wigner coefficients for uq gl n

the algebraic structure of zero curvature representations and application to coupled kdv systems the author first establishes an algebraic structure related to zero curvature representations and propose a new approach for calculating symmetry algebras of integrable systems then he deduces a hierarchy of nonisospectral flows associated with coupled kdv systems from a spectral problem with the laurent polynomial dependent form of the spectral parameter furthermore the commutator relations of lax operators corresponding to isospectral and nonisospectral flows are worked out according to this algebraic structure and thus a symmetry algebra for coupled kdv systems is engendered from this general theory

k symmetries and tau symmetries of evolution equations and their lie algebras on the basis of tu s work 1951 the author establishes a more general skeleton on k symmetries and tau symmetries of evolution equations and their lie algebraic structures and discuss carefully the corresponding symmetries and lie algebras for kdv and jaulent miodek hierarchies

hamiltonian and quasi hamiltonian structures associated with semi direct sums of lie algebras the trace variational identity is generalized to zero curvature equations associated with non semi simple lie algebras or equivalently lie algebras possessing degenerate killing forms an application of the resulting generalized variational identity to a class of semi direct sums of lie algebras in the akns case furnishes hamiltonian and quasi hamiltonian structures of the associated integrable couplings three examples of integrable couplings for the akns hierarchy are presented one hamiltonian and two quasi hamiltonian

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures