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This paper is devoted to the horizontal (``characteristic'') cohomology of systems of differential equations. Recent results on computing the horizontal cohomology via the compatibility complex are generalized. New results on the Vinogradov C-spectral sequence and the Krasil'shchik C-cohomology are obtained. As an application of general theory, the examples of an evolution equation and a p-form gauge theory are explicitly worked out.
We study a 2-parametric family of probability measures on an infinite-dimensional simplex (the Thoma simplex). These measures originate in harmonic analysis on the infinite symmetric group (S.Kerov, G.Olshanski and A.Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773-778). Our approach is to interpret them as probability distributions on a space of point configurations, i.e., as certain point stochastic processes, and to find the correlation functions of these processes. In the present paper we relate the correlation functions to the solutions of certain multidimensional moment problems. Then we calculate the first correlation function which leads to a conclusion about the support of the initial measures. In the appendix, we discuss a parallel but more elementary theory related to the well-known Poisson-Dirichlet distribution. The higher correlation functions are explicitly calculated in the subsequent paper (A.Borodin, math.RT/9804087). In the third part (A.Borodin and G.Olshanski, math.RT/9804088) we discuss some applications and relationships with the random matrix theory. The goal of our work is to understand new phenomena in noncommutative harmonic analysis which arise when the irreducible representations depend on countably many continuous parameters.
We give a summary of the results from Parts I-V (math.RT/9804086, math.RT/9804087, math.RT/9804088, math.RT/9810013, math.RT/9810014). Our work originated from harmonic analysis on the infinite symmetric group. The problem of spectral decomposition for certain representations of this group leads to a family of probability measures on an infinite-dimensional simplex, which is a kind of dual object for the infinite symmetric group. To understand the nature of these measures we interpret them as stochastic point processes on the punctured real line and compute their correlation functions. The correlation functions are given by multidimensional integrals which can be expressed in terms of a multivariate hypergeometric series (the Lauricella function of type B). It turns out that after a slight modification (`lifting') of the processes the correlation functions take a common in Random Matrix Theory (RMT) determinantal form with a certain kernel. The kernel is expressed through the classical Whittaker functions. It depends on two parameters and admits a variety of degenerations. They include the well-known in RMT sine and Bessel kernels as well as some other Bessel-type kernels which, to our best knowledge, are new. The explicit knowledge of the correlation functions enables us to derive a number of conclusions about the initial probability measures. We also study the structure of our kernel; this finally leads to a constructive description of the initial measures. We believe that this work provides a new promising connection between RMT and Representation Theory.
We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed W-algebras. We show that the q-characters give rise to a homomorphism from the Grothendieck ring of representations of a quantum affine algebra to a polynomial ring. We conjecture that the image of this homomorphism is equal to the intersection of certain "screening operators". We also discuss the connection between q-characters and Bethe Ansatz.
We suggest an hierarchy of all the results known so far about the connection of the asymptotics of combinatorial or representation theoretic problems with ``beta=2 ensembles'' arising in the random matrix theory. We show that all such results are, essentially, degenerations of one general situation arising from so-called generalized regular representations of the infinite symmetric group.
This paper is a review on recently found connection between geodesically equivalent metrics and integrable geodesic flows. Suppose two different metrics on one manifold have the same geodesics. We show that then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution and vice versa. The quantum version of this result is also true. As a corollary we have that the topology of the manifold (with two different metrics with the same geodesics) must be sufficiently simple. Also we have that the non-proportionality of the metrics at a point induces the non-proportionality of the metrics at almost all points.
We obtain explicitly all solutions of the SU(infinity) Toda field equation with the property that the associated Einstein-Weyl space admits a 2-sphere of divergence-free shear-free geodesic congruences. The solutions depend on an arbitrary holomorphic function and give rise to new hyperKahler and selfdual Einstein metrics with one dimensional isometry group. These metrics each admit a compatible hypercomplex structure with respect to which the symmetries are triholomorphic.
We consider the nonstandard constrained KP (ncKP) hierarchy which is obtained from the multi-constraint KP hierarchy by gauge transformation. The second Hamiltonian structure of the ncKP hierarchy can be simplified by factorizing the Lax operator into multiplication form, thus the generalized Miura transformation is obtained. We also discuss the free field realization of the associated W-algebra.
A new method to construct Hamiltonian functions in involution is presented. We show that on left-symmetric algebras a Nijenhuis-tensor is given in a natural manner by the usual right-multiplication. Furthermore we prove that symplectic Lie-algebras carry the structure of a Poisson-Nijenhuis manifold.
We propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parameterization of the cotangent bundle over GL(N,C). In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical $\bar{r}$-matrix. Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system. Using the dual parameterization of the cotangent bundle we also derive the algebra for the L-operator of the trigonometric Calogero-Moser system.
We integrate nonabelian Toda field equations for root systems of types A, B, C, for functions with values in any associative algebra. The solution is expressed via quasideterminants. In the appendix we review some results concerning nonabelian soliton equations.
In this paper we study the asymptotic behavior of the Jack rational functions as the number of variables grows to infinity. Our results generalize the results of A. Vershik and S. Kerov obtained in the Schur function case (theta=1). For theta=1/2,2 our results describe approximation of the spherical functions of the infinite-dimensional symmetric spaces $U(\infty)/O(\infty)$ and $U(2\infty)/Sp(\infty)$ by the spherical functions of the corresponding finite-dimensional symmetric spaces.
For each of the classical Lie algebras $g(n)=o(2n+1), sp(2n), o(2n)$ of type B, C, D we consider the centralizer of the subalgebra $g(n-m)$ in the universal enveloping algebra $U(g(n))$. We show that the $n$th centralizer algebra can be naturally projected onto the $(n-1)$th one, so that one can form the projective limit of the centralizer algebras as $n\to\infty$ with $m$ fixed. The main result of the paper is a precise description of this limit (or stable) centralizer algebra, denoted by $A_m$. We explicitly construct an algebra isomorphism $A_m=Z\otimes Y_m$, where $Z$ is a commutative algebra and $Y_m$ is the so-called twisted Yangian associated to the rank $m$ classical Lie algebra of type B, C, or D. The algebra $Z$ may be viewed as the algebra of virtual Laplace operators; it is isomorphic to the algebra of polynomials with countably many indeterminates. The twisted Yangian $Y_m$ (and hence the algebra $A_m$) can be described in terms of a system of generators with quadratic and linear defining relations which are conveniently presented in R-matrix form involving the so-called reflection equation. This extends the earlier work on the type A case by the second author.
Invariants of nonlinear gauge transformations of a family of nonlinear Schrödinger equations proposed by Doebner and Goldin are used to characterize the behaviour of exact solutions of these equations.
Using algebraic tools of supersymmetric quantum mechanics we construct classes of conditionally exactly solvable potentials being the supersymmetric partners of the linear or radial harmonic oscillator. With the help of the raising and lowering operators of these harmonic oscillators and the SUSY operators we construct ladder operators for these new conditionally solvable systems. It is found that these ladder operators together with the Hamilton operator form a non-linear algebra which is of quadratic and cubic type for the SUSY partners of the linear and radial harmonic oscillator, respectively.
The outlook of a simple method to generate localized (soliton-like) potentials of time-dependent Schrodinger type equations is given. The conditions are discussed for the potentials to be real and nonsingular. For the derivative Schrodinger equation also is discussed its relation to the Ishimori-II model. Some pecular soliton solutions of nonlinear Schrodinger type equations are given and discussed.
In this article we present first an algorithm for calculating the determining equations associated with so-called ``nonclassical method'' of symmetry reductions (a la Bluman and Cole) for systems of partial differentail equations. This algorithm requires significantly less computation time than that standardly used, and avoids many of the difficulties commonly encountered. The proof of correctness of the algorithm is a simple application of the theory of Grobner bases. In the second part we demonstrate some algorithms which may be used to analyse, and often to solve, the resulting systems of overdetermined nonlinear PDEs. We take as our principal example a generalised Boussinesq equation, which arises in shallow water theory. Although the equation appears to be non-integrable, we obtain an exact ``two-soliton'' solution from a nonclassical reduction.
Generalizing a construction of P. Vanhaecke, we introduce a large class of degenerate (i.e., associated to a degenerate Poisson bracket) completely integrable systems on (a dense subset of) the space $\R^{2d+n+1}$, called the generalized master systems. It turns out that certain generalized master systems (with different Poisson brackets and different Hamiltonians) determine the same Hamiltonian vector fields (and are therefore different descriptions of the same Hamiltonian system), and that the Poisson brackets of these systems are compatible. Consequently, our class of generalized master systems actually consists of a (smaller) class of completely integrable systems, and our construction yields a multi-Hamiltonian structure for these systems. As an application, we construct a multi-Hamiltonian structure for the so-called master systems introduced by D. Mumford.