Painlevé Balances and Dressing Transformations

CONTENTS Introduction Chapter I. Theta functions. General information § 1. Definition of theta functions and their simplest properties § 2. Theta functions of a single variable § 3. On Abelian tori § 4. Addition theorems for theta functions Chapter II. Theta functions of Riemann surfaces. The Jacobi inversion problem § 1. Periods of Abelian differentials on Riemann surfaces. Jacobi varieties § 2. Abel's theorem § 3. Some remarks on divisors on a Riemann surface § 4. The Jacobi inversion problem. Examples Chapter III. The Baker-Akhiezer function. Applications to non-linear equations § 1. The Baker-Akhiezer one-point function. The Kadomtsev-Petviashvili equation and equations associated with it § 2. The Baker-Akhiezer two-point function. The Schrödinger equation in a magnetic field Chapter IV. Effectivization of the formulae for the solution of KdV and KP equations. Recovery of a Riemann surface from its Jacobi variety. The problem of Riemann and the conjecture of Novikov § 1. The KdV equation. Genus or 2 § 2. The KP equation. Genus 2 and 3 § 3. The KP equation. Genus . Canonical equations of Riemann surfaces § 4. The problem of Riemann on relations between the periods of holomorphic differentials on a Riemann surface and the conjecture of Novikov Chapter V. Examples of Hamiltonian systems that are integrable in terms of two-dimensional theta functions § 1. Two-zone potentials § 2. The problem of Sophie Kovalevskaya § 3. The problems of Neumann and Jacobi. The general Garnier system § 4. Movement of a solid in an ideal fluid. Integration of the Clebsch case. A multi-dimensional solid Appendix. I.M. Krichever. The periodic non-Abelian Toda chain and its two-dimensional generalization References

We introduce and study the concept of a variety of dialgebras which is closely related to the concept of a variety of conformal algebras: Each dialgebra of a given variety embeds into an appropriate conformal algebra of the same variety. In particular, the Leibniz algebras are exactly Lie dialgebras, and each Leibniz algebra embeds into a conformal Lie algebra.

Conformal theory of the two-dimensional O( N) model with ordinary, extraordinary, and special boundary conditions

We define the quasi-compact Higgs GC-bundles over singular curves introduced in our previous paper for the Lie group SL(N). The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of GC at marked points of the curves. We demonstrate that in particular cases, this construction leads to the classical integrable systems of the Hitchin type. The examples of the systems are analogs of the classical Calogero-Sutherland systems related to a simple complex Lie group GC with two types of interacting spin variables. These type models were introduced previously by Feher and Pusztai. We construct the Lax operators of the systems as the Higgs fields defined over a singular rational curve. We also construct the hierarchy of independent integrals of motion. Then we pass to a fixed point set of real involution related to one of the complex structures on the moduli space of the Higgs bundles. We prove that the number of independent integrals of motion is equal to the half of dimension of the fixed point set. The latter is a phase space of a real completely integrable system. We construct the classical r-matrix depending on the spectral parameter on a real singular curve, and in this way we prove the complete integrability of the system. We present three equivalent descriptions of the system and establish their equivalence.

This paper is a survey of results (partly obtained by the authors) on homogeneous spaces of Lie groups \(G\) with a compact stabilizer subgroup \(H\), on which every \(G\)-invariant distribution is integrable. It is proved that the condition of integrability is necessary and sufficient for every invariant inner metric to be (holonomic) Finsler on such a space. As a corollary of the obtained results, we assert that the class of homogeneous spaces with invariant non-holonomic Riemannian metrics (in other terms, sub-Riemannian or Carnot–Caratheodory metrics), which were actively studied last 3 decades after Gromov’s work, is rather broad. On the other hand, the class of homogeneous spaces with integrable invariant distributions includes Cartan’s symmetric spaces as well as isotropy irreducible, in particular, strictly isotropy irreducible, homogeneous spaces, which have been classified in simply connected case in the papers by Wang and Ziller (respectively, by Manturov, Wolf and Kramer). Special attention is paid to the case, when the Lie groups \(G\) and \(H\) are connected. Then the integrability condition of the invariant distributions is equivalent to a purely algebraic condition, that for the Lie algebra \(h\) of the subgroup \(H\), any \(ad(h)\)-invariant vector subspace in the Lie algebra \(g\) of the Lie group \(G\) is a Lie subalgebra; such Lie subalgebra \(h\subset g\) is called a strong subalgebra. The first author proved that a simply connected and compact space \(G/H\) with this condition is isomorphic to a direct product of strictly isotropy irreducible homogeneous spaces. In line with this, the second author recently found several non-compact simply connected homogeneous spaces with this condition, which are not isomorphic to such direct products. These results are naturally related to the structure questions of a class of general homogeneous locally compact spaces with an inner metric. This class is exactly the closure in the Gromov–Hausdorff sense of the class of homogeneous manifolds with an inner metric. Any such manifold is isometric to some homogeneous manifold \(G/H\) with \(G\)-invariant (may be, non-holonomic) Finsler metric. The authors give fairly detailed survey of the existing methods of the search of geodesics, i.e., locally shortest arcs, on such manifolds (particularly, with invariant non-holonomic Riemannian metrics), non-holonomic metric geometry and its relations with the geometric group theory, \(CR\)-manifolds, thermodynamics, etc. Some unsolved problems are suggested.

Weights for symmetric and general linear groups

Reductive homogeneous Lorentzian manifolds

We compute the Borel equivariant cohomology ring of the left K-action on a homogeneous space G=H , where G is a connected Lie group, H and K are closed connected subgroups, and 2 as well as the torsion primes of the Lie groups are units of the coefficient ring.As a special case, this gives the singular cohomology rings of biquotients H n G=K.This depends on a version of the Eilenberg-Moore theorem developed in the appendix, where a novel multiplicative structure on the two-sided bar construction B.A 00 ; A; A 0 / is defined, valid when A 00A !A is a pair of maps of homotopy Gerstenhaber algebras.16U80, 55N91, 57T15, 57T30; 16E45, 55T20, 57T35 Homogeneous spaces, which can be realized as coset spaces G=H for G a transitively acting Lie group and H the stabilizer of a point, are arguably the most highly symmetric, most canonical, and most thoroughly investigated objects of study in differential geometry after Lie groups.A generalization of perennial interest, which offers many interesting examples in positive-curvature geometry, is the class of biquotients, the orbit spaces KnG=H of G under free left-right actions by products K H of two closed subgroups.The cohomology rings of a Lie group G over Q and those finite fields F p for which H .GI Z/ lacks p-torsion have been known to be exterior algebras since the fundamental 1941 work of Hopf [17], and that of a homogeneous space G=H with connected stabilizer H has been known over R since work of Henri Cartan from 1950 [9]: H .G=H / Tor H .BG/ .R; H .BH//: In his 1952 dissertation [6, Section 30], Borel used the Serre spectral sequence of the Borel fibration G ! G=H !BH to show the same ring isomorphism also holds when H is of maximal rank, with F p coefficients if H .GI Z/ and H .H I Z/ lack p-torsion and with Z coefficients if they are torsion free.Beyond this, however, ring isomorphisms proved hard to find.Starting with Baum's 1962 dissertation, a program began to obtain a more general result, additively at least, using the then-new Eilenberg-Moore spectral sequence of the fibration G=H !BH !BG.Cartan's result implies this spectral sequence's collapse over R, and Baum proved collapse under hypotheses covering the best-studied homogeneous spaces.Subsequent work of authors including Baum [2], Larry Smith [30], Gugenheim and May [15],

Jac(S) is a very important but quite special example, which, at the same time, is an algebraic variety. From Kodaira embedding theorem, one easily infers the following fact known already to Frobenius.

The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective subma- nifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the con- ventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective vari- able differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., (external parameter)-dependent Hartree-Fock (HF) theory. Toward such an ultimate goal, the -HF theory has been recon- structed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent ex- pansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a -dependent potential with a -periodicity. A bilinear equation for the -HF theory has been transcribed onto the corresponding -function using the regu- lar representation for the group and the Schur-polynomials. The -HF SCF theory on an infinite-dimensional Fock space F1 leads to a dynamics on an infinite-dimensional Grass- mannian Gr1 and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr1 which is affiliated with the group manifold obtained by reducting gl(1) to sl(N) and su(N). As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model.

We classify possible boundary conditions of a 6d Dirac fermion Ψ on a rectangle under the requirement that the 4d Lorentz structure is maintained, and derive the profiles and spectrum of the zero modes and nonzero KK modes under the two specific boundary conditions, (i) 4d-chirality positive components being zero at the boundaries and (ii) internal chirality positive components being zero at the boundaries. In the case of (i), twofold degenerated chiral zero modes appear which are localized towards specific directions of the rectangle pointed by an angle parameter θ. This leads to an implication for a new direction of pursuing the origin of three generations in the matter fields of the standard model, even though triple-degenerated zero modes are not realized in the six dimensions. When such 6d fermions couple with a 6d scalar with a vacuum expectation value, θ contributes to a mass matrix of zero-mode fermions consisting of Yukawa interactions. The emergence of the angle parameter θ originates from a rotational symmetry in the degenerated chiral zero modes on the rectangle extra dimensions since they do not feel the boundaries. In the case of (ii), this rotational symmetry is promoted to the two-dimensional conformal symmetry though no chiral massless zero mode appears. We also discuss the correspondence between our model on a rectangle and orbifold models in some details.

On Symmetric Cauchy–Riemann Manifolds

The sign-regularity properties of Green's functions for a special class of mixed boundary-value problems

In this research, a new analytical solution for finding the buckling loads of thin orthotropic rectangular plates with different boundary conditions is derived. The methods currently known in the literature for finding the buckling loads of plates are mainly numerical. Although some plates with specific boundary conditions have analytical solutions, a comprehensive analytical method providing analytical solutions that fit all possible combinations of boundary conditions is lacking. The solution method in this study is based on the development of a static solution for a plate. The physical meaning of buckling is the loss of stiffness, and it is found as the value of the in-plane loading intensity at which a zero force on the plate surface will generate infinite displacement. The solution is obtained in series form, and the coefficients are solved to match the edge conditions. The solution presented for this plate will fit different boundary conditions of the deflection, slope, shear force, and bending moment along the common edges of neighboring plates. Using this new method, exact buckling loads and the buckling modes of many new cases of classical boundary conditions are found (cases involving clamped, simply supported, guided, and free boundaries). Results are given for several stiffness ratios in both directions of the plate, as well as for unidirectional and several cases of bidirectional loading.

Model development and analysis of the evolution of coal permeability under different boundary conditions