| Preface |
| Background Notations |
| Dynamical systems with homogeneous configuration spaces / 1： |
| Dynamical systems with symmetries |
| Poisson structures / 1.1： |
| Lie group actions on manifolds / 1.2： |
| Symplectic structures on coadjoint representation orbits / 1.3： |
| Hamiltonian group actions / 1.4： |
| Invariant Hamiltonian systems with homogeneous configuration spaces / 1.5： |
| The existence of a maximal involutive set of functions on the orbits of semi-simple elements of a semi-simple Lie algebra / 2： |
| The algebra of invariant polynomials on a semi-simple Lie algebra / 2.1： |
| Semi-simple element orbits / 2.2： |
| Maximal involutive sets of functions on semi-simple elements orbits / 2.3： |
| The integrability criterion and spherical pairs of Lie groups / 3： |
| Criterion of integrability / 3.1： |
| Spherical pairs of complex Lie groups / 3.2： |
| Interpolation property of spherical pairs of compact Lie groups / 4： |
| Properties of spherical pairs of compact Lie algebras / 4.1： |
| Points in general position / 4.2： |
| Spherical pairs of classical simple Lie groups / 5： |
| Preliminary remarks / 5.1： |
| Involutions of simple Lie algebras / 5.2： |
| Spherical pairs of classical Lie algebras / 5.3： |
| Classification of spherical pairs of the exceptional simple Lie algebras / 6： |
| Classification of spherical pairs of semi-simple Lie groups / 7： |
| Spherical pairs of semi-simple Lie algebras / 7.1： |
| Geometric quantization and integrable dynamical systems |
| Connections on line bundles |
| Line bundles |
| Equivalence classes of line bundles |
| Connections |
| The integrality condition |
| Hermitian structures |
| Equivalence classes of line bundles with connections / 1.6： |
| Holomorphic line bundles / 1.7： |
| Derivations / 1.8： |
| Tensor products, square roots and invariant Hermitian structures / 1.9： |
| Parallel transport on a line bundle with a connection / 1.10： |
| Parallel transport and derivations / 1.11： |
| Flat partial connections |
| Flat F-connections |
| Maps of line bundles |
| Line bundles of differential forms |
| Geometric quantization |
| Polarizations |
| Geometric quantization and reduction |
| Introduction |
| Hamiltonian reduction |
| Quantization / 4.3： |
| Examples: geometric quantization of the oscillator type Hamiltonian systems |
| Geometric quantization of the generalized n-dimensional harmonic oscillator |
| Geometric quantization of geodesic flow on a sphere |
| Geometric quantization of the multidimensional Kepler problem |
| Geometric quantization of the MIC-Kepler problem / 5.4： |
| Structures on manifolds and algebraic integrability of dynamical systems |
| Poisson structures and dynamical systems with symmetries |
| Preliminary notes |
| Poisson structures on manifolds |
| Casimir functions and involution |
| Casimir functions associated with classical expansions of semi-simple Lie algebras |
| Alder-Kostant-Symes and Mishchenko-Fomenko theorems |
| The reduction method and Poisson structures on dual spaces of semi-direct sums of Lie algebras |
| The mapping canonicity of symplectic structures |
| Momentum mapping |
| Poisson structures on dual spaces of semi-direct sums of Lie algebras |
| Canonical mappings / 2.4： |
| Nonlinear Neumann type dynamical systems as integrable flows on coadjoint orbits of Lie groups |
| The Neumann problem |
| The Lie-Poisson bracket associated with the ad-semidirect sum of Lie algebras |
| The canonical symplectic structure on T(S[superscript n-1]) and its diffeomorphisms / 3.3： |
| An involutive system of integrals for the Neumann dynamical system on the sphere S[superscript n-1] / 3.4： |
| Generalized Neumann-Bogoliubov dynamical systems / 3.5： |
| Abelian integrals, integrable dynamical systems, and their Lax type representations |
| The Neumann-Rosochatius-Bogoliubov Hamiltonian system |
| Conservation laws |
| Lax type representation |
| Dual momentum mappings and their applications |
| Preliminaries |
| Dualities |
| The Neumann-Rosochatius system |
| The Lie algebraic setting of Benney-Kaup dynamical systems and associated via Moser Neumann-Bogoliubov oscillatory flows |
| The Novikov-Lax finite-dimensional invariant reductions on nonlocal submanifolds / 6.1： |
| The Moser map and its associated dual moment maps into loop Lie algebras / 6.3： |
| The finite-dimensional Moser type of reduction of modified Boussinesq and super-Korteweg-de Vries Hamiltonian systems via the gradient-holonomic algorithm and dual moment maps |
| The Moser type of finite-dimensional reduction of a Boussinesq hydrodynamic system and its Lie-algebraic integrability / 7.2： |
| The Neumann type of oscillatory super-Hamiltonian systems on the sphere S[superscript N] and their Lie algebraic super-integrability / 7.3： |
| Lax-type of flows on Grassmann manifolds and dual momentum mappings / 8： |
| Symplectic structures on loop Grassmann manifolds / 8.1： |
| An intrinsic loop Grassmannian structure and dual momentum mappings / 8.3： |
| On the geometric structure of integrable flows in Grassmann manifolds / 9： |
| Centrally extended symplectic structures and integrable flows on the loop Grassmann manifolds / 9.1： |
| Algebraic methods of quantum statistical mechanics and their applications |
| Current algebra representation formalism in nonrelativistic quantum mechanics |
| The current algebra in nonrelativistic quantum mechanics |
| Current algebra representations |
| Bogoliubov-Araki generating functional |
| Lie current algebra, Hamiltonian operator, and Bogoliubov functional equations |
| Hamiltonian operator |
| Gibbs states and the Kubo-Martin-Schwinger conditions |
| Stable states and the KMS condition |
| Functional-operator representations of the current Lie algebra |
| The Bogoliubov-Bloch functional equation / 2.5： |
| The reconstruction via Araki of the Hamiltonian operator and the Bogoluibov functional equation / 2.6： |
| Functional-operator solutions of the Bogoliubov functional equations / 2.7： |
| A generalized Virasoro algebra / 2.8： |
| The secondary quantization method and the spectrum of quantum excitations of a nonlinear Schrodinger type dynamical system |
| Preliminary notions |
| The second quantization representation |
| A generalized nonlinear Schrodinger type quantum dynamical system |
| The quantum inverse spectral transform method |
| The scattering operator |
| Eigenvalue states of the nonlinear Schrodinger type model / 3.6： |
| Quantum excitations of a bose gas with a positive momentum / 3.7： |
| Unitary representations of the generalized Virasoro algebra |
| Verma modules over the generalized Virasoro current algebra |
| Unitary irreducible modules with highest weight |
| Algebraic and differential geometric aspects of the integrability of nonlinear dynamical systems on infinite-dimensional functional manifolds |
| The current Lie algebra on S[superscript 1] and its functional representations |
| Basic notations |
| Associated cohomology complexes and their properties |
| Differential geometry analysis on real jet manifolds |
| Differential geometry analysis on jet supermanifolds |
| Euler variational derivative, external differential and tensors on infinite-dimensional functional spaces |
| Implectic operators and Poisson structures |
| Dynamical systems and bi-Hamiltonicity |
| The equivalence of dynamical systems as the Backlund transformation |
| The current Lie algebra on a cycle as a symmetry subalgebra of compatibly bi-Hamiltonian nonlinear dynamical systems on an axis |
| The Hamiltonicity of nonlinear dynamical systems on infinite-dimensional functional manifolds |
| A Lie-algebraic algorithm for investigating integrability |
| The gradient holonomic algorithm and Lax type representation |
| An adjoint Lax type equation and conservation laws |
| Lax type representation: differential algebraic approach |
| Lax type representation: geometric approach |
| Lagrangian and Hamiltonian formalisms for reduced infinite-dimensional dynamical systems with symmetries |
| General setting |
| Lagrangian reduction |
| Symplectic analysis and Hamiltonian fields |
| Discrete dynamical systems. One generalization |
| Non-isospectrally integrable dynamical systems: the generalized asymptotic structure of conservation laws |
| A nonstandard reduction problem |
| Lagrangian and Hamiltonian analysis of dynamical systems on functional manifolds. The Poisson-Dirac bracket / 3.8： |
| Conclusions / 3.9： |
| The algebraic structure of the gradient-holonomic algorithm for Lax type integrable nonlinear dynamical systems |
| Algebraic structure of the Lax type integrable dynamical system |
| The periodic problem and canonical variational relationships |
| The spectral gradient structure of Lax integrable many-dimensional nonlinear dynamical systems on operator manifolds / 4.4： |
| The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra |
| An effective Maurer-Cartan one-form construction |
| General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann algebra |
| Cartan's invariant geometric object structure of the gradient-holonomic algorithm for Lax integrable nonlinear dynamical systems in partial derivatives |
| A loop algebra and the Yang-Baxter structure / 5.5： |
| The transfer matrix properties / 5.6： |
| The algebraic structure of the gradient-holonomic algorithm for the Lax-type nonlinear dynamical systems: the reduction via Dirac and the canonical quantization procedure |
| The generalized R-structure hierarchy |
| The Dirac quantization procedure for Moser induced finite-dimensional Neumann type Hamiltonian systems |
| The scalar and operator integrable Hamiltonian systems via the algebraic gradient-holonomic algorithm / 6.4： |
| Concluding remarks / 6.5： |
| Hamiltonian structures of hydrodynamical Benny type dynamical systems and their associated Boltzmann-Vlasov kinetic equations on an axis |
| The Boltzmann equation and an associated moment problem |
| A nonlinear completely integrable Schrodinger type dynamical system approximation |
| The complete integrability of a Benney type hydrodynamical system associated with a Boltzmann-Vlasov equation / 7.4： |
| Conclusion / 7.5： |
| Appendix |
| Basic definitions, examples / .1： |
| The tangent Lie algebra / .2： |
| Lie subgroups / .3： |
| Lie algebras / .4： |
| Cartan subalgebras / .5： |
| Semi-simple complex Lie algebras / .6： |
| References |
| Preface |
| Background Notations |
| Dynamical systems with homogeneous configuration spaces / 1： |
図書
