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: |