Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems

'Background of the theory of Lie algebras and Lie groups and their representations.-
1.1 Lie algebras and Lie groups.- 1.1.1 Basic definitions.- 1.1.2 Contractions and deformations.- 1.1.3 Functional algebras.-
1.2 ?-graded Lie algebras and their classification.- 1.2.1 Definitions.- 1.2.2 Semisimple, nilpotent and solvable Lie algebras. The Levi-Malcev theorem.- 1.2.3 Simple Lie algebras of finite growth: Classification and Dynkin-Coxeter diagrams.- 1.2.4 Root systems and the Weylgroup.- 1.2.5 A parametrization and ordering of roots of simple finite-dimensional Lie algebras.- 1.2.6 The real forms of complex simple Lie algebras.-
1.3 sl(2)-subalgebras of Lie algebras.- 1.3.1 Embeddings of sl(2) into Lie algebras.- 1.3.2 Infinite-dimensional graded Lie algebras corresponding to embeddings of sl(2) into simple finite-dimensional Lie algebras.- 1.3.3 Explicit realization of simple finite-dimensional Lie algebras for the principal embedding of sl(2).-
1.4 The structure of representations.- 1.4.1 Terminology.- 1.4.2 The adjoint representation.- 1.4.3 The regular representation and Casimir operators.- 1.4.4 Bases in the space of representation.- 1.4.5 Fundamental representations.-
1.5 A parametrization of simple Lie groups.-
1.6 The highest vectors of irreducible representations of semisimple Lie groups.- 1.6.1 Generalities.- 1.6.2 Expression for the highest matrix elements in terms of the adjoint representation.- 1.6.3 A formal expression for the highest matrix elements of the fundamental representations.- 1.6.4 Recurrence relations for the highest matrix elements of the fundamental representations.- 1.6.5 The highest matrix elements of irreducible representations expressed via generalized Euler angles.-
1.7 Superalgebras and superspaces.- 1.7.1 Superspaces.- 1.7.2 Classical Lie superalgebras.- Representations of complex semisimple Lie groups and their real forms.-
2.1 Infinitesimal shift operators on semisimple Lie groups.- 2.1.1 General expression of infinitesimal operators.- 2.1.2 The asymptotic domain.-
2.2 Casimir operators and the spectrum of their eigenvalues.- 2.2.1 General formulation of the problem.- 2.2.2 Quadratic Casimir operators.- 2.2.3 Construction of Casimir operators for semisimple Lie groups.-
2.3 Representations of semisimple Lie groups.- 2.3.1 Integral form of realization of operator-irreducible representations.- 2.3.2 The matrix elements of finite transformations.-
2.4 Intertwining operators and the invariant bilinear form.- 2.4.1 Intertwining operators and problems of reducibility, equivalence and unitarity of representations.- 2.4.2 Construction of intertwining operators.- 2.4.3 The invariant Hermitian form.-
2.5 Harmonic analysis on semisimple Lie groups.- 2.5.1 General method.- 2.5.2 Characters of operator-irreducible representations.- 2.5.3 Plancherel measure of the principal continuous series of unitary representations.-
2.6 Whittaker vectors.- A general method of integrating two-dimensional nonlinear systems.-
3.1 General method.- 3.1.1 Lax-type representation.- 3.1.2 Examples.- 3.1.3 Construction of solutions.-
3.2 Systems generated by the local part of an arbitrary graded Lie algebra.- 3.2.1 Exactly integrable systems.- 3.2.2 Systems associated with infinite-dimensional Lie algebras.- 3.2.3 Hamiltonian formalism.- 3.2.4 Solutions of exactly integrable systems (Goursát problem).-
3.3 Generalization for systems with fermionic fields.-
3.4 Lax-type representation as a realization of self-duality of cylindrically-symmetric gauge fields.- Integration of nonlinear dynamical systems associated with finite-dimensional Lie algebras.-
4.1 The generalized (finite nonperiodic) Toda lattice.- 4.1.1 Preliminaries.- 4.1.2 Construction of exact solutions on the base of the general scheme of Chapter 3.- 4.1.3 Examples.- 4.1.4 Construction of solutions without appealing to the Lax-type representation.- 4.1.4.1 Symmetry pro