Abstract
The generalized theory of the ℛ-structure on affine operator Lie algebras is used to construct a complete theory of Lax integrable nonlinear dynamical systems in multidimensions. The operator bi-Hamiltonian structures and their functional reductions are discussed in great detail in the examples of operator Korteweg–de Vries and Benney–Kaup dynamical systems. As an important by-product of the developed algebraic theory, the Dirac canonical quantization problem is solved almost completely for the Neumann–Bogoliubov-type oscillatory dynamical system on spheres, associated via Moser with the spectral moment map on an affine associative metrized Lie coalgebra with a one-parameter gauge two-cocycle. Some remarks are given on the problem of extending the developed algebraic theory to the case of Lax integrable dynamical systems on discrete manifolds.
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