Abstract
The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures.
Highlights
IntroductionMuch work [21,24,25,26,27,28,29] has been devoted to the finite-dimensional representations of the reduced pre-Lie algebraic structures called the Balinsky–Novikov algebras
A left pre-Lie algebra ( A, +, ◦) is a vector space A over an algebraically closed field F with a bilinear map ◦ : A ⊗ A → A, satisfying the relation ( a ◦ b) ◦ c − a ◦ (b ◦ c) = ( a ◦ c) ◦ b − a ◦ (c ◦ b) (1)for any a, b, c ∈ A
We show that the well-known Balinsky–Novikov algebraic pre-Lie algebraic structures, obtained in [17,20] as a condition for a matrix differential system to be Hamiltonian and in [55,56,57,58] as that on a flat torsion free left-invariant affine connection on affine manifolds, affine structures and convex homogeneous cones, arise as a derivation on the Lie-algebra associated with a differential loop algebra
Summary
Much work [21,24,25,26,27,28,29] has been devoted to the finite-dimensional representations of the reduced pre-Lie algebraic structures called the Balinsky–Novikov algebras Their importance for constructing integrable multi-component nonlinear Camassa–Holm type dynamical systems on functional manifolds was demonstrated by Strachan and Szablikowski [30]. We have devised a simple algorithm, based on the Lie–Poisson structure analysis on the adjoint space to toroidal Lie algebras, rigged with non-associated and noncommutative algebras, which enables singling out new algebraic pre-Lie algebraic structures, containing the corresponding Hamiltonian operators, which generate integrable multi-component and multidimensional dynamical systems The theory of these systems was recently started in [32,33,34,35,36,37,38,39,40,41,42] and developed in [43,44]. Using the theory of Zelmanov [59], we introduce and describe Balinsky–Novikov type algebras in detail, including their fermionic version and related multiplicative and Lie structures
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