Semi-direct Sums Of Lie Algebras Research Articles

Jordan–Kronecker invariants for semidirect sums defined by standard representation of orthogonal or symplectic Lie algebras

We calculate Jordan–Kronecker invariants for semi-direct sums of Lie algebras so(n) and sp(n) with several copies of R n with respect to their standard representation.

Integrable couplings, bi-integrable couplings and their Hamiltonian structures of the Giachetti-Johnson soliton hierarchy

On the basis of zero curvature equations from semi-direct sums of Lie algebras, we construct integrable couplings of the Giachetti–Johnson hierarchy of soliton equations. We also establish Hamiltonian structures of the resulting integrable couplings by the variational identity. Moreover, we obtain bi-integrable couplings of the Giachetti–Johnson hierarchy and their Hamiltonian structures by applying a class of non-semisimple matrix loop algebras consisting of triangular block matrices. Copyright © 2014 John Wiley & Sons, Ltd.

Tri-Integrable Couplings of the Giachetti-Johnson Soliton Hierarchy as well as Their Hamiltonian Structure

Based on zero curvature equations from semidirect sums of Lie algebras, we construct tri-integrable couplings of the Giachetti-Johnson (GJ) hierarchy of soliton equations and establish Hamiltonian structures of the resulting tri-integrable couplings by the variational identity.

AN ALGEBRAIC STRUCTURE OF ZERO CURVATURE REPRESENTATIONS ASSOCIATED WITH COUPLED INTEGRABLE COUPLINGS AND APPLICATIONS TO τ-SYMMETRY ALGEBRAS

We establish an algebraic structure for zero curvature representations of coupled integrable couplings. The adopted zero curvature representations are associated with Lie algebras possessing two sub-Lie algebras in form of semi-direct sums of Lie algebras. By applying the presented algebraic structures to the AKNS systems, we give an approach for generating τ-symmetry algebras of coupled integrable couplings.

Methods for constructing complete involutive sets of polynomials on semidirect sums of Lie algebras

Methods for constructing complete involutive sets of polynomials on semidirect sums of Lie algebras

The integrable coupling system of a 3 × 3 discrete matrix spectral problem

Integrable coupling with six potentials is first proposed by coupling a given 3 × 3 discrete matrix spectral problem. It is shown that coupled system of integrable equations can possess zero curvature representations and recursion operators, which yield infinitely many commuting symmetries. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebras, the Hamiltonian form is deduced for the lattice equations in the resulting hierarchy. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian system.

Liouville Integrable System and Associated Integrable Coupling

Liouville integrable discrete integrable system is derived based on discrete isospectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.

Completely integrable Hamiltonian systems on semidirect sums of Lie algebras

The complete integrability of Hamiltonian systems arising on Lie algebras which have the form of a direct sum is investigated. For algebras in these classes Sadetov's method takes a simpler form: the isomorphism between the algebra arising at the second step of Sadetov's approach and the stationary subalgebra of a generic element can be written out explicitly. The explicit form of this isomorphism is presented, as well as explicit formulae for polynomials in complete systems for the algebras , and . For the algebras the degrees of the resulting polynomial functions are analysed.Bibliography: 15 titles.

COUPLING INTEGRABLE COUPLINGS

Integrable couplings are presented by coupling given integrable couplings. It is shown that such coupled integrable couplings can possess zero curvature representations and recursion operators, which yield infinitely many commuting symmetries. The presented zero curvature equations are associated with Lie algebras, each of which has two sub-Lie algebras in form of semi-direct sums of Lie algebras.

Quasi-Hamiltonian Structure Associated with an Integrable Coupling System

Starting from a spectral problem, a corresponding soliton hierarchy is proposed, and we construct an integrable coupling system with five dependent variables for the hierarchy by using a class of semi-direct sums of Lie algebras. Moreover, it is shown that the coupling system possesses quasi-Hamiltionian structures, and that infinitely many conserved quantities are obtained.

Discrete integrable system and its integrable coupling

This paper derives new discrete integrable system based on discrete isospectral problem. It shows that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.

The Application of Discrete Variational Identity on Semi-Direct Sums of Lie Algebras

With non-semisimple Lie algebras, the trace identity was generalized to discrete spectral problems. Then the corresponding discrete variational identity was used to a class of semi-direct sums of Lie algebras in a lattice hierarchy case and obtained Hamiltonian structures for the associated integrable couplings of the lattice hierarchy. It is a powerful tool for exploring Hamiltonian structures of discrete soliton equations.

The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems

An algebraic structure of discrete zero curvature equations is established for integrable coupling systems associated with semi-direct sums of Lie algebras. As an application example of this algebraic structure, a τ-symmetry algebra for the Volterra lattice integrable couplings is engendered from this theory.

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

New non-isospectral integrable couplings of the AKNS system

By means of semi-direct sums of Lie algebras and the compatibility condition of Lax pair, an efficient way to construct non-isospectral integrable couplings is proposed. As its applications, new non-isospectral integrable couplings of the AKNS system are obtained. The method can be devoted to constructing non-isospectral integrable couplings of other nonlinear evolution equations.

THE ALGEBRAIC STRUCTURE OF ZERO CURVATURE REPRESENTATIONS ASSOCIATED WITH INTEGRABLE COUPLINGS

The commutator of enlarged vector fields was explicitly computed for integrable coupling systems associated with semidirect sums of Lie algebras. An algebraic structure of zero curvature representations is then established for such integrable coupling systems. As an application example of this algebraic structure, the commutation relations of Lax operators corresponding to the enlarged isospectral and nonisospectral AKNS flows are worked out, and thus a τ-symmetry algebra for the AKNS integrable couplings is engendered from this theory.

Constructing integrable coupling systems of Hamiltonian lattice equations using semi-direct sums of Lie algebras

In this article, by considering a discrete isospectral problem, a hierarchy of Hamiltonian lattice equations are derived. Two types of semi-direct sums of Lie algebras are proposed, using which a practicable way to construct discrete integrable couplings is introduced. As an application, two kinds of discrete integrable couplings of the resulting system are worked out.

A discrete variational identity on semi-direct sums of Lie algebras

The discrete variational identity under general bilinear forms on semi-direct sums of Lie algebras is established. The constant γ involved in the variational identity is determined through the corresponding solution to the stationary discrete zero-curvature equation. An application of the resulting variational identity to a class of semi-direct sums of Lie algebras in the Volterra lattice case furnishes Hamiltonian structures for the associated integrable couplings of the Volterra lattice hierarchy.

Integrable decomposition of a hierarchy of soliton equations and integrable coupling system by semidirect sums of Lie algebras

Staring from a new spectral problem, a hierarchy of the soliton equations is derived. It is shown that the associated hierarchies are infinite-dimensional integrable Hamiltonian systems. By the procedure of nonlinearization of the Lax pairs, the integrable decomposition of the whole soliton hierarchy is given. Further, we construct two integrable coupling systems for the hierarchy by the conception of semidirect sums of Lie algebras.

Discrete integrable couplings associated with modified Korteweg–de Vries lattice and two hierarchies of discrete soliton equations

A direct way to construct integrable couplings for discrete systems is presented by use of two semi-direct sum Lie algebras. As their applications, the discrete integrable couplings associated with modified Korteweg–de Vries (m-KdV) lattice and two hierarchies of discrete soliton equations are developed. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards the complete classification of integrable couplings.