Quasigraded lie algebras, Kostant-Adler scheme, and integrable hierarchies

Abstract

Using special "anisotropic" quasigraded Lie algebras, we obtain a number of new hierarchies of integrable nonlinear equations in partial derivatives admitting zero-curvature representations. Among them are an anisotropic deformation of the Heisenberg magnet hierarchy, a matrix and vector generalization of the Landau-Lifshitz hierarchies, new types of matrix and vector anisotropic chiral-field hierarchies, and other types of "anisotropic" hierarchies.

Quasigraded Lie Algebras and Modified Toda Field Equations

“…to contain also infinite linear combination of the elements of its basis (X j α λ j ) * = X −j α λ −j . Under such agreement all subsequent consideration will be correct and consistent (see [18]). …”

“…In the present paper we combine our previous results from [11,16,17,18] with ideas of [19] and define a new type of the quasigraded Lie algebras g pr A admitting Kostant-Adler-Symes scheme and coinciding with deformations of the loop algebras in the principal gradation. More definitely, it turned out that for a special choice of the matrices A (that depends on the classical matrix Lie algebras g) it is possible to define "principal" subalgebras g pr A ⊂ g A in the analogous way as for the case of ordinary loop algebras [20,21].…”

Classical R-Operators and Integrable Generalizations of Thirring Equations

“…Hence, such approach permits [14] to construct using Lie algebra g the three types of integrable equations: two types of equations with U -V pair belonging to the same Lie subalgebras g ± and the third type of equations with U -operator belonging to g + and V -operator belonging to g − (or vise verse). The latter equations are sometimes called "negative flows" of integrable hierarchies.…”

“…In the case when the R-operator is of Kostant-Adler-Symes type, i.e. R = P + − P − , where P ± are projection operators onto subalgebras g R ± = g ± and g + ∩ g − = 0 we re-obtain the results of [10] (see also [14]) as a partial case of our construction. In the cases of more complicated R-operators our scheme is new and generalizes the approach of [10].…”

Special Quasigraded Lie Algebras and Integrable Hamiltonian Systems

“…In the series of our previous papers [9][10][11][12][13][14] we have proposed for the role of "Kostant-Adler admissible" Lie algebras special quasigraded Lie algebras g F with the natural decomposition g F = g + F + g − F realized as a loop Lie algebra g(u −1 , u) with the Lie bracket, deformed by some cocycle F . We have developed a theory of these algebras, theory of the corresponding integrable systems and soliton hierarchies for the cases when g is a classical matrix Lie algebra and cocycle F has the special form: F (X, Y ) = XAY − Y AX, where A is some fixed numerical matrix.…”