Lie Algebras Of Characteristic Research Articles

On the classification of nonlinear integrable three-dimensional chains via characteristic Lie algebras

Продолжена работа по описанию интегрируемых нелинейных цепочек с тремя независимыми переменными вида $u^j_{n+1,x}=u^j_{n,x}+f(u^{j+1}_{n},u^{j}_n,u^j_{n+1 },u^{j-1}_{n+1})$ по признаку наличия иерархии редукций, интегрируемых в смысле Дарбу. В основе классификационного алгоритма лежит хорошо известный факт, что характеристические алгебры интегрируемых по Дарбу систем имеют конечную размерность. В работе использована характеристическая алгебра по направлению $x$, структура которой для данного класса моделей определяется некоторым полиномом $P(\lambda)$, степень которого для известных примеров не превосходит 3. Предполагается, что $P(\lambda)=\lambda^2$, в этом случае классификационная задача сводится к отысканию восьми неизвестных функций одной переменной. Получен достаточно узкий класс претендентов на интегрируемость.

On the classification of nonlinear integrable three-dimensional chains via characteristic Lie algebras

On the classification of nonlinear integrable three-dimensional chains via characteristic Lie algebras

Characteristic Lie algebras of integrable differential-difference equations in 3D

The purpose of this article is to discuss an algebraic method for studying integrable differential-difference lattices with two discrete and one continuous independent variables. The main idea is based on the hypothesis that any integrable equation of this type admits an infinite sequence of reductions, which are Darboux integrable systems of differential-difference equations with two independent variables. To detect the Darboux integrability property of the reductions the authors use the characteristic Lie–Rinehart algebras. The hypothesis is confirmed by integrable three-dimensional lattices from the list given by Ferapontov et al (2015 Int. Math. Res. Not. 2015 4933). The required reductions are found for these lattices. Characteristic algebras corresponding to small-order reductions are described, integrals are calculated in both directions, which proves Darboux integrability. The approach might be useful for the integrable classification in 3D.

Narrow Positively Graded Lie Algebras

The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n1 and n2 from our list, that are positive parts of the affine Kac–Moody algebras A1(1) and A2(2), respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2].

Lie Algebras of Slow Growth and Klein-Gordon PDE

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms: Hence the Lie algebras $\chi (\sinh {u})$ and χ(eu + e− 2u) are slowly linearly growing Lie algebras with average growth rates $\frac {3}{2}$ and $\frac {4}{3}$ respectively.

Complete list of Darboux integrable chains of the form t1x=tx+d(t,t1)

We study differential-difference equation (d/dx)t(n+1,x)=f(t(n,x),t(n+1,x),(d/dx)t(n,x)) with unknown t(n,x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, {t(n+k,x)}k=−∞∞, {(dk/dxk)t(n,x)}k=1∞, such that DxF=0 and DI=I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n)=p(n+1). Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function f is of the special form f(u,v,w)=w+g(u,v).

On the classification of Darboux integrable chains

We study a differential-difference equation of the form tx(n+1)=f(t(n),t(n+1),tx(n)) with unknown t=t(n,x) depending on x and n. The equation is called a Darboux integrable if there exist functions F (called an x-integral) and I (called an n-integral), both of a finite number of variables x,t(n),t(n±1),t(n±2),…,tx(n),txx(n),…, such that DxF=0 and DI=I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n)=p(n+1). The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f(x,y,z)=z+d(x,y).

On the characteristic Lie algebras for equations u xy = f(u, u x )

A new approach to classification of integrable nonlinear equations is proposed. The method is based on description of the structure of the characteristic algebra. A basis of the characteristic algebra is constructed for the sinh-Gordon equation.

Characteristic Lie algebra and classification of semidiscrete models

We study characteristic Lie algebras of semi-discrete chains and attempt to use this notion to classify Darboux-integrable chains.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Characteristic Algebras of Fully Discrete Hyperbolic Type Equations

The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An eective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.

On the integrability of hyperbolic systems of riccati-type equations

We consider equations of the form Uxy = U * Ux, where U(x, y) is a function taking values in an arbitrary finite-dimensional algebra T over the field ℂ. We show that every such equation can be naturally associated with two characteristic Lie algebras, Lx and Ly. We define the notion of a ℤ-graded Lie algebraB corresponding to a given equation. We prove that for every equation under consideration, the corresponding algebraB can be taken as a direct sum of the vector spaces Lx and Ly if we define the commutators of the elements from Lx and Ly by means of the zero-curvature relations. Assuming that the algebra T has no left ideals, we classify the equations of the specified type associated with the finite-dimensional characteristic Lie algebras Lx and Ly. All of these equations are Darboux-integrable.

Descending chain conditions for Lie algebras of prime characteristic

The theme of this paper is the study of descending chain conditions for subideals of infinite-dimensional Lie algebras of prime characteristic, complementing the results of the second author [8] on Lie algebras of characteristic zero. Robinson [5] proved that every group satisfying the minimal condition for subnormal subgroups is a finite extension of a Z-group, where a Z-group is a group in which normality is transitive. As a corollary to the proof it follows that every group satisfying the minimal condition for 2-step subnormal subgroups must in addition satisfy the minimal condition for all subnormal subgroups. Lie algebra analogues of these and related results are proved in [8] for fields of characteristic zero. The proofs use properties of the Baer radical of a Lie algebra (Hartley [3]) which are not valid in prime characteristic. For about five years in question has remained open, whether the analogy extends to Lie algebras of prime characteristic: the best that was known until recently was that the minimal condition for 3-step subideals implies that for all subideals. We answer the question here, as follows. Over fields of prime characteristic it remains true that every Lie algebra satisfying the minimal condition for subideals is a finite-dimensional extension of a IL-algebra. However, it is not true that the minimal condition for 2-step subideals implies that for all subideals. In fact a structural analysis of Lie algebras with the minimal