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The grading operators for all nonequivalent Z-gradations of classical Lie algebras are represented in the explicit block matrix form. The explicit form of the corresponding nonabelian Toda equations is given., Comment: LaTeX2e, 15 pages, to be published in the Memorial volume dedicated to M. V. Saveliev, Dubna, 1999

We associate to an arbitrary $\mathbb Z$-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional extension of these equations is given. The generalisation of the associated Redheffer--Reid differential systems appears in a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established. Within this context the integrability problem for those equations is studied. As an illustration, some examples of the integrable multidimensional Riccati-type equations related to the maximally nonabelian Toda systems are given., Comment: LaTeX2e, 18 pages

A detailed consideration of the maximally nonabelian Toda systems based on the classical semisimple Lie groups is given. The explicit expressions for the general solution of the corresponding equations are obtained., Comment: 28 pages, LaTeX file. A few references and appendix B are added; this version will appear in Nuclear Physics B

We discuss a Lie algebraic and differential geometry construction of solutions to some multidimensional nonlinear integrable systems describing diagonal metrics on Riemannian manifolds, in particular those of zero and constant curvature. Here some special solutions to the Lam\'e and Bourlet type equations, determining by n arbitrary functions of one variable are obtained in an explicit form. For the case when the sum of the diagonal elements of the metric is a constant, these solutions are expressed as a product of the Jacobi elliptic functions and are determined by 2n arbitrary constants., Comment: LaTeX file, 22 pages, to appear in the proceedings of the international conference "Selected Topics of Theoretical and Modern Mathematical Physics (SIMI-96)", September 22-29, 1996, Tbilisi, Georgia

Some special solutions to the multidimensional Lam\'e and Bourlet type equations are constructed in an explicit form., Comment: 12 pages, LaTeX file

On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed., Comment: 29 pages, LaTeX file

On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed., Comment: Talk given at the IInd International Sakharov Conference on Physics, May, 20--24, 1996, Moscow, Russia, 5 pages, LaTeX

We investigate higher grading integrable generalizations of the affine Toda systems. The extra fields, associated to non zero grade generators, obey field equations of the Dirac type and are regarded as matter fields. The models possess soliton configurations, which can be interpreted as particles of the theory, on the same footing as those associated to fundamental fields. A special subclass of these models is remarkable. They possess a $U(1)$ Noether current which, after a special gauge fixing of the conformal symmetry, is proportional to a topological current. This leads to the confinement of the matter field inside the solitons, which can be regarded as a one dimensional bag model for QCD. These models are also relevent to the study of electron self--localization in (quasi)-one-dimensional electron--phonon systems., Comment: 51 pages, LaTeX

In the present paper we obtain some integrable generalisations of the Toda system generated by flat connection forms taking values in higher ${\bf Z}$--grading subspaces of a simple Lie algebra, and construct their general solutions. One may think of our systems as describing some new fields of the matter type coupled to the standard Toda systems. This is of special interest in nonabelian Toda theories where the latter involve black hole target space metrics. We also give a derivation of our conformal system on the base of the Hamiltonian reduction of the WZNW model; and discuss a relation between abelian and nonabelian systems generated by a gauge transformation that maps the first grading description to the second. The latter involves grades larger than one., Comment: 24 pages, latex, no figures; Expanded version accepted for publication in Nuclear Physics B

In the present paper we obtain some integrable generalisations of the continuous Toda system generated by a flat connection form taking values in higher grading subspaces of the algebra of the area--preserving diffeomorphism of the torus $T^2$, and construct their general solutions. The grading condition which we use here, imposed on the connection, can be realised in terms of some holomorphic distributions on the corresponding homogeneous spaces., Comment: 12 pages, no figures, LaTeX file

In the present note we suggest an affinization of a theorem by Kostrikin et.al. about the decomposition of some complex simple Lie algebras ${\cal G}$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac-Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r, \, C_{2^m}, D_r,\, G_2,\, E_7,\, E_8$ can be decomposed into the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed., Comment: 16 pages, LaTeX, no figures

In the present note we give an explicit integration of some two--dimensionalised Lotka--Volterra type equations associated with simple Lie algebras, other than the familiar $A_n$ case, possessing a representation without branching. This allows us, in particular, to treat the first fundamental representations of $A_r$, $B_r$, $C_r$, and $G_2$ on the same footing., Comment: 3 pages LATEX file

The present paper describes the $W$--geometry of the Abelian finite non-periodic (conformal) Toda systems associated with the $B,C$ and $D$ series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Pl\"ucker embedding of the $A$-case to the flag manifolds associated with the fundamental representations of $B_n$, $C_n$ and $D_n$, and a direct proof that the corresponding K\"ahler potentials satisfy the system of two--dimensional finite non-periodic (conformal) Toda equations. It is shown that the $W$--geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) of $CP^N$ with appropriate choices of $N$. In addition, these W-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Pl\"ucker embedding. These conditions are automatically fulfiled when Toda equations hold., Comment: 30 pages, no figure

In the present paper we give a differential geometry formulation of the basic dynamical principle of the group--algebraic approach \cite{LeS92} --- the grading condition --- in terms of some holomorphic distributions on flag manifolds associated with the parabolic subgroups of a complex Lie group; and a derivation of the corresponding nonlinear integrable systems, and their general solutions. Moreover, the reality condition for these solutions is introduced. For the case of the simple Lie groups endowed with the canonical gradation, when the systems in question are reduced to the abelian Toda equations, we obtain the generalised Pl\"ucker representation for the pseudo--metrics specified by the K\"ahler metrics on the flag manifolds related to the maximal nonsemisimple parabolic subgroups; and the generalised infinitesimal Pl\"ucker formulas for the Ricci curvature tensors of these pseudo--metrics. In accordance with these formulas, the fundamental forms of the pseudo--metrics and the Ricci curvature tensors are expressed directly in terms of the abelian Toda fields, which have here the sense of K\"ahler potentials., Comment: 41 pages, AMSLaTeX file

On an equation associated with the contact Lie algebras/ Mikhail V. Saveliev/ In the framework of a Lie algebraic approach we study a nonlinear equation associated with the contact Lie algebra ${\bf K}K_m$, that seems to be relavant for some solvable models of field theory and gravity in higher dimensions. LPTENS--93/43., Comment: 4 pages, no figure; LPTENS--93/43

We discuss an infinite--dimensional k\"ahlerian manifold associated with the area--preserving diffeomorphisms on two--dimensional torus, and, correspondingly, with a continuous limit of the $A_r$--Toda system. In particular, a continuous limit of the $A_r$--Grassmannians and a related Pl\"ucker type formula are introduced as relevant notions for $W_{\infty}$--geometry of the self--dual Einstein space with the rotational Killing vector., Comment: 6 pages, no figure report\# ETH-TH/93-21