Generalized Lie-algebraic structures related to integrable dispersionless dynamical systems and their application

Abstract

Our review is devoted to Lie-algebraic structures and integrability properties of an interesting class of nonlinear dynamical systems called the dispersionless heavenly equations, which were initiated by Plebanski and later analyzed in a series of articles. The AKS-algebraic and related $\mathcal{R}$-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie--Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly equations under consideration. Moreover, all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a fascinating Lagrange--d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax--Sato equations is also discussed. We pay a special attention to a generalization of the devised Lie-algebraic scheme to a case of loop Lie superalgebras of superconformal diffeomorphisms of the $1|N$-dimensional supertorus. This scheme is applied to constructing the Lax--Sato integrable supersymmetric analogs of the Liouville and Mikhalev-Pavlov heavenly equation for every $N\in\mathbb{N}\backslash\lbrace 4;5\rbrace.$