Cohomology for physicists and engineers

Description

A Mathematican based package SUPERLie for the study of Lie (super)algebras and their cohomology is offered. Among applications we find: (1) calculation of the Rieн mannian tensors needed to write Supergravity Equations for any Nнextended Minkowski superspace and to classify possible models for these superspaces; (2) the possibility to study stability of nonholonomic systems (ballbearings, gyroscopes, electroнmechanical devices like rotor collector with a gliding contact; differential games or pursuit problems; waves in plasma, etc.); (3) description of the analogue of the curvature tensor for nonlinear nonholoн nomic constraints and the fields of solids or their surfaces, e.g., cones, as in optimal control; (4) a new method for the study of integrability of differential equations (numerical methods can provide with individual solutions but are unable, generally, to study qualitative behavior, e.g., stability); (5) lists of ``natural'' operators, i.e., the operators between sections of tensor fields (or jets) invariant with respect to the group of diffeomorphism or its subgroup. For nonholonomic systems the formula for the analog of curvature tensor (the sign of whose components determines stability) is new. SUPERLIE makes it possible to determine (1) Lie algebras via defining relations, from Cartan matrix, realize via vector fields, as polynomials with Poisson or contact (Legendre) bracket, etc., (2) various modules over these Lie algebras (tensors, with vacuum vector, etc.), (3) list central extensions and deformations. All the above problems can be expressed in terms of Lie algebra cohomology.