MATHEMATICA-aided Study Of Lie AlgebrasAnd Their Cohomology - From Supergravity ToBallbearings And Magnetic Hydrodynamics

Abstract

We describe applications of a MATHEMATICA-based package for the study of Lie algebras and their cohomology such as (1) the possibility to write Supergravity Equations for any TV-extended Minkowski superspace and to find out the possible models for these super spaces; (2) the possibility of studying stability of nonholonomic systems (ballbearings, gyroscopes, electromechanical devices like a rotor collector with a gliding contact; waves in plasma, etc.); (3) description of the analogue of the curvature tensor for nonlinear nonholonomic 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 dynamical systems. The above problems are particular instances of the general problem to compute cohomology or homology of the given Lie algebra or superalgebra with various coefficients. The package SuperLie makes it possible to determine (1) Lie algebras via defining relations, from the Cartan matrix, realized 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 and even (4) back up the Leites conjecture (an analog of Kostrikin-Shafarevich conjecture) classifying simple Lie algebras over the algebraically closed field of characteristic 2 with new examples. For the details see references.