Lutfullin M. Realizations of Low-Dimensional Lie Algebras and Invariant System of Nonlinear Differential Equations

We have obtained the complete classification of realizations of real solvable Lie algebras of dimension no greater than 4 in the space of arbitrary finite number of variables. We have classified realizations of algebra AO(3) in n-dimensional complex space. We have constructed the complete list of realizations of algebra AE(3) in the space of 3 independent and n dependent complex variables. We have found all inequivalent realizations of Lorentz algebra in the space of arbitrary finite number of complex variables. This result has been used for describing all covariant realizations of Poincare algebra AP(1,3) in the space of 4 real independent and n complex dependent variables. We have found the functional basis of first-order differential invariant for realizations of real solvable three- and four-dimensional Lie algebras. We have described the general form of ordinary differential equations that are invariant with respect to these algebras. We have constructed the normal system that are invariantwith respect to solvable three- and four-dimensional Lie algebras. For one of realizations of Euclidean algebra AE(3) we have constructed the functional basis of first-order differential invariants and we have found the general form of the invariant system of partial differential equations. We have constructed a number of new exact solutions of Maxwell equations for vector-potential. We perform separating of variables in the Schrodinger–Maxwell equations are constructed.