THE METHOD OF NONCOMMUTATIVE INTEGRATION FOR LINEAR-DIFFERENTIAL EQUATIONS - FUNCTIONAL ALGEBRAS AND NONCOMMUTATIVE DIMENSIONAL REDUCTION

The method of noncommutative integration for linear partial differenti al equations [1] is extended to the case of the so-called functional a lgebras for which the commutators of their generators are nonlinear fu nctions of the same generators. The linear functions correspond to Lie algebras, whereas the quadratics are associated with the so-called qu adratic algebras having wide applications in quantum field theory. A n ontrivial example of integration of the Klein-Gordon equation in a cur ved space not allowing separation of variables is considered. A classi fication of four- and five-dimensional quadratic algebras is performed . A method of dimensional reduction for noncommutatively integrable ma ny-dimensional partial differential equations is suggested. Generally, the reduced equation has a complicated functional symmetry algebra. T he method permits integration of the reduced equation without the use of the explicit form of its functional algebra.