On the infinite dimensional hidden symmetries. I. Infinite dimensional geometry of $q_R$-conformal symmetries

Abstract

This paper opens the series of articles supplemental to the series (hep-th/9405050,q-alg/9610026,q-alg/9611003,q-alg/9611019,funct-an/9611003), which also lies in lines of general ideology exposed in the review (mp_arc/96-477). The main purpose of the activity, which has its origin and motivation presumably in the author's applied researches on the interactively controlled systems (i.e.the controlled systems, in which the control is coupled with unknown or uncompletely known feedbacks), is to explicate the essentially infinite-dimensional aspects of the hidden symmetries, which appear in the representation theory of the finite dimensional Lie algebras and related algebraic structures. The series is organized as a sequence of topics, which illustarate this basic idea on the simple and tame examples without superfluous difficulties and details as well as in the previous series. Many of objects, which will appear, are somehow related to ones discussed earlier. However, the matherial will be treated more geometrically, presumably, from the points of view of the infinite dimensional geometry, an infinite dimensional version of the nonlinear geometric algebra and the infinite dimensional noncommutative geometry. The intent reader will see some ideological and concrete similarities between the subject of the paper and the infinite dimensional geometric picture for a second quantized string.