  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.
