ar X iv : f un ct - a n / 96 12 00 4 v 2 1 1 Ja n 19 97 ON THE INFINITE – DIMENSIONAL HIDDEN

The infinite dimensional geometry of the qR–conformal symmetries (the hidden symmetries in the Verma modules over sl(2,C) generated by the spin 2 tensor operators) is discussed. This paper opens the series of articles supplemental to [1], which also lie in lines of the general ideology exposed in the review [2]. The main purpose of further activity, which has its origin and motivation presumably in the author’s applied researches [3] 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 series [1]. Many of objects, which will appear, are somehow related to ones discussed previously. However, the matherial will be treated more geometrically, presumably, from the points of view of the infinite dimensional geometry (cf.[4]), an infinite dimensional version of the nonlinear geometric algebra (cf.[5]) and the infinite dimensional noncommutative geometry (cf.[6]). The intent reader will see a lot of ideological similarities between the subject of the paper and the infinite dimensional geometric picture for a second quantized string (see [7] and numerous refs wherein). 1. Unitary HS–pseudorepresentations of D̃iff+(S ), infinite dimensional geometry of qR–conformal symmetries and the related topics Definition 1A. A unitary HS–pseudorepresentation of the Lie group G in the Hilbert space H is a homomorphism of G into U(H)/U0(H), where U(H) is the