ON THE INFINITE-DIMENSIONAL HIDDEN SYMMETRIES. II. qR-CONFORMAL MODULAR FUNCTORS.

The article is devoted to the qR-conformal modular functors, which being “deformations” of the conformal modular functors (the projective representations of the category Train(Diff+(S)), the train of the group Diff+(S) of all orientation preserving diffeomorphisms of a circle) in the class of all projective modular functors (the projective representations of the category Train(PSL(2, R)), the train of the projective group PSL(2, R)), may be regarded as their “Berezin quantizations”. This paper being the continuation of the first part [1] belongs to the series of articles supplemental to [2], and also lies in lines of the general ideology exposed in the review [3]. The main purpose of the activity, which has its origin and motivation presumably in the author’s applied researches [4] 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 infinitedimensional aspects of the hidden symmetries, which appear in the representation theory of the finite dimensional Lie algebras and related algebraic structures. The relations between the control and the representation theories will be discussed in [5]. The present 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] but from a bit more geometric point of view. On the other hand this concrete article is placed at the crossing of two very different ideologies of hidden symmetries (however, the ideological differences may be rather subtle in practice). The used version of the first was developed by G.Segal [6], M.Kontsevich, K.Gawedzki [7], M.Atiyah, G.Moore and N.Seiberg [8], Yu.A.Neretin [9; and refs wherein], E.Witten [10] and others [11]. This ideology, which was formulated most clearly in purely mathematical fashion by G.Segal (pioneered the considered version), E.Witten, M.Atiyah and to a certain extent by Yu.A.Neretin, may be characterized as a formal search of hidden symmetries on the abstract level and is related to the direct problems of representation theory. This ideology underlies the approaches of J.Mickellson and D.P.Zhelobenko [12; and refs wherein] to the representation theory of reductive Lie algebras and partially penetrates the