Noncommutative geometry of quantum complex upper half plane and discrete subgroup of a non-compact quantum group

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The final aim of this research is to establish a reasonable theoretical framework of the quantified version of the classical theory of the modular functions on the basis of the quantum group SUィイD2qィエD2(1,1) of the non-compact type and its quantum modular subgroup SLィイD2qィエD2(2,Z). The difficulty is that, even in the classical case, the modular group SL(2,Z) is Zarishi dense in SL(2,R) 【similar or equal】 SU(1,1) so that, for the purpose of describing the algebra of functions on SL(2,Z) in terms of the algebra of functions on SL(2,R) 【similar or equal】 SU(1,1), we are obliged to work in the framework of functional analysis. In view of these considerations, we started to have a trial of investigating the deformations of finite dimensional Hopf algebras and bialgebras using the language of algebraic geometry trying to study the possibilities of quantizing the finite groups which are regarded to be the typical examples of discrete groups. The author published a survey article on the above general considerations concerning the functional analytical aspects together with the perspective of quantum theory of automorphic functions. The author also published an announcement concerned with the algebraic geometrical studies of finite dimensional Hopf and bialgebras. Meanwhile, the author and Dr.Hajac published a paper discovering the new type of compact quantum group describing the quantum symmetry of the noncommatative2-torus DTィイD32(/)qィエD3 and its ambient compact quantum group Uq(2), qCィイD1x∋ィエD1 having two deformation parameters.