On modular double of semisimple quantum groups

Abstract: In this note we propose a construction of the Hopf algebra of a complex analog of devided powers of the Weyl generators of a semisimple simply-laced quantum group. Here we consider the generators as positive, self-adjoint operators. In particular, we generalize the Lusztig relations on the usual divided powers of generators of a quantum group to the case of complex devided powers of generators. These relations, some of which were known present the complete set of defining relations. As a by-product result, the pure algebraic definition of the Faddeev modular double in the case of semisimple simply-laced quantum groups is formulated. Finally, we introduce an infinite dimension version of the Gelfand-Zetlin finite-dimensional representation of the modular double $M_{q,\tilde{q}}(\mathfrak{gl}(N))$.