The quantum duality principle

Abstract

The "quantum duality principle" states that the quantization of a Lie bialgebra - via a quantum universal enveloping algebra (in short. QUEA) - also provides a quantization of the dual Lie bialgebra (through its associated formal Poisson group) - via a quantum formal series Hopf algbra (QFSHA) - and, conversely, a QFSHA associated to a Lie bialgebra (via its associated formal Poisson group) yields a QUEA for the dual Lie bialgebra as well; more in detail, there exist functors QUEA → QfSHA and QFSHA → QHEA, inverse to each other, such that in both cases the Lie bialgebra associated to the target object is the dual of that of the source object. Such a result was claimed true by Drinfeld, but seems to be unproved in the literature: I give here a thorough detailed proof of it.