The global quantum duality principle

Details The global quantum duality principle The global quantum duality principle

1999-12-22

arxiv.org/abs/math/9912186v1

Mathematics

Quantum Algebra

AMS-TeX file, 34 pages

Scientific paper

The "quantum duality principle" states that the quantization of a Lie bialgebra --- via a quantum universal enveloping algebra (QUEA) --- provides also a quantization of the dual Lie bialgebra (through its associated formal Poisson group) --- via a quantum formal series Hopf algebra (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. Such a result was claimed true by Drinfeld (and proved by the author, in math.QA/9909071), and does hold in the framework of topological Hopf algebras, hence it is essentially "local" in nature. We give here a {\sl global} formulation of this principle, dealing with standard Hopf algebras and with usual (i.e. non-formal) Poisson groups. Essentially, quantum groups "of $q$-type" (say "a' la Lusztig-Jimbo"), defined over the ring of Laurent polynomials in $q$, are used instead of quantum groups "of $h$-type" (say "a' la Drinfeld"), defined over the ring of formal series in $h$; so quantum function algebras (QFA) take place of QFSHA, and function algebras over Poisson groups (the latter depending on an infinitesimal datum --- their tangent Lie bialgebra --- and on a global datum --- such as their fundamental group) take place of formal groups. The relevant examples of the special linear group, the Euclidean group and the Heisenberg group are studied in detail, describing explicitely the various objects entering the picture.

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