Double quantum groups and Iwasawa decomposition

Abstract

The double quantum groups Cq[D(G)]=Cq[G] time sign closed Cq[G] are the Hopf algebras underlying the complex quantum groups of which the simplest example is the quantum Lorentz group. They are nonstandard quantizations of the double groupG×G. We construct a corresponding quantized universal enveloping algebraUq(d(g)) and prove that the pairing between Cq[D(G)] andUq(d(g)) is nondegenerate. We analyze the representation theory of these Cq[D(G)], give a detailed version of the Iwasawa decomposition proved by Podles and Woronowicz for the quantum Lorentz group, and show that Cq[D(G)] is noetherian. Finally we outline how to construct more general nonstandard quantum groups using quantum double groups and their generalizations. © 1997 Academic Press.