Double Quantum Groups and Iwasawa Decomposition Original Research Article

The double quantum groups imageq[D(G)] = imageq[G] bowtie imageq[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(image(image)) and prove that the pairing between imageq[D(G)] andUq(image(image)) is nondegenerate. We analyze the representation theory of these imageq[D(G)], give a detailed version of the Iwasawa decomposition proved by Podles and Woronowicz for the quantum Lorentz group, and show that imageq[D(G)] is noetherian. Finally we outline how to construct more general nonstandard quantum groups using quantum double groups and their generalizations.