A Hilbert Basis Theorem for Quantum Groups

A noncommutative version of the Hilbert basis theorem is usedto show that certain R-symmetric algebras S_R(V) are Noetherian.This result applies in particular to the coordinate ring ofquantum matrices A_R(V) associated with an R-matrix R operatingon the tensor square of a vector space V, to show that, undera natural set of hypotheses on R, the algebra A_R(V) is Noetherianand its augmentation ideal has a polynormal set of generators.As a corollary we deduce that these properties hold for thegeneric quantized function algebras R_qG] over any field ofcharacteristic zero, for G an arbitrary connected, simply connected,semisimple group over C. That R_qG] is Noetherian recovers aresult due to Joseph 10], with a different proof.1991 MathematicsSubject Classification 17B37, 16P40.