ISRAEL 3OURNAL OF MATHEMATICS 96 (1996), 97-113 GROBNER-SHIRSHOV BASES FOR QUANTUM ENVELOPING ALGEBRAS BY L. BOKUT'* Institute of Mathematics, Novosibirsk 630090, Russia e-mail: bokut@math.nsk.su AND P. MALCOLMSON Department of Mathematics, Wayne State University Detroit MI 48202, USA e-mail: petem@math, wayne, edu ABSTRACT We give a method for finding Gr6bner-Shirshov bases for the quantum enveloping algebras of Drinfel'd and Jimbo, show how the methods can be applied to Kac-Moody algebras, and explicitly find the bases for quantum enveloping algebras of type AN (for qS ~ 1). 1. Introduction Given a free algebra F over a field k and a set of relations S c_ F, the problem of reducing a given element f E F with respect to S involves computational difficulties, mainly because the reduction procedure may need to make f more complicated before some relations of S can "take effect". The main problem * Supported by the Russian Fund of Fundamental Research and the Soros Fund of Fundamental Research. This author wishes to thank the Department of Mathematics of Wayne State University for its hospitality. Received June 25, 1995 97