ABSTRACT
This chapter shows that some effective methods stemming from Commutative Algebra could be applied to most relevant quantum groups. A common property of many quantum groups is that its elements can be represented as polynomials in a finite number of variables, i.e., they have a ‘Poincare-Birkhoff-Witt’ theorem. Some quantum groups can be included, from a computational viewpoint, in the theoretic development. The chapter provides a division theorem in order to rewrite most of Kandri-Rody and Weispfenning’s results in a more algebraic context. The main idea is that the division algorithm would be based upon the assignment of an exponent for each non-zero element of the quantum group. Using this exponent, the basic topics in computational commutative algebra can be developed. The chapter iterates Ore extensions, which include universal enveloping algebras of finite dimensional Lie algebras, Weyl algebras, and quantum matrices. It discusses the effective computation of Gelfand-Kirillov dimension.
