The already broad range of applications of ring theory has been enhanced in the eighties by the increasing interest in algebraic structures of considerable complexity, the so-called class of quantum groups. One of the fundamental properties of quantum groups is that they are modelled by associative coordinate rings possessing a canonical basis, which allows for the use of algorithmic structures based on Groebner bases to study them. This book develops these methods in a self-contained way, concentrating on an in-depth study of the notion of a vast class of non-commutative rings (encompassing most quantum groups), the so-called Poincaré-Birkhoff-Witt rings. We include algorithms which treat essential aspects like ideals and (bi)modules, the calculation of homological dimension and of the Gelfand-Kirillov dimension, the Hilbert-Samuel polynomial, primality tests for prime ideals, etc.
Introduction. 1: Generalities on rings. 1. Rings and ideals. 2. Modules and chain conditions. 3. Ore extensions. 4. Factorization. 5. Other examples. 6. Quantum groups. 2: Gröbner basis computationalgorithms. 1. Admissible orders. 2. Left Poincaré-Birkhoff-Witt Rings. 3. Examples. 4. The Division Algorithm. 5. Gröbner bases for left ideals. 6. Buchberger's Algorithm. 7. Reduced Gröbner Bases. 8. Poincaré-Birkhoff-Witt rings. 9. Effective computations for two-sided ideals. 3: Poincaré-Birkhoff-Witt Algebras. 1. Bounding quantum relations. 2. Misordering. 3. The Diamond Lemma. 4. Poincaré-Birkhoff-Witt Theorems. 5. Examples. 6. Iterated Ore Extensions. 4: First applications. 1. Applications to left ideals. 2. Cyclic finite-dimensional modules. 3. Elimination. 4. Graded and filtered algebras. 5. The omega-filtration of a PBW algebra. 6. Homogeneous Gröbner bases. 7. Homogenization. 5: Gröbnerbases for modules. 1. Gröbner bases and syzygies. 2. Computation of the syzygy module. 3. Admissible orders in stable subsets. 4. Gröbner bases for modules. 5. Gröbner bases for subbimodules. 6. Elementary applications of Gröbner bases for modules. 7. Graded and filtered modules. 8. The omega-filtration of a module. 9. Homogeneous Gröbner bases. 10. Homogenization. 6:Syzygies and applications. 1. Syzygies for modules. 2. Intersections. 3. Applications to finitely presented modules. 4. Schreyer's order. 5. Free resolutions. 6. Computation of Hom and Ext. 7: TheGelfand-Kirillov dimension and the Hilbert polynomial. 1. The Gelfand-Kirillov dimension. 2. The Hilbert function of a stable subset. 3.The Hilbert function of a module over a PBW algebra. 4. The Gelfand-Kirillov dimension of PBW algebras. 8: Primality. 1. Localization. 2. The Ore condition and syzygies. 3. A primality test. 4. The primality test in iterated differential operator rings. 5. The primality test in coordinate rings of quantum spaces. Index. References.