Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups / Edition 1

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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 Poincar -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. The text is mainly aimed at postgraduate students and researchers interested in ring theory, the algebraic theory of differential equations, symbolic calculus and their applications.

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Product Details

  • ISBN-13: 9789048163281
  • Publisher: Springer Netherlands
  • Publication date: 12/7/2010
  • Series: Mathematical Modelling: Theory and Applications Series, #17
  • Edition description: Softcover reprint of hardcover 1st ed. 2003
  • Edition number: 1
  • Pages: 300
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.66 (d)

Table of Contents

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 computation algorithms. 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: Gröbner bases 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: The Gelfand-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.

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