## Noncommutative Gröbner Bases and Filtered-Graded TransferThis self-contained monograph is the first to feature the intersection of the structure theory of noncommutative associative algebras and the algorithmic aspect of Groebner basis theory. A double filtered-graded transfer of data in using noncommutative Groebner bases leads to effective exploitation of the solutions to several structural-computational problems, e.g., an algorithmic recognition of quadric solvable polynomial algebras, computation of GK-dimension and multiplicity for modules, and elimination of variables in noncommutative setting. All topics included deal with algebras of ( q-)differential operators as well as some other operator algebras, enveloping algebras of Lie algebras, typical quantum algebras, and many of their deformations. |

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### Contents

5 | |

2 Skew Polynomial Rings | 9 |

3 Zfiltrations and their Associated Graded Structures | 13 |

4 Homogenization and Dehomogenization of Zgraded Rings | 25 |

Classical and Modern | 28 |

1 Left Monomial Orderings and Left Admissible Systems | 34 |

2 Left Quasizero Elements and a Left Division Algorithm | 38 |

3 Left Gröbner Bases in a Left Admissible System | 42 |

4 GKdimension Computation and Elimination of Variables Homogeneous Case | 115 |

5 GKdimension Computation and Elimination of Variables Linear Case | 119 |

6 The ?gr
filtration on a Quadric Solvable Polynomial Algebra | 124 |

7 GKdimension Computation and Elimination of Variables General Quadric Case | 126 |

8 Finite Dimensional Cyclic Modules | 130 |

1 The Multiplicity eM of a Module M | 134 |

2 Computation of eM | 135 |

3 Computation of GKdimM ?k N and eM ?k N | 144 |

4 Left Gröbner Bases in Various Contexts | 45 |

5 Left Selements and Buchberger Theorem | 48 |

6 Left Dickson Systems and Left GNoetherian Algebras | 54 |

7 Solvable Polynomial Algebras | 58 |

8 No Left Monomial Ordering Existing on ?kx1 xn with chark 0 | 61 |

1 PBW Bases of Finitely Generated Algebras | 68 |

2 Quadric Solvable Polynomial Algebras | 73 |

3 Associated Homogeneous Defining Relations of Algebras | 81 |

4 A Remark on Recognizable Properties of Algebras via Gröbner Bases | 89 |

1 FilteredGraded Transfer of Left Admissible Systems | 92 |

2 FilteredGraded Transfer of Left Gröbner Bases | 97 |

3 FilteredGraded Transfer of Left Dickson Systems | 100 |

4 FilteredGraded Transfer Applied to Quadric Solvable Polynomial Algebras | 103 |

1 Gröbner Bases in Homogeneous Solvable Polynomial Algebras | 108 |

2 The Hilbert Function of AL | 110 |

3 The Hilbert Polynomial of AL | 112 |

4 An Application to Anq1 qn | 148 |

1 Some Operator Algebras | 154 |

2 Holonomic Functions | 156 |

3 Automatic Proving of Holonomic Function Identities | 162 |

4 ExtensionContraction of theFiniteness | 164 |

5 The ?Holonomicity | 168 |

A is Auslander Regular with K0A ? ? | 176 |

2 The A ?grfiltration on Modules | 177 |

gldim An | 181 |

K0A ? ? | 186 |

Blank Page | 198 |

199 | |

200 | |

Blank Page | 201 |

### Other editions - View all

Noncommutative Gröbner Bases and Filtered-Graded Transfer, Issue 1795 Huishi Li,Herve Pajot Limited preview - 2002 |

### Common terms and phrases

A-module A(ka algebra with respect An(k associated graded CH.I Lemma CH.V coefficients commutative polynomial computation coordinate subspace defining relations denoted Dickson system element of degree enveloping algebra Example filtered filtered-graded transfer finite dimensional fj}jej following holds forms a k-basis free algebra free semigroup function identities g-filtered g-filtration GK.dim graded algebra graded monomial ordering graded ring grie grier grlea Gröbner basis theory Gröbner representation Hence Hilbert polynomial holonomic function homogeneous element homogeneous solvable polynomial isomorphism iterated skew polynomial Jacobi sums k-algebra k-span left admissible system left Gröbner basis left ideal LM(a LM(f LM(g LM(r LM(vus LT(L Noetherian noncommutative Gröbner nonzero polynomial ring Proof Proposition Let Proposition With notation quadric solvable polynomial quasi-zero element ring homomorphism S-element solvable polynomial algebra standard filtration standard monomials subset two-sided ideal variables Weyl algebra X-gr