Noncommutative Gröbner Bases and Filtered-Graded Transfer

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Springer, Oct 20, 2004 - Mathematics - 202 pages
This 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

1 Algebras and their Defining Relations
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
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199
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200
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201
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