Noncommutative Gröbner Bases and Filtered-Graded Transfer

201 Pages·2002·0.831 MB·English

by  Huishi Li (auth.)

1795
Lecture Notes in Mathematics
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
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Berlin
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Huishi Li
Noncommutative
GröbnerBasesand
Filtered-Graded Transfer
1 3
Author
HuishiLI
DepartmentofMathematics
BilkentUniversity
P.O.Box217
06533Ankara
Turkey
e-mail:[email protected]
http://www.fen.bilkent.edu.tr/˜huishi
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MathematicsSubjectClassification(2000):16Z05,68W30,16W70,16S99
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Acknowledgement
On April 5th, 1995, on the way from Bielefeld to Xi’an, I stopped at Beijing
and met my old friend Dr. Luo Yunlun (Lao Luo). Drinking Chinese green tea,
Lao Luo introduced Computational Algebra to me. It was the first time I had
heard of the notion of a Gro¨bner basis. Soon after our meeting, Lao Luo sent
me some basic material on commutative computational algebra and a tutorial
book introducing Macaulay that enabled me to start my computational algebra
seminar combined with my elementary algebraic geometry course given at the
Shaanxi Normal University in the spring of 1996. Without this unforgettable
story, I would not have had the idea of working on the algebraic-algorithmic
aspect. Thank you, Lao Luo.
I wish to express my thanks to the referees for their valuable remarks on im-
proving the notes.
IamgratefultotheLNMeditorsfortheirhelpfulsuggestionswhileIwasprepar-
ing the manuscript.
Huishi Li

Contents
Introduction 1
CHAPTER I
Basic Structural Tricks and Examples 5
1. Algebras and their Defining Relations 5
2. Skew Polynomial Rings 9
3. ZZ-filtrations and their Associated Graded Structures 13
4. Homogenization and Dehomogenization of ZZ-graded Rings 25
5. Some Algebras: Classical and Modern 28
CHAPTER II
Gr¨obner Bases in Associative Algebras 33
1. (Left) Monomial Orderings and (Left) Admissible Systems 34
2. (Left) Quasi-zero Elements and a (Left) Division Algorithm 38
3. (Left) Gro¨bner Bases in a (Left) Admissible System 42
4. (Left) Gro¨bner Bases in Various Contexts 45
5. (Left) S-elements and Buchberger Theorem 48
6. (Left) Dickson Systems and (Left) G-Noetherian Algebras 54
7. Solvable Polynomial Algebras 58
8. No (Left) Monomial Ordering Existing on ∆(k[x1,...,xn])
with chark >0 61
CHAPTER III
Gr¨obner Bases and Basic Algebraic-Algorithmic
Structures 67
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 Gro¨bner Bases 89
viii Contents
CHAPTER IV
Filtered-Graded Transfer of Gr¨obner Bases 91
1. Filtered-Graded Transfer of (Left) Admissible Systems 92
2. Filtered-Graded Transfer of (Left) Gro¨bner Bases 97
3. Filtered-Graded Transfer of (Left) Dickson Systems 100
4. Filtered-Graded Transfer Applied to Quadric Solvable
Polynomial Algebras 103
CHAPTER V
GK-dimension of Modules over Quadric Solvable
Polynomial Algebras and Elimination of Variables 107
1. Gro¨bner Bases in Homogeneous Solvable Polynomial Algebras 108
2. The Hilbert Function of A/L 110
3. The Hilbert Polynomial of A/L 112
4. GK-dimension Computation and Elimination of Variables
(Homogeneous Case ) 115
5. GK-dimension Computation and Elimination of Variables
(Linear Case) 119
6. The (cid:1) -filtration on a Quadric Solvable Polynomial Algebra 124
gr
7. GK-dimension Computation and Elimination of Variables
(General Quadric Case) 126
8. Finite Dimensional Cyclic Modules 130
CHAPTER VI
Multiplicity Computation of Modules over Quadric
Solvable Polynomial Algebras 133
1. The Multiplicity e(M) of a Module M 134
2. Computation of e(M) 135
3. Computation of GK.dim(M ⊗ N) and e(M ⊗ N) 144
k k
4. An Application to An(q1,...,qn) 148
CHAPTER VII
(∂-)Holonomic Modules and Functions over Quadric
Solvable Polynomial Algebras 153
1. Some Operator Algebras 154
2. Holonomic Functions 156
3. Automatic Proving of Holonomic Function Identities 162
4. Extension/Contraction of the ∂-finiteness 164
5. The ∂-holonomicity 168
Contents ix
CHAPTER VIII
Regularity and K -group of Quadric Solvable
0
Polynomial Algebras 175
∼
1. Tame Case: A is Auslander Regular with K0(A)=ZZ 176
2. The (cid:1) -filtration on Modules 177
gr
3. General Case: gl.dimA≤n 181
∼
4. General Case: K0(A)=ZZ 186
References 187
Index 195
Introduction
It is well-known that because of the successful implementation of Buchberger’s
algorithm on computer (1965, 1985), the commutative Gro¨bner basis theory
has been very powerful in both pure and applied commutative mathematics. It
is equally well-known that Buchberger’s algorithm has been generalized to the
noncommutative area via noncommutative Gr¨obner bases in various contexts,
for instance,
• theGro¨bnerbasesforone-sidedidealsinalgebrasoflinearpartialdifferential
operatorswithpolynomialcoefficients(orWeylalgebras)introducedin([Gal]
1985), the Gro¨bner bases for one-sided ideals in the enveloping algebras of
finitedimensionalLiealgebrasintroducedin([AL]1988),andmoregenerally,
theGro¨bnerbasistheorydevelopedin([K-RW]1990)foralargeclassofnon-
commutative polynomial-like algebras, i.e., the class of solvable polynomial
algebras,
• the Gro¨bner basis theory for two-sided ideals in a noncommutative free alge-
bra developed in ([Berg] 1978) and in ([Mor1-2] 1986, 1994),
• and the Gro¨bner basis theory introduced in ([Gr] 1993) for two-sided ideals
inanalgebrawhichpossiblyhasnoidentitybutpossiblyhasdivisorsofzero.
From now on in this monograph, all strings of references will be listed alphabet-
ically.
Supportedbyseveralnoncommutativecomputeralgebrasystems(seeanincom-
plete list of websites at the end of this introduction for the references), noncom-
mutativeGro¨bnerbaseshavebeeneffectivelyusedinmanycontextsofpureand
applied algebra, such as the algorithmic study of PDE’s, the automatic proof
of functional identities, the representation of algebras etc. (see [ACG1–2], [CS],
[Gr], [LC1–2], [Oak1–2], [SST], [Tak1–3]).
H.Li:LNM1795,pp.1–4,2002.
(cid:1)c Springer-VerlagBerlinHeidelberg2002