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Non–commutative Computer Algebra for polynomial algebras: Gr¨obnerbases, applications and implementation Viktor Levandovskyy Vom Fachbereich Mathematik der Universit¨atKaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat) genehmigte Dissertation 1. Gutachter: Prof. Dr. G.-M. Greuel 2. Gutachter: Prof. Dr. Yu. Drozd Vollzug der Promotion: 08.06.2005 D 386 iii Viktor Levandovskyy. ”Non-commutative Computer Algebra for polynomial algebras: Gr¨obnerbases, applications and implementation”. Abstract. Non-commutative polynomial algebras appear in a wide range of applications, from quantum groups and theoretical physics to linear differential and difference equations. In the thesis, we have developed a framework, unifying many impor- tant algebras in the classes of G– and GR–algebras and studied their ring– theoretic properties. Let A be a G–algebra in n variables. We establish necessary and sufficient conditions for A to have a Poincar´e–Birkhoff–Witt (PBW) basis. Further on, we show that besides the existence of a PBW ba- sis, A shares some other properties with the commutative polynomial ring K[x1, . , xn]. In particular, A is a Noetherian integral domain of Gel’fand– Kirillov dimension n. Both Krull and global homological dimension of A are bounded by n; we provide examples of G–algebras where these inequalities are strict. Finally, we prove that A is Auslander–regular and a Cohen– Macaulay algebra. In order to perform symbolic computations with modules over GR– algebras, we generalize Gr¨obnerbases theory, develop and respectively en- hance new and existing algorithms. We unite the most fundamental algo- rithms in a suite of applications, called ”Gr¨obnerbasics” in the literature. Furthermore, we discuss algorithms appearing in the non-commutative case only, among others two–sided Gr¨obnerbases for bimodules, annihilators of left modules and operations with opposite algebras. An important role in Representation Theory is played by various sub- algebras, like the center and the Gel’fand–Zetlin subalgebra. We discuss their properties and their relations to Gr¨obnerbases, and briefly comment some aspects of their computation. We proceed with these subalgebras in the chapter devoted to the algorithmic study of morphisms between GR– algebras. We provide new results and algorithms for computing the preim- age of a left ideal under a morphism of GR–algebras and show both merits and limitations of several methods that we propose. We use this technique for the computation of the kernel of a morphism, decomposition of a mod- ule into central characters and algebraic dependence of pairwise commuting elements. We give an algorithm for computing the set of one–dimensional representations of a G–algebra A, and prove, moreover, that if the set of finite dimensional representations of A over a ground field K is not empty, then the homological dimension of A equals n. iv All the algorithms are implemented in a kernel extension Plural of the computer algebra system Singular. We discuss the efficiency of com- putations and provide a comparison with other computer algebra systems. We propose a collection of benchmarks for testing the performance of al- gorithms; the comparison of timings shows that our implementation out- performs all of the modern systems with the combination of both broad functionality and fast implementation. In the thesis, there are many new non-trivial examples, and also the solutions to various problems, arising in different fields of mathematics. All of them were obtained with the developed theory and the implementation in Plural, most of them are treated computationally in this thesis for the first time. Viktor Levandovskyy. ”Nichtkommutative Computeralgebra f¨ur Polynomalgebren: Gr¨obnerbasen, Anwendungen und Implementierung”. Zusammenfassung. Nichtkommutative Polynomalgebren entstehen in vielen verschiedenen Anwendungen, von Quantengruppen und Theoretis- cher Physik bis zu linearen Differentiellen und Differenzengleichungen. In der Arbeit wurde ein Rahmen entwickelt, in dem viele wichtige Alge- bren der Klassen G– und GR–Algebren zusammengef¨uhrtund ihre ringth- eoretischen Eigenschaften untersucht wurden. Sei A eine G–Algebra mit n Variablen. Es werden notwendige und hinreichende Bedingungen daf¨ur angegeben, daß A eine Poincar´e–Birkhoff–Witt(PBW) Basis besitzt. Es wird gezeigt, dass A neben der Existenz einer PBW Basis, weitere Eigen- schaften mit dem kommutativen Polynomring K[x1, . , xn] gemeinsam hat. A ist ein Noetherscher Integrit¨atsbereich der Gel’fand–Kirillov–Dimension n. Die Krull– und die globale homologische Dimension von A sind durch n beschr¨ankt; es werden Beispiele von G–Algebren gegeben, bei denen diese Ungleichheiten strikt sind. Schließlich wurde bewiesen, dass A eine Auslander–regul¨areund eine Cohen–Macaulay Algebra ist. F¨ursymbolische Berechnungen mit Moduln ¨uber GR–Algebren, wurde die Gr¨obnerbasentheorie verallgemeinert, neue und bestehende Algorith- men werden entwickelt und verbessert. Wir verbinden die grundlegendsten Algorithmen mit einer Reihe von Anwendungen, welche man in der Lit- eratur als ”Gr¨obnerbasics” bezeichnet. Weiterhin wurden Algorithmen v diskutiert, die nur im nichtkommutativen Fall existieren, darunter zwei- seitige Gr¨obnerbasenf¨urBimoduln, Annihilatoren von Linksmoduln und Operationen mit entgegengesetzten Algebren. Eine wichtige Rolle in der Darstellungstheorie spielen die verschiedenen Unteralgebren, wie z.B. das Zentrum und die Gel’fand–Zetlin Unteralgebra. Die Eigenschaften und ihre Beziehungen zu Gr¨obnerbasenwurden untersucht und einige Aspekte ihrer Berechnung diskutiert. Im Kapitel ¨uber algorithmische Studien von Morphismen zwischen GR–Algebren wurde die Untersuchung zu diesen Un- teralgebren fortgesetzt. Es wurden neue Ergebnisse and Algorithmen zur Berechnung des Urbilds eines Linksideals unter einem Morphismus von GR– Algebren erzielt. Die Ergebnisse und Begrenzungen verschiedener Meth- oden, die vorgeschlagen wurden, wurden gezeigt. Diese Technik wurde auch f¨urdie Berechnung des Kerns eines Morphismus, die Zerlegung eines Moduls in zentrale Charaktere und die algebraische Abh¨angigkeit von paar- weise kommutierenden Elementen verwendet. Es wurde ein Algorithmus f¨urdie Berechnung von eindimensionalen Darstellungen einer G–Algebra A erstellt. Es wurde bewiesen, dass die homologische Dimension von A ¨uber einem K¨orper K gleich der Anzahl der Variablen von A ist, falls es endlich–dimensionale Darstellungen von A existieren. Alle Algorithmen wurden in eine Kern-Erweiterung Plural des Com- puteralgebrasystem Singular implementiert. Die Effizienz der Berechnun- gen wurde diskutiert und ein Vergleich mit anderen Computeralgebrasys- temen erstellt. Es wurde eine Reihe von Benchmarks zum Test der Leis- tung von Algorithmen vorgeschlagen. Der Zeitvergleich zeigt, dass unsere Implementierung alle modernen Systeme hinsichtlich der Kombination von Funktionalit¨atund Geschwindigkeit ¨ubertrifft. In der Arbeit gibt es eine Vielzahl von nichttrivialen Beispielen und L¨osungenzu verschiedenen Problemen aus verschiedensten Bereichen der Mathematik. All wurden mit Hilfe der entwickelten Theorie und der Im- plementation in Plural erzielt, die meisten von ihnen wurden in dieser Arbeit zum ersten Mal berechnet. Contents Introduction ix Chapter 1. From NDC Towards G–algebras 1 1. Gr¨obnerBases on Free Associative Algebras 2 2. Non–degeneracy Conditions and PBW Theorem 6 3. Introduction to G–algebras 11 4. Filtrations and Properties of G–algebras. 18 5. Opposite algebras 24 6. The Ubiquity of GR–algebras 27 7. Applications of Non–degeneracy Conditions 28 8. Conclusion and Future Work 43 Chapter 2. Gr¨obnerbases in G–algebras 45 1. Left Gr¨obnerBases 45 2. Gr¨obnerbasics I 54 3. Two–sided Gr¨obnerBases and GR–algebras 61 4. Syzygies and Free Resolutions 67 5. Gr¨obnerbasics II 78 6. Conclusion and Future Work 92 Chapter 3. Morphisms of GR–algebras 95 1. Subalgebras and Morphisms 95 2. Morphisms from Commutative Algebras to GR–algebras 102 3. Central Character Decomposition of the Module 111 4. Morphisms between GR–algebras 116 5. Conclusion and Future Work 129 Chapter 4. Implementation in the system Singular:Plural 131 1. Singular and Plural: history and development 131 2. Aspects of Implementation 137 3. Timings, Performance and Experience 141 4. Download ans Support 144 5. Conclusion and Future Work 145 Chapter 5. Small Atlas of Important Algebras 147 1. Universal Enveloping Algebras of Lie Algebras 147 vii viii CONTENTS 2. Quantum Algebras 149 Chapter 6. Singular:Plural User Manual 151 1. Getting Started with Plural 151 2. Data Types (plural) 152 3. Functions (plural) 170 4. Mathematical Background (plural) 195 5. Plural Libraries 201 Appendix A. Noncommutative Algebra: Examples 240 A.1. Left and two-sided Groebner bases 240 A.2. Right Groebner bases and syzygies 242 Conclusion and Future Work 245 Bibliography 247 Introduction This thesis is devoted to symbolic computations in non–commutative algebras with PBW bases (G–algebras). These algebras already have their own history as well as symbolic methods, algorithms and implementations. Yet it is still a long way towards the acceptance and wide usage of these methods by the community of scientists, comparable to the years which were gone before the Buchberger’s algorithm for computing a Gr¨obnerbasis of an ideal in a commutative ring become accepted by everybody. G–algebras appear in many fields of science, from Non–commutative Algebra, Ring Theory and Representation Theory
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