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Algebraic Properties of Ore Extensions and their Commutative Subrings Johan Richter Faculty of Engineering Centre for Mathematical Sciences Mathematics Mathematics Centre for Mathematical Sciences Lund University Box 118 SE-221 00 Lund Sweden http://www.maths.lth.se/ Doctoral Theses in Mathematical Sciences 2014:3 ISSN 1404-0034 ISBN 978-91-7623-068-8 LUTFMA-1049-2014 (cid:13) Johan Richter, 2014 Printed in Sweden by Media–Tryck, Lund 2014 Preface Thisthesisisbasedonsixpapers(A–F).InPartIofthethesiswegiveanintroduc- tion to the subject and present a summary of the results found in thesix papers. PartIIconsistsofthepapersthemselves,whicharethefollowing: A. J. Richter, S. D.Silvestrov, On algebraiccurves for commutingelementsinq- Heisenbergalgebras,J.Gen. Lie. T.Appl. 3(2009),no. 4,321–328. B. J. Richter, S.D. Silvestrov, Burchnall-Chaundy annihilating polynomials for commutingelementsinOreextensionrings,J.Phys.: Conf. Ser. 342(2012) C. J.Öinert,J.Richter,S.D.Silvestrov,Maximalcommutativesubringsandsim- plicityofOreextensions,J.AlgebraApppl. 12,1250192(2013),arXiv:1111.1292 (2011) D. J.Richter,Burchnall-ChaundytheoryforOreextensions,inSpringerProceed- ings in Mathematics & Statistics, Vol. 85, ed: Abdenacer Makhlouf et al: Algebra,GeometryandMathematicalPhysics,arXiv:1309.4415 E. J.Richter,ANoteon“ACombinatorialProofofAssociativityofOreExtensions”, DiscreteMathematics,Volumes315–316,6February2014,Pages156–157 F. J. Richter, S.D.Silvestrov, CentralizersinOre extensionsofpolynomialrings, International Electronic Journal of Algebra, Volume 15 (2014), 196–207, arXiv:1308.3430 iii Contents Preface iii Acknowledgements ix I Introduction and summary 1 1 Introduction 3 1.1 Notationandconventions . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Oreextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Differentialoperatorrings . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 GeneralOreextensions . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Nystedt’sproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Centralizers inOreextensions . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Burchnall-Chaundy theory. . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 AlgorithmicBurchnall-Chaundy theory . . . . . . . . . . . . . 14 1.5 Simplicityandmaximalcommutativity . . . . . . . . . . . . . . . . . . 16 1.5.1 Motivationfromoperatoralgebrasanddynamicalsystems . 16 1.5.2 SimplicityofOreextensions . . . . . . . . . . . . . . . . . . . . 18 2 Summaryofthethesis 21 2.1 OverviewofPaperA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 OverviewofPaperB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 OverviewofPaperC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 OverviewofPaperD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 OverviewofPaperE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 OverviewofPaperF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 II Scientific papers 29 A Onalgebraiccurvesforcommutingelementsinq-Heisenbergalgebras 33 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 A.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A.3 R ismaximalcommutative . . . . . . . . . . . . . . . . . . . . . . . . . 37 0 v CONTENTS A.4 Annihilatingpolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A.5 TheeliminantwhentheelementsbelongtoR . . . . . . . . . . . . . 39 0 A.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 B Burchnall-Chaundyannihilating polynomials for commutingelements inOreextensionrings 47 B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 B.2 ExtensionofBurchnall–Chaundy theorytoOreextensions . . . . . . 49 B.2.1 Determinantpolynomial . . . . . . . . . . . . . . . . . . . . . . 50 B.2.2 Theresultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 B.3 Recursiveconstructionofthematrixoftheresultant . . . . . . . . . 52 B.3.1 TheHeisenbergalgebracase. . . . . . . . . . . . . . . . . . . . 52 B.4 Necessityofinjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 B.5 Thecaseδ=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 B.5.1 Theleadingcoefficientsingeneral . . . . . . . . . . . . . . . . 56 B.6 Thelower-ordercoefficients . . . . . . . . . . . . . . . . . . . . . . . . 57 B.7 Annihilating polynomials for elements in a specific commutative subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 C MaximalcommutativesubringsandsimplicityofOreextensions 71 C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 C.2 Oreextensions. Definitionsandnotations . . . . . . . . . . . . . . . . 73 C.3 ThecentralizerandmaximalcommutativityofRinR[x;σ,δ] . . . 74 C.3.1 Skewpolynomialrings . . . . . . . . . . . . . . . . . . . . . . . 76 C.3.2 Differentialpolynomialrings . . . . . . . . . . . . . . . . . . . 77 C.4 ThecenterofR[x;σ,δ] . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 C.5 SimplicityconditionsforR[x;σ,δ] . . . . . . . . . . . . . . . . . . . . 80 C.5.1 Differentialpolynomialrings . . . . . . . . . . . . . . . . . . . 83 D Burchnall-ChaundytheoryforOreextensions 95 D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 D.1.1 Notationandconventions . . . . . . . . . . . . . . . . . . . . . 95 D.2 Burchnall-Chaundy theoryfordifferentialoperatorrings . . . . . . . 97 D.3 Burchnall-Chaundy theoryforOreextensions . . . . . . . . . . . . . . 101 E ANoteon“ACombinatorialProofofAssociativityofOreExtensions” 109 E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 E.2 Theproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 vi CONTENTS F CentralizersinOreextensionsofpolynomialrings 117 F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 F.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 F.3 Centralizers arefreeK[P]-modules . . . . . . . . . . . . . . . . . . . . 120 F.4 Centralizers arecommutative. . . . . . . . . . . . . . . . . . . . . . . . 120 F.5 Singlygeneratedcentralizers . . . . . . . . . . . . . . . . . . . . . . . . 122 vii Acknowledgements IwouldliketobeginbythankingmysupervisorSergeiSilvestrovforconvincingme toapplytothePhD-programmeandforourcooperationovertheseyears. Iwould also like tothank myco-supervisor Johan Öinert forgoodcooperation andmany stimulatingdiscussions. Myotherco-supervisor,AnnaTorstensson,hascontributed interestingcommentsonPaperCinthethesis. VictorUfnarovskishowedmethatmathcanbefunandcreativewhileIwasin highschool. Forthathehasmygratitude. A big collectivethank yougoes toall myco-workers attheCentre forMathe- maticalSciences. Youhaveallcontributedtoastimulating workplaceandmany ofyouhaveparticipatedinfundiscussionsoverlunchorcoffee. TheworkinthisthesishasbeenfinanciallysupportedbytheLannerandDahlgren funds,theSwedishResearchCouncil,theSwedishFoundationforInternationalCo- operation in Research and Higher Education (STINT), the Crafoord Foundation, theRoyalPhysiographicSocietyinLund,theRoyalSwedishAcademyofSciences, the Nordforsk Research Network “Operator algebras and Dynamics”, the Danish National Research Foundation (DNRF) and the Mathematisches Forschungsinsti- tutOberwolfach. Last,butnotleast,myfamilyhasbeenaconstantsourceofloveandsupport. Thankyou. ix CONTENTS x