Algebraic Properties of Ore Extensions and Their Commutative Subrings

139 Pages·2014·0.692 MB·English

by  Johan Richter

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
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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
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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.
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