Some Criteria for Solvable and Supersolvable Leibniz Algebras

70 Pages·2016·0.29 MB·English

by  Bethany Nicole Turner

ABSTRACT
TURNER,BETHANYNICOLE.SomeCriteriaforSolvableandSupersolvableLeibnizAlgebras.
(UnderthedirectionofErnestStitzingerandKailashMisra.)
LeibnizalgebrasaregeneralizationsofLiealgebras.SincetheintroductionofLeibniz
algebrasin1993byJean-LouisLoday,manyresultsforLiealgebrashavebeengeneralized
to the Leibniz case, such as Lie’s Theorem, Engel’s Theorem, Cartan’s Criterion and the
LeviDecomposition.Since2008,motivatedbygrouptheory,DavidTowershasdefined
c-ideals,c-sections,andcompletionsofmaximalsubalgebrasofLiealgebras.Hehasused
these,aswellasCartansubalgebrasandCAP-subalgebras,tocharacterizesolvableand
supersolvableLiealgebras.
Weintroducedefinitionsforc-ideals,CAP-subalgebras,c-sectionsandcompletions
formaximalsubalgebrasofLeibnizalgebras.Wedeveloppropertiesofthesesubalgebras
ofLeibnizalgebras.Wethengiveseveralcharacterizationsofsolvableandsupersolvable
Leibnizalgebrasbasedonthebehaviorofthesesubalgebras.
©Copyright2016byBethanyNicoleTurner
AllRightsReserved
SomeCriteriaforSolvableandSupersolvableLeibnizAlgebras
by
BethanyNicoleTurner
AdissertationsubmittedtotheGraduateFacultyof
NorthCarolinaStateUniversity
inpartialfulfillmentofthe
requirementsfortheDegreeof
DoctorofPhilosophy
Mathematics
Raleigh,NorthCarolina
2016
APPROVEDBY:
ErnestStitzinger KailashMisra
Co-chairofAdvisoryCommittee Co-chairofAdvisoryCommittee
NathanReading MohanPutcha
PaulHuffman
DEDICATION
Totheoneunderthebed.
ii
ACKNOWLEDGEMENTS
Severalothersmadethisworkpossible,andIextendmythankstoallofthem,including:
Dr.Misraforinspiringperseveranceandhelpingmetoarticulate;Dr.Stitzingerforkeeping
mepointedintherightmathematicaldirection;andDr.Towersforleadingtheway.Ialso
thankthosemanygiantswhohavemadetheirshoulderscomfortable.
Thisworkexistswithsupportfromfamily,friendsandteachers.Ithankmyhusband
Padraicforhelpingmebeherenow,mymotherLindaforhertirelessbeliefinmyability,my
fatherDarrellforteachingmehowtoaskquestions,andPanforallthesoothingpurring.
iii
TABLEOFCONTENTS
Chapter1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter3 SubalgebraProperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 C-supplementedSubalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 C-ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 One-dimensionalsubalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 C-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 CAP-subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Chapter4 SolvablilityCriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 C-ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 C-Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 CAP-subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter5 SupersolvabilityCriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1 C-ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 CAP-subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
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CHAPTER
1
INTRODUCTION
Liealgebras,whicharosethoughthestudyofLiegroups,havebeenwell-studiedsincetheir
introductioninthemid-nineteenthcentury.Sincethentheyhavefoundapplicationsin
bothphysicsandappliedmathematics.
VariousgeneralizationsofLiealgebrashavebeenstudied.Forexample,in1955Malcev
algebraswereintroducedin[Mal55].Leibnizalgebrasweredefinedin1993byJean-Louis
Loday[Lod93].ThemajordifferenceisthatLeibnizalgebrasarenotantisymmetric.Since
thattime,analoguesofmanymajorLiealgebraresultshavebeenprovenforLeibnizalgebras.
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CHAPTER1. INTRODUCTION
AmongtheseareLie’sTheorem,Engel’sTheorem,Levidecomposition,andCartan’scriterion
[Dem13]. Much of the research into Leibniz algebras is focused on determining which
propertiesofLiealgebrascanbegeneralized,asin[Bar11]and[Bat13].
Animportantclassofnon-LieLeibnizalgebrasarethecyclicLeibnizalgebras.Inrecent
years,methodshavebeendevelopedforcomputingmaximalsubalgebras,Cartansubalge-
bras,minimalideals,andFrattinisubalgebrasofcyclicalgebras[McA14].ThecyclicLeibniz
algebrashavebeenclassifiedoverthecomplexnumbersin[SS14].Theseresultsareused
throughoutthisworktoconstructnon-Lieexamples.
ThisworkgeneralizessomeresultsforLiealgebrasthatwereoriginallymotivatedby
grouptheory.CompletionsofmaximalsubgroupsweredefinedbyW.E.Deskinsin[Des59],
andfurtherstudiedin[Des90],[BBE92]and[MB89].Deskinsdefinedthenormalindexofa
maximalsubgroup,andprovedthatafinitegroupissolvableifandonlyifallitsmaximal
subgroupshaveprimepowernormalindex.Ballester-Bolinchesusedcompletionsandthe
normalindextocharacterizesupersolvablegroups.
Thegrouptheoryconceptsofc-supplementedsubgroups,c-normalsubgroups,and
c-sections were defined and studied by Y. Wang and others in [BB00], [Wan] and [WS].
Wangprovedin[Wan]thatafinitegroupissolvableifandonlyifallitsmaximalsubgroups
arec-normal.In[WS],WangandShirongdescribedtherelationshipbetweenc-normal
subgroups,c-sectionsandthenormalindex.
Since 2008, D. Towers has defined analogues of these group theory concepts for Lie
algebras,andusedthemtocharacterizesolvabilityandsupersolvability.Hedefinedc-ideals
forLiealgebrasin[Tow09],motivatedbyc-normalsubgroups.HeprovedthataLiealgebra
issolvableifandonlyifallitsmaximalsubalgebrasarec-ideals.In[Tow11]hedefinedthe
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CHAPTER1. INTRODUCTION
idealindexofamaximalsubalgebra,andprovedthataLiealgebraissolvableifandonlyif
theidealindexofeverymaximalsubalgebraequalsitscodimension.Hedefinedc-sections
ofmaximalsubalgebrasin[Tow15a].
CAP-subalgebrasofLiealgebrashavebeenstudiedin[HO70]and[Sti72].In[Tow15b],
Towers gave some properties of CAP-subalgebras in Lie algebras, and proved that a Lie
algebraissolvableifandonlyifeachofitsmaximalsubalgebrasisaCAP-subalgebra.He
alsoprovedthatinasupersolvableLiealgebra,everysubalgebraisaCAP-subalgebra.
Theprimarypurposeofthisworkistogeneralize,wheneverpossible,theresultson
solvableandsupersolvableLiealgebrasdescribedabovetotheLeibnizcase.Throughout,
wealsodeveloppropertiesoftherelevantsubalgebras,andrelationshipsbetweenthem.
Wheneverpossible,weemployatechniqueofgeneralizationoftheLiealgebraproofs.
WegivebasicLeibnizalgebradefinitionsinChapter2.Wethenintroducedefinitionsof
c-ideals,c-sections,completions,andCAP-subalgebrasfortheLeibnizcase,illustrating
withnon-LieLeibnizexamples.
In Chapter 3 we develop properties of c-ideals, c-sections, completions, and CAP-
subalgebrasinLeibnizalgebras.Wecharacterizec-idealsandc-sectionsinsimpleLeibniz
algebras,whichdiffersfromtheLiecase.Wealsodefinetheidealindexη(A:M),andthe
c-sectiondimensionη∗(A:M)formaximalsubalgebras,andgiveanequationrelatingthese
quantities.Wegiveconditionsunderwhicheveryone-dimensionalsubalgebraisac-ideal.
InChapter4wegiveseveralcharacterizationsofsolvableLeibnizalgebras.Weprove
thateachofthefollowingconditionsisequivalenttosolvabilityinaLeibnizalgebra:(i)
Everymaximalsubalgebraisac-ideal,(ii)Everymaximalsubalgebrahasidealindexequal
toitscodimension,(iii)Everymaximalsubalgebrahastrivialc-section,(iv)Everymaximal
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CHAPTER1. INTRODUCTION
subalgebrahasanabelianidealcompletion,and(v)EverymaximalsubalgebraisaCAP-
subalgebra.WeprovethataLeibnizalgebraissolvableifeveryCartansubalgebraisac-ideal,
andwepresentaLiecounterexampletotheconverse.
In Chapter 5 we give characterizations of supersolvable Leibniz algebras. We show
thataLeibnizalgebraissupersolvableifandonlyifeverymaximalsubalgebrasatisfies
η(A:M)=1.WepresentoneresultwhichwehavenotseenforLiealgebras,namelythat
LeibnizalgebrasaresupersolvableifandonlyifalltheirsubalgebrasareCAP-subalgebras.
We prove that one sufficient condition for supersolvability given by Towers is in fact a
characterizationofsupersolvabilityinthenon-LieLeibnizcase.
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