Theory of linear G-difference equations

arXiv:q-alg/9712044v117Dec1997THETHEORYOFLINEARG-DIFFERENCEEQUATIONSPERK.JAKOBSENVALENTINV.LYCHAGINAbstract.Weintroducethenotionofdifferenceequationdefinedonastruc-turedset.Thesymmetrygroupofthestructuredeterminesthesetofdif-ferenceoperators.Allmainnotionsinthetheoryofdifferenceequationsareintroducedasinvariantsoftheactionofthesymmetrygroup.Linearequationsaremodulesovertheskewgroupalgebra,solutionsaremorphismsrelatingagivenequationtootherequations,symmetriesofanequationaremoduleen-domorphismsandconservedstructuresareinvariantsinthetensoralgebraofthegivenequation.Weshowthattheequationsandtheirsolutionscanbedescribedthroughrepresentationsoftheisotropygroupofthesymmetrygroupoftheunderlyingset.Werelateournotionofdifferenceequationandsolutionstosystemsofclassicaldifferenceequationsandtheirsolutionsandshowthatoutnotionsinclusetheseasaspecialcase.Date:December17l,1997.1991MathematicsSubjectClassification.Primary:39A05;Secondary:39A70.Keywordsandphrases.Finitedifferenceequations,modules,morphisms,categories.12PERK.JAKOBSENVALENTINV.LYCHAGINContents1.Introduction22.Themainnotionsinthetheoryoffinitetypedifferenceequationsonaset42.1.ThealgebraofG-differenceoperators52.2.LinearG-differenceequationsoffinitetypeandsolutions53.ThestructureoftheCategoryofGF-differenceequations63.1.ThealgebraofGF-differenceequations73.2.GF-differenceequationsasmodulesofsectionsinvectorbundles93.3.ThegeometricdescriptionofA-morphisms163.4.ThegeneralstructureofGFE194.TheequivalenceTheorem205.TheprojectionformulaforGF-differenceequations236.CoordinatedescriptionofGF-differenceequations256.1.Coordinatedescriptionoftensoroperations266.2.CoordinatedescriptionofA-modulemorphismsandsolutions287.Invariantstructures287.1.Conservedquantities297.2.Self-dualequations297.3.Solutionsandcompositionprinciples308.Moduledescriptionofclassicaldifferenceequations308.1.Themoduleofdifferenceoperators318.2.Compositionofdifferenceoperators328.3.Modulescorrespondingtodifferenceoperators348.4.Classicalsolutions358.5.Modulescorrespondingtosystemsofdifferenceequations35References361.IntroductionLetusconsiderageneralsecondorderdifferenceequationoftheformaifi+1+bifi+cifi−1=0IntroducethesimplegraphSconsistingofvertices{xi}i∈Zandedges{{xi,xi+1}}i∈Z.LetF(S)betheR-algebraofR-valuedfunctionsonthegraphS.Thenthese-quences{ai},{bi},{ci}and{fi}areallelementsinF(S).Denotetheseelementsbya,b,c,andf.LetsbetheoperatoroflefttranslationonthelatticeS,sxi=xi−1.ThensactsonF(S)inthenaturalway(sf)(xi)=f(s−1xi)Define△=as+be+cs−1whereeactsastheidentityonS.Then△actsonF(S)asaR-linearoperatorandouroriginalequationcanbewritten△(f)=0Inordertounderstandwhat△isinalgebraictermsweneedtointroducesomenewnotions.LetG=Aut(S)betheautomorphismgroupofthegraphS.ThisTHETHEORYOFLINEARG-DIFFERENCEEQUATIONS3groupactsonF(S)inthenaturalway(gf)(xi)=f(g−1xi).LetAbethesetoffiniteformallinearcombinationsofelementsinGwithcoefficientsinF(S).A={Xgagg|ag∈F(S)}OnthesetAwedefineadditionandscalarmultiplicationwithelementsr∈Rcomponentwise.Productisdefinedinthefollowingway(ag)(bg′)=(agb)(gg′)WiththeseoperationsAisaR-algebra.A=F(S)[G]istheskewgroupalgebraofGoverF(S).ThisalgebraactsonF(S)through(Xgagg)f=Xgagg(f)Usingthesenotionsweobservethatourclassicaldifferenceoperator△=as+be+cs−1isanelementoftheskewgroupalgebraA.ItisnowevidentthatwecaninterpretallelementsinAasdifferenceoperatorsoverS.WewillinfactdefineAtobethealgebraofG-differenceoperatorsoverS.ThismeansthatthenotionofdifferenceoperatorisdefinedintermsofthesymmetriesoftheunderlyinggraphS.ThegroupofsymmetriesofSmeasuresthearbitrarinessinthedescriptionofS.Withoutthisarbitrarinessdifferenceoperatorscouldnotexist,inatotallyasym-metricalspacewithtrivialsymmetrygrouptherecouldbenodifferenceoperatorsandasaconsequencenodifferenceequations.WewillinthispapergeneralizethesesimpleobservationsandconsiderasetSandagroupGactingonS.ForanysuchgroupactionthereexistssomestructureonSsuchthatGisasubgroupofthefullautomorphismgroupofthisstructure.IfthesetisfinitethenthegroupisactuallythefullgroupofautomorphismsofthespaceS.ThealgebraofscalardifferenceoperatorsonSwillbetheskewgroupalgebraA=F(S)[G]whereF(S)willbethealgebraofF-valuedfunctionsdefinedonS.DifferenceequationsonSandtheirsolutionsmustbeinvariantobjectsundertheactionofthegroupG.Iftheyarenotinvarianttheirdescriptionandsolutionswilldependonthearbitrarinessinthespecificationoftheunderlyingspace.TheKleinErlangerprogramingeometryhasshownthatthebuildingblocksofthegeometryofasetwithagroupactionaretheinvariantsofthegroup.Geometricalobjectsandtheirrelationsareconstructedfrominvariants.Inthiswaythegeometrywillnotdependonthearbitrarinessoftheunderlyingspace.WhatweproposeinthispaperisinthespiritoftheErlangerprogramingeometry.Weproposethatthebuildingblocksofthetheoryofdifferenceequationsonafinitespacewithsomestructurearetheinvariantsofthegroupofautomorphismsactingonthespace.Thealgebraofdifferenceoperatorswillbetheskewgroupalgebra,A,ofGandallmainnotionsinthetheoryofdifferenceequationswillbedefinedintermsofinvariants.AlineardifferenceequationsEwillbeanA-module,symmetriesofEwillbeA-endomorphismsofE.Allconserv