; TeX output 2001.06.04:1801 XIZlgE3K`y cmdunh10TwistedGroupRingsK`y cmr10BoGdoUUPareigisjMathematischesUUInstitut rcUniversitatUUMGunchenw!ITheresienstraeUU39b8000UUMGunchen2,Germany#^ٓRcmr71)Kw:+- cmcsc101.IntroductionThisLpapGerdealswith(twisted)Hopfalgebraformsofgrouprings,i.e.withmHopfalgebras b> cmmi10H=overmabaseringK,suchthatforasuitableextensionLofKLʫtheHopfalgebraLc !", cmsy10  0ercmmi7K xHeisisomorphictoagroupringLGoverL.OThese|HopfalgebrasarisefromanadjointsituationasasolutionofauniversalUUprobleminuniversalalgebra.OneofthewellknownexamplesofapairofadjointfunctorsistheunitsfunctorU:K-Algb!GrassoGciatingthegroupofunitsorinvert-ibleRelementsU(A)witheachK-algebraA,anditsleftadjointfunctorK- q:Gr5OQ!K-Alg{aassoGciating'witheachgroupGitsgroupringKG.bThegroup\ringKGactuallyturnsouttobGeacocommutative\Hopfalgebra.W*eUUaregoingtogeneralizethissituation.Let!F{bGea nitegroup.]aW*econsiderthecategoryFc-GrofgroupsonwhichFiactsbyautomorphismsandgrouphomomorphismscompatiblewiththeactionofFc.<:F*urthermoreletLbGeaGaloisextensionofthecommutativeUUbaseringK qwithGaloisgroupFc.qThenthereisafunctor]v:K-Alg7!Fc-Grde ned_by(A)^:=U(Lꕷ K HA).|The_actionofFuon(A)isgivenby the٩GaloisactionofF=8onL.InthispapGertheGaloisextensionLofK *ffxs ^1|sMayUU8,1989* XisassumedtobGeacommutativeringandfreeasaK-module.Our rst resultUUwillbGethatpossessesaleftadjointfunctor!EqCKqȫ:Fc-GrC`!K-AlgS:"BTheץK-algebrasKqȫ(G)orsimplyKGwillbGecalledtwistedgrouprings.They=shouldnotbGeconfusedwithcrossedproductgrouprings,skewÊgrouprings,smashproGducts,orsimilarconstructions.A/ThesetwistedgroupaEringsdoingeneralnotevenallowacanonicalmapfromGintoKqȱG.Aswewillsee,Ntherewill,however,bGeacanonicalmapG 0Z!(KqȱG)^nwhereUUnistheorderofFc.!DTheftwistedgroupringswillturnouttobGecocommutativefHopfalgebras.'TheyQqevenhavethepropGertytocoincideafterbaseringextensiontoرLwiththeordinarygroupringL= KqȱGT͍S+3S=nRL KG. QThisisanisomorphismJofHopfalgebras.QThusKqȱGisa(twisted)L-formofthegroupUUringKG.In[2]westudiedtheconstructionandgeneraltheoryof(twisted)formspofgroupringsKGfora xedgroupG.W*eshowedpthattheformsareʔinonetoonecorrespGondencewiththeFc-GaloisextensionsofKwhereFɫ=K:Aut=(G)._UsingthespGeci cknowledgeofallquadraticextensionsofK(under minorrestrictions),9wewereabletodescribGeallHopfalgebraforms\ofK"V cmbx10Z,ޱKC3|s,KC4|s,and\KC6ϫbygeneratorsandrelations.7AKformHưofKGwhichcorrespGondstotheFc-GaloisextensionLunder[2Thm.5]isccertainlysplitbyL,-i.e.xL> K aHT͍+3=f"L KKGcasHopfalgebras,-butitKmayalsobGesplitbyadi erent,MevensmallerGaloisextensionL^O!cmsy70ofK.SotZwewillnotmakeuseofthecorrespGondencebetweentZtheformsandGalois%extensions.9W*ewill xaGaloisextensionLofKEAandthenstudythe/HopfalgebraformsHa-ofKGforallgroupsGwhicharesplitbytheGaloisUUextensionL.TheogroupelementsgHinKGareconnectedwithcertainelementsinKqȱGbyformulaswhichgeneralizetheEulerformulasandfunctionalequationsUUforthetrigonometricfunctions4⍍͍ exp=cosc+8isin; cosP=<$K1Kwfe (֍2 -(exp+exp1:2s=X ㉱[٫(s)e;T=X ti;xiTL[٫(yis)=X tִixiTLtr "(yiTLs):3PЫ(4)'W*e rstshowthatthematrixB8=( @L;i ):=([٫(yiTL)jx2Fc;i=1;:::;n)g^istheinverseg^matrixofC.W*eobservethatPi[٫(xiTL)(yi)=[٫(P ;i/xiTL^1 M(yi))7=[٫(㐴e;@L0ncmsy5Zcmr51 )=㐴e;@L1 =; A̫.wehavedescribGedthealgebrastructureofLsolelyintermsofthematrixCŬ(anditsinverseBq).xThedescriptionoftheFc-actionusesthe0`rightregularactionofFonitself.TheactionisthusgivenbytheinducedpGermutationsofthecolumnsofC.5ObservethatthecoGecientsofRCnandBZëareinL,thecoGecientsoftheunit,themultiplicationandtheFc-action,however,areallelementsinKt(sincetheyareobtainedfromtheUUtrace).'W*eUUconsidertheisomorphism&H?LnT͍8+38=̱L8 KnT͍ﲷ+3ﲫ=}L LT΍~h2!L (KFc)T͍_+3_=2L KFT͍ $+3 $=JLF^where5ձL^F ۵isthesetofn-tuplesofelementsfromL,<"indexedbythesetFc,oruthesetofmapsfromFutoL.'MThisisomorphismsends(liTLji=1;:::;n)uto(P ;i/liTL[٫(xi)j"2Fc)=(P ;iliTL i; ),i.e.#CitiismultiplicationbyC ontheright.TheinversemapismultiplicationbyBq.?>F*ora;b9=2L^F ?weget(afb)B=aBi?bBq,where\?denotesthemultiplicationonL^n swithmultiplicationcoGecients( ^ zk;Zij )andL^F zhascomponentwisemultiplication.PSimilarlythe `TmapisalsocompatiblewiththeFc-action,onL^F ;`compGonentwiseandonL^nӫwithUUcoGecients^uǴk;Zi; .'ThecoGecientsascalculatedinLemma1canalsobeusedtodescribea ;K-algebrastructureonS^n Fforany(non-commutative)K-algebraSȫand3 Xan XactionofFlonS^n cjustbyapplyingtheisomorphismL K VST͍Z+3Z= ۱K^nɀ K ST͍۷+3۫=GS^n .IfBSϫisanalgebraoftheformST͍۷+3۫=LX( Tc,thenBwecanevenusetheUUisomorphismofFc-algebrasqK(L8 Tc)nT͍8+38=(L T)Fwheret(Lwc Tc)^F uhascompGonentwiseopGerationsand(Lwc Tc)^nhasopGerations withNcoGecientsfromLemma1.^TheisomorphismisstillgivenbythematricesUUBƫandC.'_3.TheConstructionofaTwistedGroupRingW*e΄wanttoconstructtheK-algebraKqȱGforanFc-groupGbyus- ingٽde ninggeneratorsandrelationsofG.W*e rsthaveٽtoloGokatfreealgebras.xLetXjbGeasetandGfޫbethefreemonoidonX.WThenthemonoidringJcLGfyandthefree(non-commutative)JcL-algebraLhXicoincide.n!Thisis)6duetothefactthattheunderlyingfunctorfromthecategoryAlgofnon-commutativeKL-algebrastothecategorySetofsetsfactorsthroughthecategory!Monofmonoids.,ThefreeconstructionsarethecorrespGondingleftUUadjointfunctors.AT similarTJargumentcanbGeusedinaslightlymorecomplicatedsetting.LetF6bGea nitegroup.FQAnyofthecategoriesSet,Mon,AlgnamedabovecanObGeusedtoconstructanewcategoryFc-Set,PٱF-Mon,F-AlgOofob8jectson2SwhichFactsasautomorphisms(actuallypairsconsistingofanob8jectplusanaction)andmorphismswhicharecompatiblewiththeFc-action.ItiseasytoseethattheleftadjointfunctorsoftheunderlyingfunctorsAlgB !BSet,=Alg!Mon,=andMon!SetcanbGeusedasleftadjointfunctors KfortheunderlyingfunctorsFc-Alg (!F-Set,F-Alg (!F-Mon,andUUFc-Mon q!F-Set,i.e.qthediagrams%+3m)Fc-CatX1FP$': cmti10fr}'eel2!4Fc-CatX2vU3#bU3#sMCatj+1FPfr}'eel2!«Cat62commute,lThenweimpGorttherelationsofGwrittenaspairsKofelementsofGf/ .TqTheirdi erencesgeneratetheidealJAƫofLGftobGefactoredouttoobtain(arepresentationof)LG.ThisidealwillbetransferredtotheidealMofLhSij iviathegivenisomorphism.qThecentralcypGointofthecalculationwillbGethatthegeneratorsofMzarealreadyde ned>overKIZ(insteadofL),wi.e.(theyareelementsofKhSij iwhichgenerateQanidealM^0PkKhSij i.fThusweobtainaK-algebraKqȱGk:=KhSij i=qM^0:withUUthepropGertyL8 K KqȱGT͍+3= UNLG.؍MoredpreciselyletGbGeanFc-groupgeneratedasanF-monoidbythe setY=ĝfgiTLji2I1|sg.Then thereisasurjectiveFc-homomorphism:Gf Z4!:G,.wherejGf 2isthefreeFc-monoidonY8.|A=relationforGisanpairofelements(r1|s;r2)o2Gff|FGf鎫withn(r1|s)=(r2).ThensetofallrelationshforGisanFc-submonoidofGft(EGf/ .I#Letfrjī=(rjg13;rjg2)jjY2I2|sgbGeageneratingsetoftheFc-submonoidofrelations. SThenGcanbeconsideredK%asaquotientofGfzEmoGdulothecongruencerelationgeneratedbyUUfrj6g.؍W*econsidertwoelementsr:andsofGf=(writtenwithnon-negativepGowers]andFc-multiplesofthegeneratorsinY8).[LetJbetheidealgen-eratedf"byf[٫(rG)D(s)j>2Fcgf"inLGf/ ..ThenM3:=(J9)isgeneratedbyMڟ Xf([٫(rG)ti(s))jȷ2[Fcg.}W*ewanttochangethisgeneratingsettoaset ofUUK-linearcombinationsoftheSij .qF*orthispurpGoseweform ߍM(fX 7@L2F! @L;i ([٫(rG)8(s))ji=1;:::;ng:'5Lemma'2.TheidealMinLhSij igeneratedbyf([٫(rG)uķ(s))j+2_RFcgis+alsogeneratedbythesetfP ;@L2F/ @L;i ([٫(rG)m(s))jiz=1;:::;ngKhSij i.Proof:Sincethematrix( @L;i )isinvertible,thissetgeneratesthesameideal眱MinLhSij i.(W*eshowthatthecoGecientsoftheproGductsoftheSij 'sUUareallinK.qLetr5=1|s(g1)8:::gpR(gp).ThenUUweobtainEN пҍ+X 7@L2F"' @L;i ([٫(H*1|s(g1)8:::gpR(gp))׍J=ɢX 7@L2F螱 @L;i ([1|s(g1))8:::g(pR(gp))J=ɢX 7@L2FX t螴i1 ;::: ;i O \cmmi5p05ұ @L;i i1 ;@L1 Si1 ;1п8:::g8 ip2Դ;@Lp{Sip2Դ;pE[J=_yX ti1 ;::: ;ipi(X 7@L2F! @L;i i1 ;@L1X8:::g8 ip2Դ;@Lp{)Si1 ;1п:::Sip2Դ;pr:DcdSinceUUthecoGecientsk!X 7 %@L2F20 @L;i i1 ;@L1X8:::g8 ip2Դ;@Lp~=ɢX 7@L2F螱[٫(yiTL)1|s(xi1;l):::pR(xip )areUUthetraceofcertainelementstheyareinK.qǷtuuTheoremz3.LetGbGeanFc-groupwithF-monoidgeneratorsfgiTLji2I1|sgand\Fc-monoidrelationsfriTLji2I2|sg\generatingallF-monoidrelationsofG.LetטM^0bGetheidealoftheK-algebraKhSij ji=1;:::;n;jY2I1|siטgeneratedbyUUtheset#68+fX 7@L2F! @L;i ([٫(rjg13)8(rjg23))ji=1;:::;n;jY2I2|sg΍where{([٫(gj6)):=P USiTG i; Sij .3ThenKqȱG:=KhSij i=qM^0isaHopfalgebraandUUanL-formoftheHopfalgebraKG,i.e.qǫ:L8 K KGT͍+3= UNL K KqȱG.Proof:ByEhypGothesistheFc-groupGisrepresentedasaquotientofafreeʱFc-monoidbycertainrelations.&Ifr`=(r1|s;r2)isarelationinGthen \ Xr1s="r2in%{Ghence[٫(r1|s)=(r2|s)%{forall}ٷ2Fc.9Itiseasytoseethat thegroupalgebraKGisthequotientKGf/ =J9^0ՃwhereGf @1isthefreeFc-monoidL2generatedbyfgiTLjib2I1|sgL2andJ9^0istheidealofKGf {Rgeneratedby\f[٫(ri1P) (ri2P)ji̷2I2|s;,2Fcg.If\weidentifyL K AKGandLG,L#H KKhSij iJandLhSiji,moregenerallythetensorproGductwithLwiththemultiplication\withL,^thenwegetanalgebraisomorphism:L=Է KGT͍{+3{=L8 KhSij i=qM^0T,UUsinceLisfaithfully at. xThediagonalonLhSij i=LM^0:isinducedby:LG!LhSiji=LM^0T.W*eUUhave9QY2(sk+Bl}X)4=(X 7@L2F! @L;k 8([٫(glȫ)))׍4=ɢX 7@L2F螱 @L;k 8([٫(glȫ))8 ((glȫ))=sJX ti;jT;; i; jT;  @L;k 8sil 8sjgl鍍4=X ti;j㉱ zkij sil 8sjgl= where_sij WistheresidueclassofSij .NSincethecoGecients ^ zk;ZijareinK{thediagonalisalreadyde nedinKhsij ih=KhSij i=qM^0T.Inasimilarwayweobtain"(sij )M=P ßڱ @L;i C="iK(indepGendentofj!)calsode nedonKhsij i.SinceQ|Lhsij iisabialgebra(isomorphictoLG)andL=Kisfaithfully at,wegetthatKhsij iisalsoabialgebra.=ItisabialgebraformofKG.=By[2 RemarkfollowingThm.1]abialgebraformofaHopfalgebraisalreadyaUUHopfalgebra.qǷtu⥍TheHopfalgebraKqȱG:=Khsij iiscalleda-twiste}'d!groupring.<=W*ehave,!constructedthe-twistedgroupringbygeneratorssij 7andrelationsusingUUtheformulaǍtEsij =ɢX 7@L2F螱[٫(yiTL)(gj6)֖inUULG.qEachrelationrrforGinducesnrelationsinKqȱGi`ؕX 7_ @L2Fq @L;i ([٫(r1|s)8(r2|s))=0:"֖Thesegeneratorsandrelationsareobtainedfromthede ningFc-genera-torsandrelationsforG.5OEachgeneratorgj֘(infacteachelement)ofGanditsvFc-orbitinGKGvcorrespGondstoafamily(sij ji=1;:::;n)vinKqȱGby thegivenformula.YThediagonalofKqȱGappliedtoanelementofsuchaafamilycanbGeexpressedsolelybytheelementsofthesamefamily(and k XcoGecientsKinK)sowede nea-gr}'oup-likesequenceKofaHopfalgebra H%StoUUbGeann-tuple(siTL)inHsuchthat;ՍD(sk됫)=X ti;j㉱ zkij si, 8sjand.ͱ"(siTL)="iV}holds.bCorollarUTy4.LetGbGeanFc-group.7[LetLbeacommutativeringwhichcontainsisubringsKL^0:LisuchthatL^0::K isanFc-Galoisextension.ThenqthegroupalgebraLGhasgeneratorsfsilji=1;:::;n;lQ2IgqsuchthatUUforalllx2I;э4](sk+Bl}X)=X㉱ zkij sil 8sjglandE"(sil)="iTL:;ՍTheUUrelationsforthefsij garethoseofTheorem3.BProof:followsimmediatelyfromtheisomorphismLGT͍÷+3ë=䤱L L0 lرL^0* KKGT͍+3= UNL8 L0 DZL^0 K KqȱG.qǷtuv+DS4.TwistedGroupRingsasAdjointFunctorsIn7;thisparagraphweshallidentifyLGandL KqȱG7;viatheisomor-phism>asconstructedabGove..F*urthermore>weviewKGandKqȱGassubalgebrasUUinLG.qInparticularwehavetheequations)Aa[٫(gj6)==n X tմi=1㉱ i; sij ;sij =ɢX 7@L2F螱 @L;i[٫(gj6);6whereUUfgiTLji2I1|sgisageneratingsetofGasanFc-monoid.ATheorem5.LetGbGeanFc-group.D|ThenthereexistsacocommutativeHopfalgebraKqȱGandahomomorphismofFc-groups:G 1!(KG)such+(P ;@L2F/ @L;i ([٫(r1)[٫(r2)))=0.AThusEwegetadwell-de nedK-algebrahomomorphism\qdef}:KqȱG!S.,6F*orgjķ2Gwethen get(\qef)(gj6)=(\qef)(P ;i/ i;e sij )=P USiTGxij (\qef(sij ))=P USiTGxij tij =f(gj6). XT*oshowtheuniquenessof\qf˽ef tlet\qf˽bfbGeanotherextensionoffVsothat (\qef),=(\qbf)=f.\q 1bQf꟫inducesanL-algebrahomomorphismfromLG \sto*L S:whichwealsodenoteby\q.bf .gGThen\q.ef(sij )#=\q'ef (P ;R @L;i [٫(gj6)) v= _P @L;i\q e f([٫(gj6)) _=P @L;i (\qef)([٫(gj6))=P @L;i f([٫(gj6))=\qbf(P ;R @L;i [٫(gj6))=\qbf(sij )UUsothat\q5Yef S=\qbf:tuPǍTheUUuniversalpropGertyshownabGoveimpliesimmediately^CorollarUTy6.The$functorKqȫ:Fc-GrV!emK-AlgԴisleftadjointto :K-Alg7!Fc-Gr|s.qǷtuOnecouldalsohaveshownthatisanalgebraicfunctorinthesenseofĐ[4].yThentheadjointnesspropGertywouldhaveresultedfromthegen-eralwtheory*,xbuttheexplicitconstructionasK韫wouldnothavefollowed.F*urthermoreFtheHopfalgebrapropGertyofKqȱGwouldhaverequiredad-ditionalUUinvestigations./5.TheStructureofFUormsofGroupRingsUptonowwehaveconstructedaspGeci cversionof(twisted)L-formsofgrouprings(Theorem3),?namelythetwistedgrouprings.>W*ehave[studiedtheirstructureintermsofgeneratorsandrelationsandtheirpropGertydasaleftadjointfunctor.Nowwewanttoprovethatallformsofzgroupringsaretwistedgroupringsunderonlyminorrestrictions.F*orthisɓpurpGoseweconsiderthoseformsHofgroupringswhicharesplitbyafree(Fc-)GaloisextensionLofK,i.e.FLL3 K鱱HT͍+3=%LL KKGandweassumethatUULisconnected,i.e.qhasonlytheidempGotents0and1./Theorem7.LetL:KDbGeafreeGaloisextensionwith niteGaloisgroupFandAMletLbGeconnected.kLetGbeagroupandHKbeaK-HopfalgebrawhichJisanL-formofthegroupringKG.ThenthereisanFc-structureonUUGsuchthatH%Sisisomorphictoa-twistedgroupringKqȱG.Proof:SinceS(Ln!:K Disfaithfully atwecanidentifyH#&andKGwithspGeci cӱK-subalgebrasofLG.qCInparticulartheH-moduleLG(T͍+3= ]L KH)^isfreelygeneratedover^thebasisfxiTLg. QEachelementgR2|yGhasarepresentationg8s=ܚPލjմn%ji=1̱xiTLsi(g[٫)withuniquelydeterminedcoGecientssiTL(g[٫)2H.ThisGde nesmapssiK:G u!H.TheseGmapscorrespGondtothegeneratorssij œcomingfromthegroupgeneratorsgjGintheprecedingparagraphs./W*ede neanFc-actiononGby[٫(g)e:=Pi[٫(xiTL)si(g). )7Firstweshow[Hthat[٫(g)[HisanelementofG. SinceLisconnecteditsuces Xto.provethat[٫(g)isagroup-likeelementinLG. ByLemma1the diagonaloftheelementssiTL(g[٫)canbGederivedfromPk<2xk됫(sk(g[٫))=(g[٫))=gD g=P dijxiTLxj6si(g) sj6(g))=P dkP 1ןij(w ^ zk;Zij xk됱siTL(g) sj6(g)]`as (sk됫(g[٫)) =P/Hij ^ zk;Zij siTL(g) sj6(g).TheaugmentationisPfWkxk됱"(sk(g)) ="(g[٫)v=1=Pܱxk됱"k DhenceXw"(siTL(g)="iTL.{.SoXw(si(g)jiv=1;:::;n)Xwisa-group-likeUUsequenceinH.qThen*PW([٫(g))@=X 4k㉱[٫(xk됫)(sk(g))=X 4k㉟X tij" zkij (xk됫)siTL(g)8 sj6(g)鍍@=X tij㉱[٫(xiTLxj6)si(g)8 sj6(g)=(g)8 (g)+5andpZ"([٫(g))!=P \iP[٫(xiTL)"(si(g))!=(P ;i/"iTLxi)!=(1)=1pZimplythat(g)isUUgroup-like,hence[٫(g)2G.yNow>itiseasytoverifythatthisisanactionofF#onGbyautomor-phisms.qF*romUULemma1wegetk/!ǫ([٫(g))=(X tξiq[٫(xiTL)si(g))=!ǫ(X tξiqX k 8uǴki; xk됱siTL(g))#Í/=ٟX i;k+B;_ i;@LQ @L;k 8 k+B; űsiTL(g[٫)=X tִi㉱ i;si(g[٫)=(!)(g):IF*urthermoreUUwehave7 !I[٫(gh)+w=[٫(X thi;jqxiTLxj6si(g)sj6(h))=(X kqxk8X t Yi;j zkij siTL(g)sj6(h))鍍+w=X 4k㉱[٫(xk됫)X tni;j zkij siTL(g)sj6(h))=X ti;j[٫(X kq zkij xk됫)siTL(g)sj6(h))+w=X ti;j㉱[٫(xiTL)(xj6)si(g)sj6(h)=[٫(g)(h):9ByUUde nitionwealsohavee(g[٫)=g.soUUthatGisanFc-group.yThechoiceofFc-monoidgeneratorsfgj6gforGde nesK-algebragen-erators%siTL(gj6)forH#withtheformulasgj iq=2şPixisi(gj6)andsi(gj6)2=P ;R @L;i [٫(gj6).qTheUUrelationsaretransformedasinTheorem3.tu6TheconnectionwithsomeknownresultsshouldbGementionedhere.In[6Thm.6.4.]Nthecategoryof niteetalegroupschemesovera eldKwas@foundtobGeequivqalenttothecategoryof niteGѫ-groups,xwhereG؝isthepro niteGaloisgroupoftheseparableclosureofK,actingcontinuously XonVthe nitegroups.Thesegroupschemes,representedVbycommutative Hopfjalgebras,bGecomeconstantgroupschemesalreadyaftera niteGalois eldB_extensionL,F*i.e.kuL H^T͍/+3/=0LGB_fora nitegroupG.TheactionofGonUUGisgivenbytheactionofF*=Aut(L=K)onGviatheisomorphismZ@K-AlgS(HA;L)T͍Pط+3Pث=ZL-Algt(L8 K HA;L)T͍+3= UNL-CoalgUY(L;L K Hh)T͍Pط+3Pث=ZGroup-LikesH(L8 Hh)T͍+3= UNGroup-LikesBT(LG)T͍+3= UNG:Theorem'7isageneralizationofthistheoremtoin nitegroupsandawayfromq:the eldrequirementsforLandK.xOntheotherhandourthe-oremf2alsogeneralizespartoftheknownantiequivqalencebGetweengroupschemes?ofmultiplicativetypGeandabelianGѫ-groups[6Thm.7.3.]j`tonon-commutativeP4Hopfalgebras.bdInfactourTheorem3givesadescriptionofdtherepresentingalgebrasofgroupschemesofmultiplicativetypGebygeneratorsUUandrelationsofthecorrespGondingcharactergroup.9t$86.TwoExamples:W*e3Gwanttoillustrateourresultswithtwoexamples. LetC2 `=f1|s;2g=f;[ٷgjCbGethecyclicgroupwithtwojCelements.Assumethat2Q5isinvertibleQ5inK.eiConsidertheC2|s-GaloisextensionLj=K(i)Q5ofKwithUUi^2C=1andbasisx1=1;x2=i.qW*eUUformthematrixLE(Iq iA)]=^ 41|s(x1)12|s(x1) 41|s(x2)12|s(x2)Oͧ^Y3=^ 411 :i6i'^0 :LwithUUtheinverseUUmatrixZGul(yP* ij )=^ g1 g&fes2 "1!GП&feп2i g1 g&fes21 Y133&feп2i.^ThenUUthemultiplicationcoGecientsare ⍍?lbD zkij bX=\ o2  ğX  ôl `=15 k+Bl}X l `i l `jt\!J)==^ 410 4061)? )? )? )? 1ի0@111@0FȀ^"8where%thetwo%adjacentmaticesare( ^ z1;Zij )and( ^ z2;Zij).ThecoGecientsoftheUUactionofFonLareZIKvlbP ñuǴki; bf_=\ NX 7 Ĵ@L2FJ i;@LQ @L;k 8\!N«=^ 410 4061)?^# XandUUthecoGecientsoftheunitare("1|s;"2)=(1;0).?XF*or a-twistedgroupringKqȱGweusetheisomorphism(LG)^nT͍ g+3 g= (LG)^F ras`describGedattheendofparagraph2givenbymultiplicationbyB ثresp.C.W*egdenotethemultiplicationinthealgebra(LG)^nby?,theactionofFwithnospGecialnotation.W*estillconsiderKqȱGandKGembGedded inLG.WEachelementg"2Ginducesanelement([٫(g)j"2Fc)2(LG)^F and[thefactthatFactsbyautomorphismsinduces([٫(g))D?([٫(h))=([٫(gh)).Each8relationinGthusinducesarelationin(LG)^F,qhencenrelationsUUinLG.?XW*econstructthetwistedgroupringKqZ.˙ThegroupGo:=Zhasone(Fc-monoid)generatorg2ҫandC2SlactsonGby[٫(g))=g[ٟ^1 M.Sothereisłarelation[٫(g)g= 1ł(ifweconsiderGasamultiplicativegroup).NW*ede ne`]$(c;s):=(g[;(g))^ O1 O&fes2 $i1$iҟ&fes2 O1 &feп2i Y133&feп2i0՟^9;2-i.e.fc 7= >j1>j&fes2 n(g+Tg[ٟ^1 M)~4ands= 1>j&feп2i B\(gTg[ٟ^1 M)inLG.fTheseelementsgenerateaQK-subalgebraK[c;s]kLG.fThereQisarelationc^2]+"s^20=k1,infactthe2subalgebraisisomorphictoKqȱG=KhC(;Si=(C^2'++S^2+1;CSSC)whereKhC(;SiisthefreealgebraonC,#S(thepGolynomialringinnon-commuting`vqariablesC,S).TherelationsarisefromatranslationoftherelationUU[٫(g)g"=1to?Ym[٫(c;s)8?(c;s)=("1|s;"2)orUUusingthede nitionof?andtheopGerationofFon(LG)^nh (c;s)( zkij )(c;s)tLn=(1;0)henceUUc^2S+8s^2C=1andsc+cs=0.?XItisobviousthatthisalgebraKqȱGrepresentsthecirclegroupfunctor whichassoGciateswitheachK-algebraSJ[the"circle"f(c;s)jc;si2S;c^2N+s^2=8x1g[withmultiplication(c;s)f9(c^09;s^0)8x=(cc^04rf9ss^09;cs^0+f9sc^09)[andunit(1;0)UUwhichwestudiedin[2].The< >: UO1&\nfor5i=jk+8kP; UOb^1&\nfor5i8+n=jk+8kP; UO0&\nelse,!qՍandyEZ=tLn(1;0;:::;0):0References[0] Anonymous:qRefereesUURepGort(unpublished).[1] S.[Chase,D.K.Harrison,A.RUosenberg:tGaloisTheoryand GaloisyCohomologyofCommutativeyRings.ߤMem.Am.Math.SoGc. 52UU(1965).[2] R.HaggenmM*uller,wB.P areigis:  cmmi10Zcmr5ٓRcmr7K`y cmr10