Complete Classification of BKM Lie Superalgebras Possessing Strictly Imaginary Property

Transcription

1 Appled Mthemtcs 4: -5 DOI: 59/m4 Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety N Sthumoothy K Pydhs Rmu Isttute fo Advced study Mthemtcs Uvesty of Mds Che 6 5 Id Astct I ths ppe complete clssfct os of ll BKM Le supelges wth f te ode fte ode Ct mtces possessg Stctly Imgy Popety e gve These clssfctos lso clude ptcul the Moste BKM Le supelge Keywods Bo cheds Kc-Moody Le Supelges Stctly Imgy Roots Puely Imgy Roots Itoducto I[4] the theoy of Le supelges ws gve [5] theoy of Kc- Moody Le supelges ws desced Bocheds[] tted the study of geelzed Kc-Moody lgesgkm lges Wmoto[9] toduced BKM supelgesbkm Le supelges The exstece of specl mgy oots fo Kc-Moody lgeskm lges wee show [] the cocept of specl mgy oots ws exteded fom KM lges to GKM lges [7] I[] so me popetes of oots of GKM lges wee studed [][4] specl mgy oots of these clsses wee foud out flly [5] complete clssfct o of GKM lges possessg specl mgy oots ws foud out The oto of specl mgy oots of BKM lges ws geelzed to BKM supelges [6] cet clsses of BKM Le supelges possessg specl mgy oots wee foud out [6] I [8] complete clssfcto of BKM Le supelges possessg specl mgy oots ws gve The cocept of stctly mgy oots fo KM lges ws toduced y Kc[5][6] Cspeso[] gve complete clssfcto of KM lges possessg stctly mgy popety The cocept of puely mgy oots fo KM lges ws toduced [] thee the KM lges possessg puely mgy p opety wee completely clssfed Ag [] the cocept of puely mgy oots fom KM lges to GKM ws exteded the GKM lges possessg puely mgy popety wee completely clssfed I[4] the popetes of stctly mgy oots puely mg y oots of GKM lges wee comped Coespodg utho: sthu@yhoocomnsthumoothy Pulshed ole t Copyght Scetfc & Acdemc Pulshg All Rghts Reseved usg the clssfcto of GKM lges possessg puely mgy popety the lges whose puely mgy oots e stct ly mgy oots wee foud Co mplete clssfcto of GKM lges possessg specl mgy oots stctly mgy popety wee gve [5] The cocepts of stctly mgy oots puely mgy oots of Bocheds Kc-Moody lgesbkm lges wee exteded to BKM supelges [7] A complete clssfcto of those BKM supelges wth puely le mgy popety puely mgy popety wee gve [7] Moeove the popetes of stctly mgy oots puely mgy oots of BKM supelges wee comped the BKM supelges whose puely mgy oots e lso stctly mgy wee foud out [7] Am of ths ppe s to g ve complete clssfcto of BKM Le supelges possessg stctly mgy popety Pelmes Bsc Deftos I ths secto we efly ecll the fudmetl deftos egdg BKM Le supelges the Weyl goups oot systems s gve [9] Fo the def to of Geelzed Geelzed Ct mtxggcm oe c see[9] Defto :[9] Let I { } e fte dex set let A e el mtx Let ψ e suset I of I If A stsfes the followg codtos the A ψ s clled BKM supe mtx c ~ o

2 Appled Mthemtcs 4: -5 I c ~ c ~ c ~ 4 f the Z fo ll c ~ 5 f ψ Z fo ll Defe e e susets I m } I m > I { I Let { I the m I of I y } m Z e collecto of postve teges such tht chge of A Also ψ e : { ψ : ψ : { ψ e m fo ll I We cll m } I } ψ e ψ : { ψ m ψ : { ψ } ψ ψ Rems: } set If ψ s e mpty set the the BKM supe mtx cocdes wth the coespodg BKM mtx o GKM mtx Fo descpto of the qus- Dy dg m qdy A oe c efe to[9] A Geelzed Geelzed Ct Mtx s clled decomposle f t cot e educed to loc dgol fo m y shufflg ows colums[8] Fo the se of completeess we epet the followg fudmetls ledy exp led [7] Defto :[6] Let I e dex set A ψ e decomposle BKM supe mtx whee A ψ I I The oe oly oe of the followg thee possltes holds fo A F det A thee exsts u > such tht Au > Au u > o u Aff Co thee exsts u > such tht Au Au Au Id det A thee exsts u > such tht Au Au > u > u Refeg to the ove thee cses we sy tht A s of f te ffe o defte type espectvely wte A F A Aff o Id A espectvely Defto :[8] We sy tht BKM supe mtx A ψ s of hypeolc type f t s defte type evey pcpl sumtx of A s ethe f te o ffe type BKM supe mtx Defto 4[9]: If BKM supe mtx A decomposes s A DB whee D δ : ε dgol mtx B s symmetc mtx the A s sd to e symmetzle If A s symmetz le BKM supemtx the tg the dgol mtx D stsfyg ε ε ε > y ε we hve fo ll ψ We ssume tht A s symmetzle decomposle BKM supemtx Defto 5:[9] Fo y BKM supemtx A ψ whee A we hve t p le h Π Π Π { I} Π { I} whee stsfyg the followg elt os: h s fte dmesol co mple x vecto spce such tht dmh A Π { } h s lely depedet Π h I { h s lely depedet whee Hom } I C h C whee deotes dulty pg etwee h h Ths tple Π Π elzto of A h s clled Cll elemet of Π espectvely Π fudmetl oot o smp le oot espectvely fudmetl cooot o smp le cooot e e Moeove set { } Π I m m Π { I } We cll elemet of m Π Also dvde Π s Π : { I \ ψ } e Π esp el smp le oot esp mgy smple oot eve Π the set of ll eve smple oots : { } the set of ll odd smp le oots Let odd ψ Z s the esdue clss g mod wth elemets Defto 6:[9] A Z -gded vecto spce g g opeto clled the cet poduct g possessg the le mp []: g g x y [ x y] g s clled Le supelge f t stsfes the followg codtos:

3 N Sthumoothy et l: Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety [ g g ] g Z x y [ y x] [ x y] x y [ x [ y z ] [ y[ x z]] [[ x y] z] fo ll x y z g Defto 7:[9] The Bocheds Kc-Moody Le supelge evted ~ s BKM Le supelge o BKM supelge g A ssocted to symmetzle BKM supe mtx A ψ s the Le supelge geeted y the vecto spce h the elemets e f I stsfyg the followg elt os: ' ' [ h ] h fo h h h [ h e ] h e fo h h [ h f ] h f fo h h [ e f ] δ fo 4 e 5 f I I the d d e e 6 f I the [ f f ] ψ the e e ] [ f f ] f [ f e [ e ] 7 f Rems: As we e ssumg tht the mtx A ψ s ~ symmetzle the ssocted BKM supelge g A s smp le fo poof oe c see[6] lso[9] whch we wll deote y g A So fo BKM supemtx A ψ g A s clled BKM Le supelge o BKM supelge ssocted to A ψ I[5] Dy dgms wee defed fo Le supelges Dy dgms wee ledy exteded fom KM lges to GKM lges [] the exteded to BKM Le supelges [7] whch e g g ve elow Defto 8:[7] To evey BKM supe mtx A ψ whee A ψ I the dex set I s ssocted I wth Dy dgm S A defed s follo ws: S A hs vetces vetces e coected y mx{ } ume of les f 4 thee s ow potg towds f > If > 4 e coected y old fced edge equpped wth the odeed p Moeove f ψ the -th vetex wll e deoted y whte ccle f ψ the -th vetex wll e deoted y whte ccle wth od wtte wth petheses elow the ccle to deote the vetex coespodg to odd smp le oot th s cse f ψ the -th vetex wll e deoted y cossed ccle 4 f ψ the -th vetex wll e deoted y cossed ccle wth od wtte wth petheses elow the ccle to deote the vetex coespodg to odd smp le oot th s cse 5 f > ψ the -th vetex wll e deoted y whte ccle wth wtte wth petheses ove the ccle 6 f > ψ the -th vetex wll e deoted y whte ccle wth wtte wth petheses ove the ccle wth od wtte wth petheses elow the ccle to deote the vetex coespodg to odd smple oot ths cse Wth these deftos the Dy dgms of ll BKM supelges c e dw Some exmples of Dy dgms of BKM supelges wee dw [7] A BKM Le supelge g le KM o BKM lge hs the follo wg tul oot spce decomposto: g g whee g { X g [ h X ] h X h h} s clled the oot spce ssocted to A elemet Q s clled oot f g The ume mu lt dm g s clled the mu ltplcty of the oot A oot of g A c e expessed s m m Z whee m 's e ll ll Coespodg to whethe m 's e ll o o ll s clled postve oot o egtve oot espectvely Also m s clled the heght of s deoted y ht We deote y the set of ll oots postve oots egtve oots espectvely Also ote tht g Ce Defto 9:[9] g Cf

4 Appled Mthemtcs 4: -5 Let G e BKM Le supelge Set te suspce '' h of h stsfyg Defe the symmetc le fom follows: h : ε h h h ' h C ' '' h h h o h s '' '' '' '' '' h h : fo h h h The s o-degeete o h ths duces the le somophsm : ν h h h v ths mp ν We completely detfy h omt the symol ν the followg esults The poofs of these esults e [9] Lemm :[9] oe hs the followg: Fo ε ε ε 4If the λ λ λ h ptcul f the λ λ λ h Rem: Fo m I tems of e I poduct Π { I we hve e Π { > } } m Defto :[7] e Fo ech I h y we defe the smp le eflecto λ λ λ λ h A The Weyl goup W of g s the sugoup of e GL h geeted y the 's I Note tht e W { I } s coxete system So fo el oot e w w W Π we defe the eflecto of whee Note tht h wth espect to y λ λ λ λ h w h s the dul el oot of w w W Lemm :[9] The le fom ude the cto of the Weyl goup o h h s vt I ptcul we hve ε fo Defto :[7] The set of ll el oots of BKM Le supelge s defed s e W Π e e W { ψ } m e The the set of ll mgy oots s \ We e e m m hve Defto 4:[9] Let Q Z Q Z Q Q s clled the oot lttce the we hve Q s clled the postve oot lttcethe oot lttce Q ecomes ptlly odeed set y puttg β β Q fo β Q Now fo m Q suppot of s defed s supp { I m } If supp s coected A we sy tht supp suset of the Dy dgm of s coected Defto 5:[9] Imgy oots of BKM supelges e sclly of two types domestc-type le-type Do mestc-type mgy oot: A mgy oot whch s cougte to fudmetl oot ude the cto of the Weyl goup s clled om m d domestc-type mgy oot We deote y the set of ll do mestc-type mgy oots Ale-type mgy oot: A mgy oot whch s ot cougte to fudmetl oot ude the cto of the Weyl goup s clled l m le-type mgy oot We deote y the set of ll le-type mgy oots Sce mgy oot s ethe cougte o ot cougte to fudmetl oot ude the cto of Weyl goup ech mgy oot s ethe domestc mgy o le mgy m dom m l m We hve Lemm 6:[9] m s vt ude the cto of the Weyl goup m If the thee exsts w W stsfyg w C Fo m Theoem 7:[9] Fo symmetzle BKM supemtx A ψ f we set e : W Π e e W { ψ }

5 N Sthumoothy et l: Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety 4 dom m the coceg m : W ±Π l m e : \ m W { ± ψ } : m m \ l m dom m w W we hve the followg l m esults: W K w K m m W K W Π W { ψ } whee K { Q > supp s coected } { K supp } K ' Also y otto R R C C h R ' e { h h h I } R { h ' h R Lemm 8:[9] Fo h > I e } e I oe hs the followg: { Z s fte set Let p The set } e the mmum coted ths set let q e the mxmum ths set The p q { Z } { Z p q} c the sequece { mult } s ltelly p q symmetc the left hlf of ths sequece s mootoe odecesg Nmely p q mult mult p q p mult mult > / 4 / Stctly domestc type mgy oots stctly le type mgy oots stctly mgy oots puely mgy oots puely domestc type mgy oots puely le mgy oot wee ledy expled [7] We epet the followg deftos whch we eed hee Defto 9:[`7] A domestc-type mgy oot BKM supe lge s sd to e stctly domestc-type mgy f fo e evey ethe o s oot Let s dom m deote the set of ll stct ly s dom m s dom m domestc-type mgy oots postve egtve stctly domestc-type mgy oots espectvely Defto :[7] A le-type mgy oot BKM supe lge s sd to e stctly le-type mgy f fo evey e ethe o s oot Let s l m s l m s l m deote the set of ll stct ly le-type mgy oots postve egtve stctly le-type mgy oots espectvely Defto :[7] A mgy oot BKM supe lge s sd to e e stctly mgy f fo evey ethe o s oot The set of ll stctly mgy oots s sm sm sm sm deoted y Let deote the set of ll stctly mgy oots postve egtve stctly mgy oots espectvely Rem: As t ws otced [5] sm m m If β the β sm s semgoup Defto :[7] A BKM supe mtx A ψ s sd to hve stctly mgy popety f A sm m A If BKM supemtx stsfes stctly mgy popety we sy tht coespodg BKM Le supelge stsfes stctly mgy popety Puely le mgy oots puely domestc mgy oots wee ledy expled Sthumoothy et l9 Defto :[7] we sy tht s puely mgy f Let m fo y m m β β We sy tht the BKM supe lge g A hs the puely mgy popety f A ψ stsfes ths popety We hve m dom m l m m Smlly we sy tht egtve oot s puely mgy f s puely mgy oot Deote y pm pm A { pm A pm m { m s s The the set of ll puely mgy oots s pm pm pm puely mgy} puelymgy} We omt the poof of the followg theoem fo BKM Le supelges whch c e dectly vefed usg the poof fo KM lges ledy poved [] Theoem 4:[7]

6 5 Appled Mthemtcs 4: -5 Fo BKM Le supelges the followg esults e tue: e fo ll I the If If c If s l m s l m l m e fo ll the slm e fo ll I β fo ll β s l m l m l m β the β the d If e s semgoup slm I ddto to the ove esults we pove the followg esults fo BKM Le supelges Theoem 5: supp s dom m If \{ } m coected the dom m If \{ } e / fo ll s l m the Poof: Let dom m \{ } supp e coected The sm dom m \{ If / So y le mm 8 Let } fo ll e the > / > / > >> o So y le mm 8 we hve o Hece s stctly mgy oot Rem: Fo m the popety d of the Theoem 4 we hve sm m m Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety Rem: I[7] complete clssfct o of BKM Le supelges possessg puely mgy popety ws gve Fst we G ve the followg Results fom[] Defto :[] We sy tht the geelzed Ct mtx A hs the popety SIMmoe efly: A SIM f sm m A A Defto :[] A s sd to stsfy NC f thee exsts o susets S T { } such tht A S s ffe o def te A S T s decomposle type The followg theoem poved y Cspeso 994 s fo the defte Kc-Moody lges possessg stctly mgy popety Cspeso 994 gve complete clssfcto of Kc-Moody lges possessg stctly mgy popety Theoem :[] A GCM les SIM f oly f t stsfes the codto NC hs o pcpl sumtx coted the followg lst: The mtces of the fom 4 o wth The mtces of the followg Dy dgms of twsted ffe type: The stctly hypeolc mtces ssocted wth the Dy dgms of the fom: 4 The hypeolc 4 4 mtces ssocted wth the Dy d g ms of the fom: Fo m[] we c coclude the followg fo ffe Kc-Moody lges: 5 If lge s ffe we hve tht sm δ hece { Z \ {}} m sm \ { δ Z \ Z} whee δ s the uque mml postve mgy oot s the ode of the dgm utomophsm used to costuct the lge

7 N Sthumoothy et l: Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety 6 Ag fo the cse of mtces the fo llo wg theoem gves complete clssfcto of the o-stctly mgy oots: Theoem 4:[] Suppose fo the GCM Z A tht Γ A { K A : ± / A} The ethe Γ A 4 A { A { } Γ A Coolly 5:[] The GCM } o whee > 4 s ot SIM f oly f 4 ethe o Poposto 6:[] A GCM of ffe type s meme of SIM f oly f t s of o-twsted ffe type Rems: Fo m[7] the set of ll stct ly mgy oots of y BKM supelge s suset of set of ll puely mgy pm sm oots tht s \ my e empty set o o-empty set depedg upo the lges So fo the BKM supelges possessg puely mgy popety pm sm we vefy whethe the set \ s e mpty o ot I the pm sm pm sm cse whee \ ll puely sm pm mgy oots e stctly mgy s s lwys tue BKM supelges whch stsfy the codto pm sm \ wll e the clss of BKM lges possessg stctly mgy popetyhece the codto pm sm s equvlet to sm m whch s equvlet to SIM popety As the Cses of Specl Ad Puely Imgy Roots We Dvde The Clsses of BKM Supe lge s Ito Two Ctegoes We Dvde these BKM Le Supe lge s to Two Ctegoes Ctegoy : BKM Le supelges wthout odd ootsgkm lges oly Ctegoy : BKM Le supelges wth o-e mpty set of odd oots: We dscuss ctegoy elow Ctegoy : BKM Le supelges wthout odd oot:gkm lges oly Co mplete clssfct o of GKM lges possessg Stct ly mgy popety ws ledy gve [7] Ctegoy : BKM Le supe lge s wth o-e mpty set of odd oots: We dvde ths ctegoy to two clsses whch e Ctegoy: ClssI: BKM Le supelge s of fte o de Ct mtces w th o-empty set of odd oots Ctegoy: ClssII: BKM Le supe lge s of fte o de Ct mtces wth o-empty set of odd oots We dscuss elow these two clsses septely Ctegoy :ClssI: BKM Le supelges w th o-empty set of odd oots We clssfy these BKM supelges to thee suclsses BKM supelges wth ll smp le oots eg el w th o-empty set of odd oots: These e BKM supelges whch do ot hve y mgy oot So ths set of BKM supelges do ot possess stctly mgy popety BKM supelges ll whose smple oots e mgy w th o-empty set of odd oots: These e BKM supelges whose supemtces do ot ppe s the extesos of KM mtces So ll the dgol elemets e egtve Hece thee s o el smp le oot ll the oots e mgy lso stctly mgy BKM supelges wth fte o-zeo ume of el smple oots fte o-zeo ume of mgy smp le oots wth o-empty set of odd oots: Rem: Heefte we deote y GGX Geelzed Geelzed Ct mtx BKM supe mtx o BKM mtx We pove the followg theoem fo th s cse Theoem : Let A the symmetzle GGX d d d d c c Hee c w w Z c d x GX c x e postve teges Moeove GX s KM mtx of fte ffe o defte type of ode GGX s supemtx of f te ffe o def te type wth smple mgy oots dded to tht of GX The the follo wg esults e tue fo BKM supelges wth odd oots GX s of fte type: If l W K wth e I s tue fo ll l l fo ll the the coespodg BKM supelge stsfes SIM popety

8 7 Appled Mthemtcs 4: -5 m If W Π \ ψ W ψ W{ ψ } m m ψ { ψ } ψ { ψ } d x w > s tue fo ll wth the ove GGX fo the the coespodg BKM supelge stsfes SIM popety Hee ψ ψ s the set of ll odd oots ψ m w K W Π W { ψ } s the m w W set of ll postve mgy oots GX s of utwsted ffe type: If l W K wth e I s tue fo ll l l fo ll the the coespodg BKM Le supelge stsfes SIM popety m If W Π \ ψ ψ W{ ψ } wth d x w > the ove GGX fo s tue fo ll the the coespodg BKM Le supelge stsfes SIM popety If GX s of twsted ffe type the Stctly mgy popety does ot hold 4 If GX s of defte type the Stctly mgy popety does ot lwys hold Poof: I the usul otto let I { } wth I m { } I { } Π { } s the set of ll smple oots wth m Π { } s the set of ll smple mgy e Π { oots } s the set of ll smp le el oots Igeel ψ { I { I m o { > } o } { m Z Z I e } I e } Let GX e of f te type GCM e postve mgy oot The m m w K W Π W{ ψ } Hee w W e K { Q > I supp > } We dscuss elow Cse Cse Csec septely Cse: If l W K the we hve > l > fo l > l By Theoem 4 t s cle tht f l m the Stctly mgy popety holds W Π ψ ψ the c e m Cse: If \ wtte s Flly m fo ll Π Hee We dvde ths cse to Cse Cse CseCse Cse septely m Cse: Let We hve Sce e ε fo I ' s e lwys egtve teges s ε e lwys postve y theoem 5 t s cle tht sm Cse:Let Cse: Let wth /

9 N Sthumoothy et l: Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety 8 If y the theoem sm 5 / O the coty f The As o s el smp le oot / Ths s lso ot tue ecuse e egtve teges So y theoem sm 5 Cse: Let wth The But o Hee / ecuse s smp le el oot / fo e ψ wth Z / So / / > Hece y theoem 5 > Cse:Let We hve wth sm f > / As ll the ' s e egtve teges y theoem 5 sm W{ ψ the c e Csec: If } wtte s Hee fo ll ψ Flly We dscuss elow Csec Csec Cse septely Csec m Csec: Let We hve Sce e ε fo I ' s e lwys egtve teges s ε e lwys postve y theoem 5 t s cle tht sm Csec: Let Csec:Let wth / If the We hve But sm / o

10 9 Appled Mthemtcs 4: -5 As s el smple oot wth / Ths s ot tue ecuse e egtve teges So y theoem sm 5 Csec: Let wth The But 4 4 e o / Hee ecuse s smp le el oot / fo e ψ wth Z / So / / wth sm > f > By theoem 5 Csec:Let > We hve / As ll the ' s e egtve teges y theoem 5 Let GX e of utwsted ffe type Fo KM lges of utwsted ffe type SIM popety holds s pe Cspeso994 Fo BKM lges wth odd oots whch we get s extesos of KM lges utwsted ffe type the poof s exctly sme to cse hece SIM popety holds Let GX e of twsted ffe type As pe Cspeso994 metoed ove SIM popety does ot hold fo KM lges the sme s tue fo BKM Le supelges whch ppe s exteso of KM lges of twsted ffe type Hece SIM popety does ot hold 4Let GX e of defte type As f s defte BKM Le supelges e coceed exteso of fte sm utwsted ffe type of KM lges wll hold SIM popety whee s othe lges do ot hold The followg exmple wll llustte the ove theoem Exmple: Exteso of fte type Let A the symmetzle GGX Ths s BKM supemtx of defte type deoted y SBGA wh ch s exteso of fte type A If > > 4 the the Dy d gm c e dw s follows: F gue Dy dgm of SBGA The Weyl goup fo coespodg BKM Le supelge s W { } Hee m m W K W Π W{ w W w W W K ψ } p q p q o o o o o wth W Π m p q p q o o o o o

11 N Sthumoothy et l: Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety o o o q p q p W } { ψ o o Cse: W K The followg elt os v c e dectly vefed If wth the > > If wth the > > If wth the > > vif wth the > > vif wth the > > vif wth the > > By Theoem 4 fo m the ove esults v v v t s cle tht f tht s f geel l l the sm fo ll W K Cse : Let m W Π The follo wg elt os v c e esly vefed If m we get By Theoem 5 f > the / / whch mples sm If m we get If m we get v If m we get v If m we get d v If m we get Fo m the ove esults v v v wth > t s cle tht / fo v v v Hece y Theoem 5 SIM p opety holds I geel f > the sm fo m W Π

12 Appled Mthemtcs 4: -5 Csec: Let W{ ψ } The followg eltos v c e dect ly vefed m If we get By Theoem 5 f > the / / wh ch mp les sm m If we get m If we get m v If we get m vif we get get m v If we Fo m the ove esults v v v wth / fo t s cle tht Z v v v Hece y theoem 5 SIM p opety holds I geel fo W ψ } SIM popety holds f > { Exmple: Exteso of utwsted ffe type Let A the symmetzle GGX defte type deoted y SBGA Ths s BKM supemtx of whch s exteso of utwsted ffe type A If > 4 > 4 the the Dy dgm c e dw s fo llo ws: F gue Dy dgm of SBGA The Weyl goup of the coespodg BKM Le supelge s W { The Z m m W K W Π W{ ψ } Hee w W w W W K { o o p q p q o } o wth W Π m p q p q o o o o o o o 6 o 6 o 8 o }

13 N Sthumoothy et l: Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety 4 6 o W{ ψ } p q p q o o o o o } o o 6 o 6 o 8 o 4 6 o Cse:Let w W w K The follo wg elt os v c e esly vefed If wth the > > } If wth the > > If wth the v If > > wth v If > > wth vif the > the > 4 4 wth > > the Smlly we c fd > fo dffeet m Hece y Theoem 4 y the ove esults v othes t s cle tht sm fo ll f w W w K l l SIM p opety holds W Π W Tht s f ψ m Cse: Let { } The followg eltos v c e esly vefed m If we get > / / By Theoem 5 f the sm whch mples m If we get m If we get vif we get m Smlly we c fd fo I e 7 7 m Hece y Theoem 5 y the ove esults othes wth > t s cle tht / fo m > m W Π I geel f W Π we get sm fo

14 Appled Mthemtcs 4: -5 CseC: If W ψ } The followg { eltos v c e esly vefed m If we get By Theoem 5 f > / / the whch mples sm m If we get m If we get vif get m Smlly we c fd fo I e 7 6 we m Hece y Theoem 5 y the ove esults v othes wth > t s cle tht m W Π > we get W ψ } / fo I geel f { sm fo ClssII: BKM Le supelges of fte ode wth f te o-empty set of odd oots We dvde ths clss to thee suclsses All smple oots e mgyodd o eve Oe smple el ootodd o eve fte ume of mgy ootsodd o eve Fte ume of smple el oots fte ume of mgy oots We dscuss these cses elow All smple oots e mgyodd o eve: Fo ths clss ll the oots e mgy So these lges stsfy stctly mgy popety Oe smple el ootodd o eve fte ume of mgy ootsodd o eve: We pove the followg theoem fo ths cse Theoem : Let GGX d d c A Hee Z c the symmetzle c c d e postve teges GGX s the BKM supemtx wth oe el smp le oot fte ume of mgy oots If wth l l s tue fo ll l the the coespodg BKM Le supelge stsfes Stct ly mgy popety Poof: I the usul otto {} e I wth I {} I m { I } Π { } wth e m m Π { } Π { I } We defe N ψ ψ {} o ψ { I ψ {} { I Let 4 l m } β m } o The y Theoem > l > l > l l l whch s sme s If l the Stctly mgy popety holds Rem: Fo BKM supelges whch ppe s exteso of twsted ffe typecse exteso of defte typecse 4 exmples wee gve sm \ pm ull [7] setsecto 4 Cse sucse Rems: We hve see ove tht the cse of BKM Le supelges of fte ode wth oe smp le el oot odd o eve fte ume of mgy oots odd o eve m fo ψ { I } SIM p opety holds oly whe e ll gete th oe As coute exmple we cosde Moste Le supelge wth oe smp le el oot fte ume of smp le mgy ootsodd o eve Cosde wth l

15 N Sthumoothy et l: Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety 4 l l fo c c: mu lt plcty of the oot coespodg to - As c the coespodg BKM Le supelge does ot stsfy SIM popety We pove ths elow Cosde Moste Le supelge whch hs the followg supemtx s defed elo w: Let I { } { } e dex set cosde the Bocheds-Ct supe mtx A wth chge m c I I whee c e the coeffcets of the ellptc modul fucto q 744 q 96884q 4976q Hee c q A I 4 s the BKM supemtx I { } { } We defe ψ { I } s the el oot coespodg to the dgol elemet e the mgy oots c coespodg to the dgol elemet - We cosde y fo c the > Ths mp les does ot stsfy the stctly mgy popety fo c Hece Stctly mgy popety does ot hold fo Moste Le supelge Fte umetlest two of smple el oots fte ume of mgy oots: We pove the followg theoem fo ths cse Theoem : Let A the symmetzle GGX d ' ' d d d d c d d c c c c Z c d c d Hee e postve teges GGX s the BKM supemtx wth el smp le oots fte ume of mgy oots If wth l l l s tue fo ll the the coespodg BKM Le supelge stsfes Stctly mgy popety Poof: I the usul ottos I {} wth I e { } I m { } { } e { m { Hee Π the set of ll smple oots wth Π } the set of smp le el oot Π the set of ll smp le mgy } oots We defe ψ N ψ { I ψ { I Let e e } o ψ { I } { I m } m } o e } Π l As ll > { l > l I / e egtve teges Hece > we hve l e SIM popety holds Rems: As the cse of Moste Le supelge wth oe smp le el oot fte ume of mgy smp le oots wth the codto we c cosde BKM Le supelges wth two smp le el oots fte ume of mgy smp le oots wth the codto fo some fo I ths cse s > fo ll I m fo some I sml to Moste Le supelge s theoem SIM popety does ot hold Hece we udest tht fo the fte ode cse the SIM popety depeds o the o-dgol o-zeo etes of the coespodg BKM supemtx 4 Coclusos I ths ppe complete clssfcto of Bocheds Kc-Moody Le supelges possessg stctly mgy popety s gve Fo m ths clssfcto oe c udest tht stctly mgy popety depeds m ly o the coeffcets of the coespodg BKM supemtx Wth these fdgs dffeet complete clssfctos of Bocheds Kc-Moody Le supelges possessg specl mgy oots puely mgy oots stctly mgy oots wee septely foud out dffeet e

16 5 Appled Mthemtcs 4: -5 esech ppes I fct these clssfctos wll e vey much helpful to the eseches to exted these clsses of oot systems to othe types of f te fte dmesol Le supelges Moeove othe chctestcs of these clsses of Bocheds Kc-Moody Le supelges possessg these oot systems c lso e studed These fdgs my lso led to my othe pplctos ACKNOWLEDGEMENTS The esech hs ee fclly suppoted y the Uvesty Gts Co mmsso UGC Govt of Id though the Mo esech Poect FNo6-7/8SR The uthos NSthumoothy Pcpl Ivestgto of the Poect KPydhsPoect Fello w fo the Poect e thful to the UGC fo the sme REFERENCES [] Beett C Imgy Roots of Kc-Moody lge whose eflectos peseve oot multplctes J Alge vol 58 pp [] Bocheds RE Geelzed Kc-Moody lges J Alge vol5 pp [] Cspeso D Stctly mgy oots of Kc-Moody lges J Alge vol 68pp [4] Kc VG Le supelges AdvMth vol6 pp [5] Kc VG Ifte dmesol lges Deded's η - fucto clsscl Mous fucto the vey stge fomul Adv Mth vol pp [6] Kc VG Ifte dmesol Le lge d ed Cmdge Uvesty Pess Cmdge 99 [7] LZ-H JY-D BR-P Reltoshp etwee eflectos detemed y mgy oots the Weyl goup fo specl GKM lge J Mth Res Exp Vol9 pp [8] Mott V Sco A Relzto of Bocheds lges It J Mod Phys A vol7 pp [9] Stosh N Kzhd - Lusztg coectue fo geelzed Kc- Moody lges II Poof of the coectue Ts Ame Mth Soc vol 47 pp [] Sthumoothy N Um Mhesw A Puely mgy oots of Kc-Moody lges Comm Alge vol 4 pp [] Sthumoothy N Llly PL O the oot system of geelzed Kc-Moody lges JMds Uvesty WMY- Specl Issue Secto B : Sceces vol 5 pp 8- [] Sthumoothy N Llly PL Specl mgy oots of geelzed Kc-Moody lges Comm Alge vol pp [] Sthumoothy N Llly PL A ote o puely mgy oots of geelzed Kc-Moody lges Comm Alge vol pp [4] Sthumoothy N Llly PLO some clssess of oot systems of geelzed Kc-Moody lges Cotempoy MthemtcsAMS vol4 pp 89-4 [5] SthumoothyN LllyPL Complete clssfctos of geelzed Kc-Moody lges possessg specl mgy oots stctly mgy popety Comm Alge vol 5 pp [6] SthumoothyN LllyPL Nzee BshA Specl mgy oots of BKM Le supelges Ite Jou Pue ppled Mths vol 84 pp [7] SthumoothyN LllyPL Nzee BshAStctly mgy oots puely mgy oots of BKM Le supelgescomm Alge vol 77 pp [8] SthumoothyNPydhsK Complete clssfcto of BKM Le supelges possessg specl mgy ootsccepted fo pulcto Comm AlgeUSA6 pges- to ppe [9] MWmoto Ifte dmesol Le lges Oglly pulshed Jpese y Iwm Pulshes Toyo 999 Tslted fom the Jpese y Ke Ioh