Stuy on Root Popetes o Supe Hypebol GKM lgeb G.Uth n M.Pyn Deptment o Mthemts Phypp s College Chenn Tmlnu In. bstt: In ths ppe the Supe hypebol genelze K-Mooy lgebs o nente type s ene n the mly s lso elte. Dent ses une ths mly s stue wth one gny sple oot. The symmetzble GGCM o s gven by b wth the ontons e b n e whe b e { }. Fo the mly the popetes o oots s ompute o ent ses. The 8 non-somoph onnete Dynn gm ssote wth the Supe-hypebol genelze K-Mooy lgeb. The exstene n non exstene o puly gny oots sttly gny oots n spel gny oots o ths mly s elute wth exmples. INTRODUCTION The Genelze K-Moy lgeb bbvte s GKM lgeb ws ntoue by Bohes n []. Bennett n Cspeson elte the spel n sttly gny oots o KM lgebs n [] n []. In KM lgebs ll sple oots l n n GKM lgebs gny sple oots exsts. In GKM lgebs the Genelze Genelze Ctn Mtes GGCM extensons o nte ne n nente types o KM lgebs. Stnumoothy. The puly gny sttly gny n spel gny oots o nte ne n hypebol types o spe mles n [5]- [9]. Fo the mlyeb Xnng Song ompute the oot multpltes n lso etemne the oot stutu n [] n []. Keywos: Dynn gm l gny Roots Supe hypebol. MS MSC Coe: 7B67 PRELIMINRIES The nottons n entons use n ths ppe s n []. Denton. []: In GKM lgebs the Dynn gms s ene s ollows: To evey GGCM s ssote Dynn gm S ene s ollows : S hs n vetes n vetes n j onnete by mx { j j } numbe o lnes j. j n the s n ow pontng tows j >. I j. j > n j onnete by bol e ege equppe wth the oe p j j o nteges. I = th vetex wll be enote by whte le n = th vetex wll be enote by osse le. I = - > th vetex wll be enote by whte le wth wtten bove the le wthn the pnthess. Denton. []: We sy GGCM n j j s o Supe Hypebol type s o nente type n the Dynn gm ssote wth hs onnete pope sub gm o hypebol type whose GGCM s o oe n-. We then sy the ssote Dynn gm n the osponng GKM lgeb to be o Supe hypebol type bbvte s SH type. Denton. []: Let α be n gny oot o g whh s symmetzble GKM lgeb. We ll α spel gny oot α stses the ollowng ontons: α α ; α = α = = ; α pseves oot multpltes. Denton. []: oot α s s to be sttly gny o evey o s oot. The set o ll sttly gny oots s enote by Denton.5 [7]: oot s. s lle puly gny o ny The KM lgeb s s to hve the puly gny popety evey gny oot s puly gny. Note: In ths ppe the GGCM onssts o one gny sple oot. CLSSIFICTION OF DYNKIN DIGRMS OF GKM LGEBRS : In ths seton the GGCM e whe p q { } b s onse 56
Poposton. : The 8 non-somoph onnete Dynn gms ssote wth the GGCM o nente Supe hypebol type. Poo : The ssote Dynn gm wth the hypebol mly H s We exten the th vetex wth H n ll possble ombntons o onnete non-somoph Dynn gms etemne o the ssote GGCM o Supe hypebol type He n be psente by one o the possble 9 eges:. Tble Extene Dynn gm o Supe hypebol type 8 Cosponng GGCM Numbe o possble Dynn gms When = e b By onnetng the outh vetex to ll the othe the vetes the exsts 9 onnete Dynn gms n whh 56 somoph Dynn gms. Exlung these we get 65 onnete non-somoph Dynn gms o both the se when = n >.e In totl we get Dynn gms o both the ses. When > e b When = mong the vetes we onnet the outh vetex wth ny o the two vetes wth ent ombntons by the 9 possble eges. Theo n ths se the ssote onnete Dynn gms x 9 = n exlung the 98 somoph Dynn gms we get 5 onnete nonsomoph Dynn gms o both 57
Extene Dynn gm o Supe hypebol type 8 Cosponng GGCM Numbe o possble Dynn gms When > the se when = n >. When = When > In ths se we onnet the outh vetex nepenently to the othe the vetes by the 9 possble eges. Thus the possble numbe o onnete Dynn gms ssote wth s x 9 = 7. But by jonng the vetes n we get 8 somoph Dynn gms. Thus by eletng these somoph gms we get 9 gms when = n 9 gms when >. Thus the exsts 8 types o onnete non somoph Dynn gms ssote wth the GGCM o. PROPERTIES OF ROOTS Conse the symmetzble GGCM o ontons = e b wth the e b n whe b e { }. We hve } } n { }. { The non-egenete symmet blne om gven s { e 58
59 The unmentl letons e He W w K w whe K s gven by / { K } ; ; e e n n n e wth n e Note tht the Weyl goup s nnte. Root System o : In ths seton we suss the oot popetes o the Supe Hypebol mly. Rel Roots: ll sple l oots hve sme length. Roots o Heght : α +α α +α = -+- Cse : When sotop s gny s Cse : When = ; α +α α +α = - sotop s gny s α +α α +α = -+- Cse : When & & & sotop s gny s gny s l s l s Cse : When = ; α +α α +α = -
s gny & s sotop & s l & α +α α +α = +- =. Theo α +α s n sotop oot. α +α α +α = so tht α +α s sotop. Slly the othe ses o heght oots α +α α +α n be susse. o Roots o Heght : α +α α +α = -+- Cse : When s gny s sotop Cse : When = ; α +α α +α = - s gny s sotop α +α +α α +α +α = -6 6 s gny s sotop Slly oots o heght o ll othe ses n be elte. Roots o Heght : α +α α +α = -+8-8 Cse : When s gny s sotop Cse : When = ; α +α α +α = 8-8 8 8 s gny o s sotop o n α +α α +α = 8+- = 8 whh s l oot. 5
b Poposton.: Let = be the symmetzble GGCM o e The exsts no spel gny oot o g. whe p q { }. n Poo: Suppose K be spel gny oot o g. We hve e e be Let αα=. By the leton o gny oot enton we hve b e e ; ;... Then o spel gny oot α we hve Fom the bove equtons n we get e Then e 5 s bsu. Theo no spel gny oot exsts o g whe s symmetzble. Poposton.: The Supe hypebol GKM lgeb stses the puly gny popety. Poo: The some genelze K-Mooy lgebs n whh the puly gny oots exsts n oes not exsts. He the SHGKM lgeb gny popety. poves the puly He α s n gny sple oot n ny oot e wth α the sultnt s lso n gny oot n lso suppot o α s onnete. Theo ll gny oots puly gny n hene o ny n o ny we get s oot. Exmple : α +α stses the puly gny popety..e. α = α β = α +α ; we get α+β = - whe whh stses the puly gny popety. Poposton.: The Supe hypebol GKM lgeb stses the sttly gny popety. Poo: Sne suppot o α s onnete the ton o subtton o ny ombnton o α =& s oot. Hene o ny n o ny we get s oot. Theo the SHGKM lgeb stses the sttly gny popety. Exmple : α +α +α stses the sttly gny popety..e. α = α γ = α +α ; we get α+ γ = -6 s oot whh poves the sttly gny popety. REFERENCES []. BennettC. 99 Imgny oots o K Mooy lgeb whose letons pseve oot multpltes J. lgeb 58 pp. -67. []. BohesR.E. 988 Genelze K Mooy lgebs J.lgeb 5 pp. 5-5. []. CspesonD. 99 Sttly gny oots o K Mooy lgebs J. lgeb 68 pp. 9-. []. CHEN Hongj n LIU Bn. 997 Supe Hypebol Type K-Mooy Le lgeb System Sene n Mthemtl senes 9-. [5]. SthnumoothyN n LllyP.L. On the oot systems o genelze K-Mooy lgebs J.Ms Unvesty WMY- spel ssue Seton B:Senes 5 8-5
[6]. SthnumoothyN n LllyP.L. Spel gny oots o genelze K-Mooy lgebs Comm. lgeb pp. 7-787. [7]. SthnumoothyN n LllyP.L. note on puly gny oots o genelze KM lgebs Comm. lgeb pp. 567-58. [8]. SthnumoothyN n LllyP.L. On some lsses o oot systems o genelze KM lgebs Contemp. Mth. MS pp. 89-. [9]. SthnumoothyN n LllyP.L. 7 Complete lsstons o Genelze K-Mooy lgebs possessng spel gny oots n sttly gny popety Communtons n lgeb US58 pp. 5-. []. Xnng Song n Yngln Guo. Root Multplty o Spel Genelze K- Mooy lgeb EB Mthemtl Computton pp. 76-8. []. Xnng Song n Xox Wng Yngln Guo. Root Stutu o Spel genelze KM lgebs Mthemtl Computton pp. 8-88. 5