On the Quasi-Hyperbolic Kac-Moody Algebra QHA7 (2)

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1 Ieaoal Reeach Joual of Egeeg ad Techology (IRJET) e-issn: 9 - Volume: Iue: May- -ISSN: 9-7 O he Qua-Hyebolc Kac-Moody lgeba QH7 () Uma Mahewa., Khave. S Deame of Mahemac Quad-E-Mllah Goveme College fo Wome (uoomou), Chea Taml Nadu, Ida Deame of Mahemac M.O.P Vahav College fo Wome (uoomou), Chea Taml Nadu, Ida *** bac - I h ae, fo a ecal cla of defe whoe aocaed ymmezable ad decomoable ye of ua hyebolc Kac-Moody algeba QH () 7, we oba he comlee clafcao of he Dyk dagam aocaed o he Geealed Caa Mace of QH () GCM 7. Moeove, ome of he oee of magay oo uch a cly magay, uely magay ad ooc oo ae alo uded. a b c d e Key Wod: Kac-Moody algeba, affe, defe, Dyk dagam, ua hyebolc, cly ad uely magay oo.. INTRODUCTION Roo yem lay a val ole he ucue of Kac- Moody algeba [],[]. Kac [], oduced he coce of cly magay oo fo Kac-Moody algeba. Caeo [], obaed he comlee clafcao of Kac- Moody algeba oeg cly magay oey. ecal cla of defe ye of Kac-Moody algeba, called a exeded-hyebolc Kac-Moody algeba ad he ew coce of uely magay oo wa oduced by Shaumoohy ad Uma Mahewa [4]. Shaumoohy e. al. [-8] obaed he oo mullce fo ome acula clae of EH () ad EH (). I [9], Uma Mahewa oduced aohe ecal cla of defe ye of o-hyebolc Kac-Moody algeba called ua-hyebolc Kac-Moody algeba. I [-4], ome acula clae of defe ye ua hyebolc Kac-Moody algeba QHG,QH (), QH () 4, QH () ad QH () 7 wee codeed, he homology module uo level hee ad he ucue of he comoe of he maxmal deal uo level fou wee deemed by Uma Mahewa ad Khave. Fo ome ua affe Kac Moody algeba QC (), QG () ad QGGD () he comlee clafcao of he Dyk dagam ad ome oee of eal ad magay oo wee obaed [,7 ad8] by Uma Mahewa. The comlee clafcao of he Dyk dagam aocaed o he ua hyebolc Kac-Moody algeba QH () wa obaed ad he oee of uely magay ad cly magay oo wa uded []. I h wok, we code he acula cla of defe ye of ua-hyebolc Kac-Moody algeba QH 7 (),, IRJET Imac Faco value: 4.4 ISO 9:8 Cefed Joual Page 49 whee,,,,,a,b,c,d,e ae o-egave ege. The ma am of h wok o gve a comlee clafcao of he Dyk dagm aocaed wh QH () 7 ad o udy ome of he oee of magay oo... Pelmae I h eco, we ecall ome eceay coce of Kac- Moody algeba. ([],[9]). Defo.[] : ealzao of a max ( a ), of ak l, a le ( H, π, π v ), H a - l dmeoal comlex veco ace, π = {α,,α } ad π v = { α v,,α v } ae lealy deede ube of H* ad H eecvely, afyg α (α v ) = a fo, =,.,. π called he oo ba. Eleme of π ae called mle oo. The oo lace geeaed by π Q z. Defo.[]: The Kac-Moody algeba g() aocaed wh a GCM ) he Le algeba geeaed by ( a, he eleme e, f, =,,.., ad H wh he followg defg elao : ' [ h, h ], [ e, f [ h, e ] ( h) e [ h, f ( ade ) ( adf ) v ] ' h, h H ] ( h) f a a f e,,, N,, N The Kac-Moody algeba g() ha he oo ace decomoo g( ) g ( ) whee g Q ( ) { x g( ) /[ h, x] ( h) x, fo all h H}. eleme α, α Q called a oo f g α. Defo.[9]: Le ) be a ( a, decomoable GCM of defe ye. We defe he aocaed Dyk dagam S() o be Qua Hyebolc

2 Ieaoal Reeach Joual of Egeeg ad Techology (IRJET) e-issn: 9 - Volume: Iue: May- -ISSN: 9-7 (QH) ye f S() ha a oe coeced ubdagam of hyebolc ye wh (-) vece. The GCM of QH ye f S() of QH ye. I h cae, we ay ha he Kac- Moody algeba g() of QH ye. Defo.4[]: oo α Δ called eal, f hee ex a w W uch ha w(α) a mle oo, ad a oo whch o eal called a magay oo. magay oo γ ad o be cly magay f fo evey eal oo α, ehe α + γ o α γ a oo. magay oo α called a ooc f (α, α) =. Defo.[]: geealzed Caa max ha he oey SIM (moe befly: SIM) f Δ m () Δ m (). Defo.[4]: Le α Δ m + ad o be uely magay f fo ay β Δ m +, α+β Δ m +. GCM afe m m he uely magay oey f ( ) ( ). If afe he uely magay oey he he Kac- Moody algeba g() ha he uely magay oey.. DYNKIN DIGRMS SSOCITED WITH THE INDEFINITE TYPE OF QUSI HYPERBOLIC KC-MOODY LGEBR QH7 () I h eco, we ove he clafcao heoem whee coeced, o omohc Dyk dagam aocaed wh QH 7 () ae comleely clafed. Nex, we udy ome of he oee of oo fo ecfc famle he cla QH 7 (). Theoem.: ( Clafcao Theoem ) : Thee ae 88 coeced, o omohc Dyk dagam aocaed wh he GCM of he defe ye of ua hyebolc Kac-Moody algeba QH 7 (). Poof: The Dyk dagam of he defe ye of Kac- Moody algeba QH 7 () obaed by addg a xh veex, whch coeced o he Dyk dagam of he affe Kac-Moody algeba 7 () by, whee ca be ay oe of he e oble: Cae (): edge oed fom he xh veex o ay oe of he fve vece 7 (). Th ca be obaed he followg oble way. () 4 () 4 4 () (4) The umbe of way, he xh veex ca be oed o ay oe of he fve vece 7 () =. Code he dagam (): Hyebolc: Poble of ae, Numbe: Qua Hyebolc: Nl No Qua Hyebolc: Poble of : Numbe : Code he dagam (): Hyebolc: Poble of Numbe: Qua Hyebolc: Poble of ae, Numbe: No Qua Hyebolc: Numbe: Code he dagam (): Hyebolc: Numbe: Nl Qua Hyebolc: Poble of : Numbe: No Qua Hyebolc: Numbe: 8, Code he dagam (4): Hyebolc: Poble of : Numbe: Qua Hyebolc: Numbe: Nl No Qua Hyebolc: Numbe: 8 Theefoe h cae, we ge coeced, o omohc ua hyebolc ye of 4 Dyk dagam, hyebolc ye [9] ad 8 Dyk dagam ae o ua hyebolc ye QH () 7. Cae ( ) : Two edge ae added fom he xh veex o ay wo of he vece of he Dyk dagam of 7 (). Th ca be obaed he followg oble way., IRJET Imac Faco value: 4.4 ISO 9:8 Cefed Joual Page 4

3 Ieaoal Reeach Joual of Egeeg ad Techology (IRJET) e-issn: 9 - Volume: Iue: May- -ISSN: () () (7) (8) (9) () 4 Table 4: () The umbe of way, he xh veex ca be oed o ay wo of he fve vece 7 () 7. The followg able, how he aue ad alo he umbe of ua hyebolc ye of Dyk dagam: Table : Table : ll he eulg Dyk dagam of he dagam (9), () ad () ae o of ua hyebolc ye. Fom he above able, we ge coeced, o omohc ua hyebolc ye of Dyk dagam QH () 7. Cae (): Whe hee edge ae oed fom he xh veex o ay hee vece of he Dyk dagam () 7, we ge he followg oble fom. 4 () () (4) () Table :, IRJET Imac Faco value: 4.4 ISO 9:8 Cefed Joual Page 4

4 Ieaoal Reeach Joual of Egeeg ad Techology (IRJET) e-issn: 9 - Volume: Iue: May- -ISSN: 9-7 Table 7: () (7) (8) The oal umbe of way, he xh veex ca be oed o ay hee of he ohe fve vece () 7 9 7=. Table : ll he eulg Dyk dagam coeod o (), (), () ad (7) ae o of ua hyebolc ye. Fom he above able, we ge 4 coeced, o omohc ua hyebolc ye of Dyk dagam QH 7 (). Cae (v): Le he fou edge be oed fom he xh veex o ay fou vece of he Dyk dagam of 7 (), The oal umbe of way, he xh veex ca be oed o ay fou of he ohe fve vece 7 () 9 4 =. mog hee, he edge coecg he vece bewee o, o, o 4 by ad vay he edge bewee o by, we ge 9 coeced, o omohc ua hyebolc ye of Dyk dagam QH 7 (). Cae (v): Whe he xh veex added o all he fve vece of he Dyk dagam 7 (). Table : (9) The oal umbe of way, he xh veex ca be oed o all he fve vece 7 () 9 = 949. ll he eulg Dyk dagam coeodg o (9) ae o of ua hyebolc ye. Theefoe, fom he above fve cae we ge a oal of ( ) = 88 coeced, o omohc ua hyebolc ye of Dyk dagam QH 7 (). Poee of magay oo: Pooo.: Code he defe ye of ua hyebolc Kac-Moody algeba QH 7 (), whoe aocaed ymmezable ad decomoable GCM, IRJET Imac Faco value: 4.4 ISO 9:8 Cefed Joual Page 4

5 Ieaoal Reeach Joual of Egeeg ad Techology (IRJET) e-issn: 9 - Volume: Iue: May- -ISSN: 9-7, IRJET Imac Faco value: 4.4 ISO 9:8 Cefed Joual Page 4 e d c b a whee,,,,,a,b,c,d,e ae o-egave ege. The he Kac-Moody algeba g() coeodg o QH 7 () ha he followg oee: () The magay oo of g() afy he uely magay oey. () The magay oo of g() afy he cly magay oey. Poof: () Sce a coeced, ymmezable ad decomoable GCM, by ug coollay. [4], we ge, ) ( ) ( m m. Hece he Kac- Moody algeba g() coeodg o QH 7 () ha uely magay oey. () Sce ymmezable ad decomoable GCM, afe he eued codo gve he Theoem () []. Hece he Kac-Moody algeba g() coeodg o QH 7 () ha cly magay oey. We gve he decomoo of he ymmezable GCM fo a geeal famly QH 7 () : Fo he defe ye of ua-hyebolc Kac- Moody algeba QH 7 (), he aocaed ymmezable ad decomoable GCM e d c b a whee,,,,,a,b,c,d,e ae oegave ege. Sce ymmezable, ca be exeed a =DB whee a a / / / adb / D wh he codo, b=a/, c = a/, d= a/, e = a/. Examle : Code he defe ua-hyebolc Kac- Moody algeba QH7 () whoe aocaed ymmezable ad decomoable GCM. Sce ymmezable, =DB whee / / B ad D Hee (α,α ) =, (α,α ) =, (α,α ) =, (α 4,α 4) =, (α,α ) =, (α,α ) =, (α,α ) = (α,α ) =,(α,α ) =(α,α ) = -, (α,α 4) = (α 4,α ) =, (α,α ) = (α,α ) =, (α,α ) = (α,α ) = -, (α,α ) = (α,α ) = -, (α,α 4) = (α 4,α ) =,(α,α ) = (α,α )=, (α,α ) = (α,α )= -, (α,α 4) =(α 4,α ) = -, (α,α ) =(α,α ) =, (α 4,α ) =(α,α 4) = -, (α,α ) = (α,α )=,(α 4,α ) = (α,α 4) = -,(α,α ) = (α,α ) = ½. Le β = α +α +α +α 4+α +α, he (β, β) = - 4 <,Theefoe β a magay oo. Fo evey eal oo α, we fd ha β + α alo a oo. Theefoe β a cly magay oo. Le γ = α 4+α +α he (γ, γ) =, Hece β a ooc oo. Le β + γ = α + α +α +α 4+α +α he (β + γ, β + γ) <.Hece γ a uely magay oo. Examle : Code he defe ua-hyebolc Kac- Moody algeba QH 7 () whoe aocaed ymmezable ad decomoable GCM. Sce ymmezable, =DB whee B ad D Hee (α,α ) =, (α,α ) =, (α,α ) =, (α 4,α 4) =, (α,α ) =, (α,α ) =, (α,α ) = (α,α ) =,(α,α ) =(α,α ) = -, (α,α 4) = (α 4,α ) =, (α,α ) = (α,α ) =, (α,α ) = (α,α ) = -, (α,α ) = (α,α ) = -, (α,α 4) = (α 4,α ) =,(α,α ) = (α,α )=, (α,α ) = (α,α )= -, (α,α 4) =(α 4,α ) = -, (α,α ) =(α,α ) =, (α 4,α ) =(α,α 4) = -, (α,α ) = (α,α )=,(α 4,α ) = (α,α 4) = -,(α,α ) = (α,α ) = ½. Le β = α +α +α +α 4+α +α, he (β, β) = - <, Theefoe β a magay oo. Fo evey eal oo α, we fd ha β + α alo a oo. Hece β called a cly magay oo. Le γ = α +α +α 4+α +α he (γ, γ) = - <, he γ a aohe magay oo. Now le β + γ = α + α +α +α 4+α +α he (β + γ, β + γ) = - <.Hece β alo a uely magay oo.

6 Ieaoal Reeach Joual of Egeeg ad Techology (IRJET) e-issn: 9 - Volume: Iue: May- -ISSN: 9-7. CONCLUSIONS I h ae, he comlee clafcao of he Dyk dagam obaed fo he defe ye of uahyebolc Kac-Moody algeba QH 7 (). We ca exed h wok fuhe o comue he oo mullce fo QH 7 (). REFERENCES [] Davd Caeo, Scly Imagay Roo of Kac- Moody algeba, Joual of lgeba, vol. 8, 994,. 9-. [] Kac V.G., Ife Dmeoal Le lgeba, d ed., Cambdge Uvey Pe, Cambdge, 99. [] Moody R.V, ew cla of Le algeba, J. lgeba, vol., 98,. -. [4] Shaumoohy N. ad Uma Mahewa., Puely Imagay Roo of Kac-Moody algeba, Commucao lgeba, vol. 4(), 99, [] Shaumoohy N. ad Uma Mahewa., Roo mullce of exeded hyebolc Kac-Moody algeba, Commucao lgeba, vol. 4(4), 99, [] Shaumoohy N., Llly P.L. ad Uma Mahewa., Roo mullce of ome clae of exededhyebolc Kac-Moody ad exeded-hyebolc geealzed Kac-Moody algeba, Coemoay Mahemac MS, vol. 4, 4, [7] Shaumoohy N. Uma Mahewa. ad Llly P.L., Exeded-Hyebolc Kac-Moody EH () lgeba ucue ad Roo Mullce, Commucao lgeba, vol (), 4, [8] Shaumoohy N. ad Uma Mahewa., Sucue ad Roo Mullce fo Two clae of Exeded Hyebolc Kac-Moody lgeba EH () ad EH () fo all cae, Commucao lgeba, vol. 4,,. -. [9] Uma Mahewa., Imagay Roo ad Dyk Dagam of Qua Hybebolc Kac-Moody lgeba, Ieaoal Joual of Mahemac ad Comue lcao Reeach, vol. 4(), 4, [] Uma Mahewa. ad Khave S., udy o he Sucue of a cla of defe o-hyebolc Kac- Moody lgeba QHG, Ieaoal Joual of Mahemac ad Comue lcao Reeach, vol. 4(4), 4, 97-. [] Uma Mahewa. ad Khave S. O he Sucue of Idefe Qua-Hyebolc Kac-Moody lgeba QH (), Ieaoal Joual of lgeba, vol. 8(), 4, [] Uma Mahewa. ad Khave S., Sucue of he Qua-Hyebolc Kac-Moody lgeba QH () 4, Ieaoal Mahemacal Foum, vol. 9(), 4, [] Uma Mahewa. ad Khave S., Sudy o he Sucue of Idefe Qua-Hyebolc Kac-Moody lgeba QH () 7, Ieaoal Joual of Mahemacal Scece, vol. 4(), 4, [4] Uma Mahewa. ad Khave S., Sudy o he Sucue of Qua-Hyebolc lgeba QH (), Ieaoal Joual of Pue ad led Mahemac, vol. (),,. -8. [] Uma Mahewa., I Igh o QC () : Dyk dagam ad oee of oo, Ieaoal Reeach Joual of Egeeg ad Techology (IRJET), vol. (),, [] Uma Mahewa. ad Khave S., Sudy o he Roo Syem ad Dyk dagam aocaed wh QH (), Ieaoal Reeach Joual of Egeeg ad Techology (IRJET), vol. (),, [7] Uma Mahewa., Roo yem ad Dyk dagam fo he Geeal cla of Idefe Qua ffe Kac- Moody lgeba QG (), Ieaoal Joual of Egeeg Iovao & Reeach, vol., Iue,,. 9-. [8] Uma Mahewa., Qua ffe Geealed Kac- Moody lgeba QGGD () : Dyk dagam ad oo mullce fo a cla of QGGD (), Ida Joual of Scece ad Techology, vol. 9(),,. -. [9] Wa Zhe-Xa, Ioduco o Kac Moody lgeba, Wold Scefc Publhg Co. Pv. Ld., Sgaoe, 99., IRJET Imac Faco value: 4.4 ISO 9:8 Cefed Joual Page 44

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