A Study on Generalized Generalized Quasi hyperbolic Kac Moody algebra QHGGH of rank 10

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1 Global Joural of Mathematcal Sceces: Theory ad Practcal. ISSN Volume 9, Number 3 (7), pp Iteratoal Research Publcato House A Study o Geeralzed Geeralzed Quas (9) hyperbolc Kac Moody algebra QHGGH of rak A. Uma Maheswar Assocate Professor & Head, Departmet of Mathematcs, Quad-E-Mllath Govermet College for Wome (Autoomous), Chea - 6, Tamladu, Ida Abstract I ths work, the defte Quas Hyperbolc Geeralzed Kac-Moody algebra (9) QHGGH of rak s studed. Behavor of roots s aalyzed. Specfc propertes of roots amely, specal magary, strctly magary ad purely magary roots are dscussed. The complete classfcato of o-somorphc, (9) coected Dyk dagrams assocated wth QHGGH s obtaed. Keywords: Geeralzed eeralzed Carta Matrx, Dyk dagrams, quas hyperbolc, defte, strctly magary, purely magary. AMS MSC Code: 7B67. INTRODUCTION The costructo of Geeralzed Kac-Moody algebra was gve by Borcherds []. I [] ad [3], Beett ad Caperso troduced the specal ad strctly magary roots. Sthaumoorthy ad Uma Maheswar [9-] defed a ew class of magary root amely, purely magary root ad determed the root multplctes for the Geeralzed Kac-Moody algebras (GKM algebras) of EHA (), QAGGA () ad QAGGD3 (). Sthaumoorthy et al. defed strctly magary ad specal magary roots for fte, affe ad hyperbolc cases [4-9]. I [4,5], Xfag Sog ad et al. determed the root structure ad root multplcty for the fte GKM algebra EB. I ths paper, secto deals wth the basc deftos ad prelmary deas requred for further study; I secto 3,the complete classfcato of Dyk dagrams of

2 44 A. Uma Maheswar defte quas hyperbolc Kac Moody algebras of rak, deoted as QH (9), whch are obtaed from the hyperbolc famly of rak 9, H (9) []; I secto 4, a specfc quas hyperbolc famly of QH (9) s aalysed depth; Basc propertes of the roots are also dscussed... PRELIMINARIES For the detaled study o Kac-Moody algebras oe ca refer Wa[6]. Defto.[6]: A teger matrx A ( a ), (abbrevated as GCM) f t satsfes the followg codtos: () () () a = =,,., a = a =, =,,, a, =,,,. s a Geeralzed Carta Matrx Let us deote the dex set of A by N = {,,}. A GCM A s sad to decomposable f there exst two o-empty subsets I, J N such that I J = N ad a = a = I ad J. If A s ot decomposable, t s sad to be decomposable. Defto.[6]: A GCM A s called symmetrzable f DA s symmetrc for some dagoal matrx D = dag(q,,q), wth q> ad q s are ratoal umbers. Defto.3[6]: A realzato of a matrx s a trple ( H, v, ) where A ( a ), l s the rak of A, H s a - l dmesoal complex vector space,,..., } ad v v v {,..., } { are learly depedet subsets of H* ad H respectvely, satsfyg ( ) a for, =,.,. s called the root bass. Elemets of are called v smple roots. The root lattce geerated by s Q Defto.4 [6]: The Kac-Moody algebra g(a) assocated wth a GCM ) s the Le algebra geerated by the elemets e, f,,,... ad H wth A ( a, the followg defg relatos : ' [ h, h ] [ h, e ] ( ade ), h, h ( h) e a e ;( adf ' H;[ e, f ;[ h, f ) ] a v ], z ( h) f f The Kac-Moody algebra g(a) has the root space decomposto,,, N, g( A) g ( A) where g ( A) { x g( A) /[ h, x] ( h) x, for all h H}. Q N

3 A Study o Geeralzed Geeralzed Quas hyperbolc. 45 Defto.5 [6]: A elemet, Q s called a root f g. Let Q z, Q has a partal orderg defed by f Q +, where, Q. Let ( ( A)) deote the set of all roots of g(a) ad the set of all postve roots of g(a). We have ad. Defto.6 [6]: Defto.4[7]: Defe r Ed (H * ) as r() v,where v, v ) ad N. For each, r s a vertble lear trasformato of H* ad r s called a fudametal reflecto. Defe the Weyl group W to be the subgroup of aut(h*) geerated by {r, N}. For ay Q ad k, defe support of, wrtte as supp, by supp { N / k }. Defto.7 [6]: The Dyk dagram assocated wth the GCM A of order s deoted by S(A) : S(A) has vertces ad vertces ad are coected by max { a, a } umber of les f a. a 4 ad there s a arrow potg towards f a >. If a. a> 4, ad are coected by a bold faced edge, equpped wth the ordered par ( a, a ) of tegers. Defto.8 [6]: A root α s called real, f there exsts a w ɛw such that w(α) s a smple root, ad a root whch s ot real s called a magary root. A magary root α s called sotropc f (α,α) =. A postve magary root α s a mmal magary root (MI root, for short) f α s mmal wth respect to the partal order o H *. The symmetry of the root system meas that we eed oly to prove the results for postve magary roots. Defto.9 []: We defe a defte o-hyperbolc, GCM A= ( ) to be of a, exteded hyperbolc type f every proper, coected sub dagram of S(A) s of fte, affe or hyperbolc type. Defto. [3]: Let A= ( ), be a decomposable GCM of defte type. a, We defe the assocated Dyk dagram S (A) to be of Quas Hyperbolc (QH) type f S (A) has a proper coected sub dagram of hyperbolc type wth - vertces. The GCM A s of QH type f S (A) s of QH type. We the say the Kac Moody algebra g(a) s of QH type. Note: Every Exteded hyperbolc Dyk dagram s Quas hyperbolc coversely. but ot

4 46 A. Uma Maheswar 3. CLASSIFICATION OF DYNKIN DIAGRAMS OF RANK, DENOTED BY QHGGH (9) : Let the GGCM assocated wth the Dyk dagram QHGGH (9) obtaed from the rak 9 hyperbolc dagram H () be geerally gve as a >. a a k ad the correspodg Dyk dagram s gve by Whe k > Whe k = (-k)

5 A Study o Geeralzed Geeralzed Quas hyperbolc. 47 Case ) α s coected wth oly oe of the 9 vertces of H (9). Ths vertex ca be selected from the 9 vertces 9C ways. α ad α, ca be oed by ay of the 9 possble edges. α9 α ; ca represet ay of the e possble edges: wth (a,a), where a >. Hece we wll get 9 x 9 possble Dyk dagrams ths case. Case ) α s coected wth ay two of the 9 vertces of H (9). These two vertces ca be selected from the 9 vertces 9C ways. α, α ca be oed by ay of the 9 possble edges 9 x 9 ways. Hece there are 9 x 9 x 9C possble Dyk dagrams. Case 3) α s coected wth ay three of the 9 vertces of H (9). These three vertces ca be selected from the 9 vertces 9C3 ways. Hece there are 9 x 9 x 9 x 9C3 possble Dyk dagrams ths case. Case 4) α s coected wth ay four of the 9 vertces of H (9). These four vertces ca be selected from the 9 vertces 9C4 ways. Thus, there wll be 9 4 x 9C4 possble Dyk dagrams ths case. Case 5) α s coected wth ay fve of the 9 vertces of H (9). These fve vertces ca be selected from the 9 vertces 9C5 ways. Thus, there wll be 9 5 x 9C5 possble Dyk dagrams ths case. Case 6) α s coected wth ay sx of the 9 vertces of H (9). These sx vertces ca be selected from the 9 vertces 9C6 ways. Thus, there wll be 9 6 x 9C6 possble Dyk dagrams ths case. Case 7) α s coected wth ay seve of the 9 vertces of H (9). These seve vertces ca be selected from the 9 vertces 9C 7 ways. Thus, there wll be 9 7 x 9C7 possble Dyk dagrams ths case. Case 8) α s coected wth ay eght of the 9 vertces of H (9). These eght vertces ca be selected from the 9 vertces 9C8 ways. Thus, there wll be 9 8 x 9C8 possble Dyk dagrams ths case.

6 48 A. Uma Maheswar Case ) α s coected wth all the 9 vertces of H (9). Thus, there wll be 9 9 possble Dyk dagrams ths case. Thus, there are 9 x 9C + 9 x 9C x 9C x 9C x 9C x 9C x 9C x 9C Dyk dagrams assocated wth the Quas hyperbolc geeralzed type QHGGH (). We have proved the followg theorem : Theorem 3. (Classfcato Theorem) : Let QHGGH (9) be the famly of Quas hyperbolc Kac Moody algebras, obtaed from the rak 9 hyperbolc famly H (9). The the umber of o-somorphc coected Dyk dagrams assocated wth QHGGH (9) s ) 9 (9 9 C. 4. A SPECIAL CLASS OF QUASI HYPERBOLIC KAC-MOODY ALGEBRA QHGGH (9) : I ths secto let us study about a partcular class of quas hyperbolc Kac Moody algebra the famly QHGGH (9). Let QHGGH (9) deote the quas hyperbolc Kac- Moody algebra obtaed from H (9) whose assocated GGCM A= a, ) (, s gve by a a k The correspodg Dyk dagram s The GCM s symmetrc. Therefore, blear form <α,α>= a ad (α,α)= for =,3,,. Hece all real smple roots are of equal legth. (a,a)

7 A Study o Geeralzed Geeralzed Quas hyperbolc. 49 The o degeerate symmetrc blear form (, ) defed o the root system s gve as follows: (α,α) = -k; (α,α3)= (α3,α4)=(α4,α5)=(α5,α6)=(α6,α7)= (α7,α8)= (α6,α9)= (α3,α)= -; (α,α)= - a Roots of heght : roots. (α+α, α+α) = -k + -a Case ): Whe k = (α+α, α+α) = -a α+α s a magary root, whe a > Case ): Whe k (α+α, α+α) = -k + -a α+α s a magary root, whe a > Smlarly, α+α3, α3+α4, α4+α5, α5+α6, α6+α7, α7+α8, α3+α, α6+α9 are all real Roots of heght 3: (α+ α+ α3, α+ α+ α3) = -k + -a Case ): Whe k = (α+ α+ α3, α+ α+ α3)= -a α+ α+ α3s a magary root, whe a > Case ): Whe k (α+ α+ α3, α+ α+ α3)= -k + -a α+ α+ α3 s a magary root, whe a > (α+ α3+ α4, α+ α3+ α4) = >, whch s real. Smlarly, α+ α3+ α4, α3+ α4+ α5, α3+ α4+ α, α4+ α5+ α8, α5+ α6+ α7, α+ α3+ α, α5+ α6+ α9, α6+ α7+ α9, α6+ α7+ α8 are real roots. Proposto 4.: The quas hyperbolc geeralzed Kac Moody algebra QHGGH (9) satsfes the purely magary root. Proof: From the Dyk dagram assocated wth QHGGH (9), t s clear that ay magary root volves α+ α. Cosder the addto of ay two postve magary roots α & β Support of α + β wll be coected ad (α + β, α + β). Hece α + β s also a

8 4 A. Uma Maheswar magary root. Hece QHGGH (9) satsfes purely magary property. Example 4.:Let α = α+ α ad β = α+ α+ α3 be two magary roots. We get (α+β, α+β) = -k + -a <, sce a >. Therefore, (α+ α)+ α3 s a magary root. Proposto 4.3: The quas hyperbolc geeralzed Kac Moody algebra QHGGH (9) does ot satsfy the specal magary property. The Dyk dagram assocated wth QHGGH (9) cotas proper sub dagram of defte type ad hece o specal magary root exsts for the QHGGH (9). Remark 4.4: The quas hyperbolc geeralzed Kac Moody algebra QHGGH (9) does ot satsfy the strctly magary property. For example, let α = α3+ α5+ α be a magary root ad β = α4+ α5 be a real root. Nether α+β or α-β s a root, because the support s ot coected ad therefore, there exsts o strctly magary root for QHGGH (). CONCLUSION I ths work, the complete classfcato of o-somorphc Coected Dyk dagrams assocated wth QHGGH (9) s obtaed. Some of the propertes of root system were studed. Usg represetato theory, the structure of these GKM algebras ca be further aalyzed ad the dmesos of root spaces ca be computed. ACKNOWLEDGEMENT The author scerely ackowledges the Uversty Grats Commsso, Ida for sactog the UGC Research Award ad ths work s part of the research work doe uder the UGC Research Award Scheme. REFERENCES [] Beett, C., 993, Imagary roots of Kac Moody algebra whose reflectos preserve root multplctes, J. Algebra, 58, pp [] Borcheds, R.E., 988, Geeralzed Kac-Moody algebras, J.Algebra, 5, 5-5 [3] Casperso, D., 994, Strctly magary roots of Kac Moody algebras, J.

9 A Study o Geeralzed Geeralzed Quas hyperbolc. 4 Algebra, 68, pp. 9-. [4] Sthaumoorthy, N ad Llly, P.L.,, O the root systems of geeralzed Kac-Moody algebras, J.Madras Uversty (WMY- specal ssue) Secto B:Sceces, 5, 8-3 [5] Sthaumoorthy, N ad Llly, P.L.,, Specal magary roots of geeralzed Kac-Moody algebras, Comm. Algebra, 3, [6] Sthaumoorthy, N ad Llly, P.L., 3, A ote o purely magary roots of geeralzed Kac-Moody algebras, Comm. Algebra, 3, [7] Sthaumoorthy,N ad Llly, P.L., 4, O some classes of root systems of geeralzed Kac-Moody algebras, Cotemp. Math., AMS, 343, [8] Sthaumoorthy,N ad Llly, P.L., 7, Complete classfcatos of Geeralzed Kac-Moody algebras possessg specal magary roots ad strctly magary property, to appear Commucatos algebra (USA), 35(8), pp [9] Sthaumoorthy, N, Llly, P.L ad Uma Maheswar, A., 4, Root multplctes of some classes of exteded-hyperbolc Kac-Moody ad exteded - hyperbolc geeralzed Kac-Moody algebras, Cotemporary Mathematcs, AMS, 343, pp [] Sthaumoorthy, N ad Uma Maheswar, A., 996b, Purely magary roots of Kac-Moody algebras, Comm. Algebra, 4 (), pp [] Uma Maheswar, A., 6, Quas Affe Geeralzed Kac Moody Algebras QAGGD3(): Dyk dagrams ad root multplctes for a class of QAGGD3 (), Ida Joural of Scece ad Techology, 9 (), pp. -6 [] Uma Maheswar, A., 6, Root Multplctes for a class of Quas Affe Geeralzed Kac Moody Algebras QAGGA () of rak 4, Iteratoal Joural of Mathematcal Archves, 7 (), pp -9. [3] Uma Maheswar, A.,, A Study o Dyk dagrams of Quas hyperbolc Kac Moody algebras, Proceedgs of the Iteratoal Coferece o Mathematcs ad Computer Scece, ISBN , pp [4] Xfag Sog ad Ygl Guo., 4, Root Multplcty of a Specal Geeralzed Kac- Moody Algebra EB, Mathematcal Computato, 3, Issue 3, PP [5] Xfag Sog ad Xaox Wag Ygl Guo., 4, Root Structure of a Specal geeralzed Kac-Moody algebra, Mathematcal Computato, 3, Issue 3, PP

10 4 A. Uma Maheswar [6] Wa Zhe-Xa., 99, Itroducto to Kac-Moody Algebra, Sgapore: World Scetfc Publshg Co. Pvt. Ltd.