Root Structure of a Special Generalized Kac- Moody Algebra

Transcription

1 Mathematcal Computaton September 04, Volume, Issue, PP8-88 Root Structu of a Specal Generalzed Kac- Moody Algebra Xnfang Song, #, Xaox Wang Bass Department, Bejng Informaton Technology College, Bejng, 00070, Chna College of Computer Scence and Technology, Bejng Unversty of Technology, Bejng, 00, Chna #Emal: xf-sohu@sohucom Abstract The paper systematcally dscusses the root structu of a specal generalzed Kac-Moody algebra EB whch was extended from the fnte Kac-Moody algebra B It s extended one pont from the fnte Coxter dagram BAt the begnnng the paper defne a specal generalzed generalzed Cartan matrx (abbvated as GGCM) and a specal generalzed Kac-Moody algebra(abbvated as GK algebra) As a startng pont for the specal GK algebra t manly obtan the specal agnary root system, the latonshp between flectons determned by agnary roots and the Weyl group for the specal GMC algebra and the puly and the strctly agnary roots Keywords: Generalzed Kac-Moody Algebra, Imagnary Root, Specal Imagnary Root, Weyl Group, Strctly Imagnary Root, Puly Imagnary Root INTRODUCTION Ths paper contnues to use the concept of the specal root and the exstence of the agnary roots from [] It follows the agnary roots and the specal root for a class of generalzed Kac-Moody algebra wth rank from [] And also t follows the latonshp between flectons determned by agnary roots and the Weyl group for a specal GKM algebra In [] the author manly dscussed the generalzed Kac-Moody algebra from the extenson of fnte type A In the next secton, after a statement of the basc problem, varous stuatons nvolvng possblty knowledge a nvestgated: frst, an ently possblty model s proposed; then the cases of a fuzzy servce te wth stochastc arrvals and non fuzzy servce rule s studed; lastly, fuzzy servce rule a consdedths psent paper enhances the dege of dffculty beng obtaned from the extenson of fnte type GCM B Note: The paper also uses the symbol from [] BASIC CONCEPTS AND SYMBOL In ths part the paper gves some basc defntons (from [] ) gardng Kac-Moody algebra and GKM algebra Defnton Let A ( aj ) nn be a al n n matrx satsfyng the followng condtons: (c) ether a or a 0 ; (c) aj 0 f j and aj Z f a (c) a 0 a 0 j j A s called a generalzed generalzed Cartan matrx (abbvated as GGCM) And the Le algebra g(a) assocated wth A s called the generalzed Kac-Moody algebra (abbvated as GKM algebra) Note He assume that the ents n the GGCM a ntegers wwwvypuborg/mc

2 Defnton For a GGCM A, ( ) be a alzaton of A, whe { n } and n { } a ndependent sets Note n Q Z and Q n Z denotes the root lattce and the postve root lattce of g(a) spectably Let (sp ) be the root system (sp postve root set) of g(a) Denote { a } to be the set of al sple root and { a 0} the set of sple agnary root Denote W by the Weyl group of g(a) Defnton We defne the al (spagnary) root set of g(a)to be W( )( sp ) We defne C { 0 } to be the dual fundamental Weyl chamber and N Z {0} to be the set of natural numbers And denote R to be the set of al postve numbers Let K 0 { Q \{0} C and supp s connected} and K K0 j j In partcular the set of postve agnary roots s W-stable In [6] we have the followng propostons: Proposton : ( K) W A GGCM A s called symmetralzed f the exsts an nvertble dagonal matrx D dag( n ) wth R for all and a symmetrc matrx B such that A=DB The exsts a non-degenerate, symmetrc, nvarant blnear form (, )on We have an somorphsm defned by ( h) h ( h h ) h h and the nduced blnear form (, ) on It s clear that ( ), ( ) b j n We note that a root s agnary f and only f ( ) 0 Proposton : j j ( ) a ( ) a ( ) ( ) THE MAIN RESULTS The Imagnary Root System Of A Class Of Rank= GKM Algebra Lemma : Let a a A b b whe a, b N Then K k k k ak k k N { } { } { k k k ak k k N} { k k k k ak k k k ak k Nk k a not zero at the same te} Proof: Snce { } { } and C { 0 } Suppose k k k Q {0} 0 thus k ak k, Then ak k k ak k k 0 thus k k ak wwwvypuborg/mc

3 And K0 { Q {0} C and supp s connected} { kq {0} k ak kk ak k} It s clear that k 0 for every k K If k k 0, then k K0 k N So K0 ( j ) { k k k k ak k k ak k k N k j te} 0 and k a not zero at the same Snce K K0 j, we get the proof of lemma j For the splcty, we let p( kk k) denote the set of all k k and k satsfyng the condtons: k ak kk ak kk N k and k a not zero at the same te Then we can get K n lemma as K { } { k k k p( k k k )} follows: Lemma : Let a a A b b whe a, b N Then the Weyl group of g(a) s W { ( ) } Proof: We know thatw { ( ) } The sult can be easly obtaned n operaton by the GGCM A a a Theom : Let A b whe a, b N Then the postve agnary root set of g (A) s b ( K) W( ) W( k k k ), whe K from lemma W Proof: From Lemma we know the Weyl group of g(a) s W { ( ) } We get: ( K) { b } { k ( bk k k) k p( kk k)} ( K) { b } k k ( bk k k) p( kk k)} ( K) { b b } {{ k (bk k k) ( bk k k) p( kk k)} ( K) { b b } { k ( bk k k ) ( bk k k ) p( k k k )} ( K) { 4b b } { k (bk k bk k) ( bk k k bk) p( kk k)} ( K) { b b } { k ( bk k) ( ak k k) p( kk k)} ( ) ( K) { 4b b } { k (4bk k k ) ( bk k ) p( k k k )} K ( K) ( K) ( K) ( K) ( K) ( K) ( ) ( K) As { b b b b b b 4b b b b b b { k k k k ( bk k k ) k k k ( bk k k ) k (bk k k) ( bk k k) k ( bk k k) ( bk k k) k (bk k bk k) ( bk k k bk ) k ( bk k) ( ak k k) k (4bk k k ) ( bk k ) p( k k k )} 4 } wwwvypuborg/mc

4 So we obtan the proof of ths theom Puly Imagnary Roots and Strctly Imagnary Roots Suppose a a A b,whe ab N b The s a dagonal matrx ( b ) 0 0 D 0 ( a ) 0, wth R and a symmetrc matrx 0 0 ( a ) b ab ab B ab a a So A=DB exts Then GGCM A s symmetrzable ab a 4a Defnton : An agnary root n a GKM algebra s sad to be strctly agnary root f for every or s a root The set of all strctly agnary roots s denoted by s, ether Defnton : Let puly agnary roots s denoted by, defne to be puly agnary root f for any p The set of all Proposton : Let A ( a ) n j j be an ndecomposable symmetrzable GGCM such that a 0 j N and p jthen Proof: In [4] j Proposton 4: Consder the GKM algebra assocated wth the GGCM If both b and b Proof: In [4] p p s Then a a A b p b q wth pq and q= Theom : Let a a A b whe a b N Then b p s Proof: It s easly known that A s an ndecomposable symmetrzable GGCM And every a 0 j N and j p From proposton, we get ( A) ( A) And also A s satsfed wth proposton 4 Then theom s completed the proof Reflectons Determned By the Imagnary Roots and the Weyl Group for a Specal GKM Algebra Let g(a) be a symmetrzable GKM algebra, be an agnary root of g(a) If ( ) 0 then we defne a flecton on by ( ) ( ) ( ) for If we set ( ) ( ) then we know f a f a By proposton and we have ( ) for defnton Let be an agnary root of g(a) whch s symmetrzable GKM algebra Wecall a specal agnary root, f satsfes the followng condtons : (s) ( ) wwwvypuborg/mc j

5 (s) ( ) ( ) ( ) (s) pserves root multplctes It s clear that f s a specal agnary root, then W, whe W { W} Theom : Let a a A b whe ab N, and k k k K b k k Then W f and only f k k bk k k N Proof: ( ) Suppose k k bk k 4 N 4 N k Then ( 4 ), We have ( ) k (b 5 ab ) 0 Use the defnton of flecton of agnary root, we have Thus Thefo Thus ( ) ( ) ( ) a a k ( 5 ab) ( ) 4b b ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Snce deta 4 0ab 0, we have d = and { } s a bass of ( ) W as a flecton on ( ) Conversely let k k k K We know that k k k N wth k ak kk ak k Then ( ) A 0 whe A bk ak ak abk k abk k ak k Usng the defnton of flecton of agnary root, we have ( ) (bk abk abk ), A ( ) ( abk ak ak) () A ( ) ( abk ak ak ) () A ( ) If ths s not true, for example We assert that f W, then then ( ) ( ) from the equaton (), we have ( abk ak ak ) k ( abk k ak ak k A) ( abk k ak k ak A) 0 As we know that { } s ndependent, so we get ( abk ak ak) k 0 abkk ak akk A 0 abk k akk ak A 0 Snce k 0 abk ak ak 0 and hence A=0, whch s a contradcton Slarly, we can prove that for each wwwvypuborg/mc

6 element n -W except ( ), we get a contradcton Suppose ( ) Then ( ) ( ) ( ) () ( ) ( ) ( ) (4) From the equaton (), (), () and (4), we obtan ( abk ak ak ) ( abk ak ak) Thefo k k bk k complete the proof of theom 4 N From the theom above, we obtan the followng: a a Corollary : Let A b whe b specal agnary root of g (A) k ab N, and ( 4 b b ) k=,, Then s a REFERENCES [] BennettC, Imagnary roots of a Kac-Moody Le algebra whose flectons pserve root multplctes, Journal of Algebra 58:44-67, 99 [] L Zhenheng, Jn Ydong,Ba Rupu, Relatonshp between flectons determned by agnary roots and the Weyl [] Group for a specal GKM Algebra, Journal of Mathematcal Research and Exposton, No Page [4] N Sthanumoorthy and P L Llly, on some classes of root systems of generalzed Kac-Moody algebrasmathsoc, 004 [5] N Sthanumoorthy and P L Llly, A note on puly agnary roots of generalzed Kac-Moody algebras CommAlgebra, 00, : [6] JE Humphys, Introducton to le algebras and psentaton theory, Graduate Text of Mathematcs, 97 Sprnger-Verlag, 9 [7] VG Kac, Infnte-densonal Le algebras, rd ed,990, Combrdge Unv Ps [] [8] Zhexan Wan, Introducton to Kac-Moody algebra [M] 99Bejng:~Scence Publshers (n chnese) AUTHORS Xnfang Song, ceved her MS deges n basc mathematcs from the Harbn Normal Unversty, Chna n 006 She s a Lectur of Bejng Informaton Technology college, Chna Her search ntests a basc and appled mathematcs Xaox Wang s a master student of Bejng Unversty of Technology, Bejng, Chna Her search ntests a nformaton securty, appled mathematcs, etc Emal: Songxf@btceducn wwwvypuborg/mc