In insight into QAC2 (1) : Dynkin diagrams and properties of roots

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1 International Research Journal of Engineering and Technology (IRJET) e-issn: Volume: 03 Issue: 0 Jan-06 p-issn: In insight into QAC () : Dynkin diagrams and properties of roots A.Uma Maheswari Associate Professor, Department of Mathematics, Quaid-E-Millath Government College for Women, Chennai, Tamilnadu, India *** Abstract - In this paper the indefinite quasi affine Kac studied by Feingold & Frenkel [], Benkart, Kang & Misra Moody algebras QAC () are considered; basic ([],); Kang studied the structure of HA (), HA () and properties of real and imaginary roots are studied for HA () n ([6]-[8]);Other sub classes of the indefinite type, three specific classes of quasi affine Kac Moody namely Extended hyperbolic were introduced by algebras belonging to the family QAC () ; the short and Sthanumoorthy & Uma Maheswari ([0],[] ) and long real roots are identified; the minimal imaginary determined the structure and computed root multiplicities roots and isotropic roots up to height 3 have been up to level 5 for EHA (), EHA () ( []-[]) ; computed for these classes of Kac Moody algebras; These algebras also satisfy the purely imaginary Quasi hyperbolic and Quasi affine types were introduced property i.e. all the imaginary roots are purely by Uma Maheswari ([5],[8]). Uma Maheswari and et.al imaginary ; All the 909 non isomorphic connected studied Quasi hyperbolic algebras QHG,QHA (), QHA (), Dynkin diagrams associated with QAC (), are given in QHA () 7, QHA () 5 ([0]-[]), quasi hyperbolic Dynkin the classification theorem. It is also proved that there diagrams of rank 3 in [6] and quasi affine Kac Moody are 68 Dynkin diagrams of extended hyperbolic type, 8 algebras QAD () 3, QAG () ( [8]-[9]). In [7], the family hyperbolic type and 53 diagrams of indefinite, non QAC () was defined, the structure of a specific class of extended hyperbolic type in the classification of Dynkin QAC () was determined using homological techniques and diagrams associated with QAC (). the general form of the Dynkin diagrams associated with this family QAC () was given; Key Words: Kac Moody algebras, Dynkin diagrams, roots, real, imaginary,, quasi hyperbolic, quasi affine. AMS Classification :7B67. INTRODUCTION The subect Kac Moody algebras was developed by Kac and Moody in 968, simultaneously and independently ([5],[9]). Among the three main types of Kac Moody algebras, namely finite, affine and indefinite, the study of indefinite type of Kac Moody algebras is still wide open, whose structure is not completely understood; The Kac Moody algebras were completely classified through Dynkin digrams for the finite, affine and, hyperbolic types ([5], [5]). The root systems of the finite and affine types have been characterized ([5],[9],[5]); As far as the imaginary roots of the indefinite type Kac Moody algebras are concerned, strictly imaginary roots were studied by Casperson [3] and special imaginary roots by Bennett []; Purely imaginary roots were introduced by Sthanumoorthy and Uma Maheswari ([0],[]) ; Among the indefinite Kac Moody algebras, hyperbolic Kac Moody algebras, which are natural extensions of affine type were Here in this paper, the complete classification of non isomorphic connected Dynkin diagrams ( 909 in number) associated with QAC () is given; Also, we consider three particular classes belonging to QAC () and study the basic properties of real and imaginary roots for the associated Quasi affine Kac Moody algebras QAC ().. PRELIMINARIES DEFINITION : A realization of a matrix A ( a ) n i i, is a triple ( H,, ) where l is the rank of A, H is a n - l dimensional complex vector space, {,,..., n } {,,..., n } are linearly independent and subsets of H* and H respectively, satisfying ( i ) ai for i, =,.,n. is called the root basis. Elements of are called simple roots. The root lattice generated by n is Q i z i 06, IRJET Impact Factor value:.5 ISO 900:008 Certified Journal Page 87

2 International Research Journal of Engineering and Technology (IRJET) e-issn: Volume: 03 Issue: 0 Jan-06 p-issn: The Kac-Moody algebra g(a) associated with a GCM A ( a ) n i i, is the Lie algebra generated by the elements ei, fi, i=,, n and H with the following defining relations : ' [ h, h ] 0, ' h, h H [ e, f i [ h, e [ h, f ( ade ) ( adf i i ] ] ) v ] ( h) e ai ai i ( h) f e f i 0 0,, i i,, N i, The Kac-Moody algebra g(a) has the root space decomposition g( A) g ( A) where g Q ( A) { x g( A) /[ h, x] ( h) x, for all hh}. An element, 0 in Q is called a root if g 0. For any n k i i N Q and k define support of, written as supp, by supp { i N / k i 0}. Let ( ( A)) denote the set of all roots of g(a) and the set of all positive roots of g(a); and. Definition :[]A GCM A is called symmetrizable if DA is symmetric for some diagonal matrix D = diag(q,,qn), with qi> 0 and qi s are rational numbers. Note that, a GCM A ( a ) n i i, is symmetrizable if and only if there exists an invariant, bilinear, symmetric, nondegenerate form on g(a). Definition 3: [][5] To every GCM A is associated a Dynkin diagram S(A) defined as follows: (A) has n vertices and vertices I and are connected by max{ a i, a i } number of lines if a i.a i and there ia an arrow pointing towards I if a i >. If a i.a i>, i and are connected by a bold faced edge, equipped with the ordered pair ( a i, a i ) of integers. Definition : [7] Let A ( a ) n i i,, be an indecomposable GCM of indefinite type. We define the associated Dynkin diagram S(A) to be of Quasi Affine (QA) type if S(A) has a proper connected sub diagram of affine type with n- vertices. The GCM A is of QA type if S(A) is of QA type. We then say the Kac-Moody algebra g(a) is of QA type. Definition 5[9]:A root is called real, if there exists a w W such that w(α) is a simple root, and a root which is not real is called an imaginary root. An imaginary root α is called isotropic if (α,α) = 0. is called a minimal imaginary root (MI root, for im short) if α is minimal in im with respect to the partial order on H*. By the symmetry of the root system, it suffices to prove the results for positive imaginary roots only. Definition 6[3]: A root re imaginary if for every, im is said to be strictly or is a root. The set of all strictly imaginary roots is denoted by Definition 7[5]: A root im imaginary if for any, im sim. is called purely.the Kac Moody algebra is said to have the purely imaginary property if every imaginary root is purely imaginary. 3. CLASSIFICATION OF DYNKIN DIAGRAMS ASSOCIATED WITH QAC () In this section we explicitly give all the 909 non isomorphic connected Dynkin diagrams associated with the indefinite Quasi affine Kac-Moody algebras QAC (). Also we identify the extended hyperbolic and indefinite non extended hyperbolic diagrams in this family. It is interesting to note that this family OF Kac Moody algebras satisfy the purely imaginary property. The general representation of the GCM associated with a b c QAC () l 0 is A, a,b,c,l,m,n are non m n 0 negative integers not all 0 simultaneously; Note that A is symmetrizable if bn=cm and am=bl. i.e. A = DB, where a b c D = 0 l/ a 0 0, B= a a / l a / l m/ b 0 b a / l b / m b / m n/ c c 0 b / m c / n It can also be represented as 0 l m im 0 n a b c Theorem (ClassificationTheorem):. There are 909 non isomorphic, connected Dynkin diagrams associated with the indefinite Quasi affine Kac-Moody algebra QAC (). 06, IRJET Impact Factor value:.5 ISO 900:008 Certified Journal Page 875

3 International Research Journal of Engineering and Technology (IRJET) e-issn: Volume: 03 Issue: 0 Jan-06 p-issn: Proof. Consider the affine Kac-Moody algebra C () with 3 vertices whose Dynkin diagram is given by vertices can be chosen from the three vertices in possible ways. ( i.e.(,) and (,) can be oined or (,) and (3,) can be oined.) 3 Add the fourth vertex, which is connected to atleast one of the three vertices of C (). and Here denotes the 9 possible connections of vertex with the other 3 vertices. Note that oining (,) and (3,) will result in a isomorphic set of diagrams with (,), (,) connections. i.e. the two Dynkin diagrams,,,,,,,,, and The fourth vertex can be connected to the existing three vertices by one the following cases: Case i) Fourth vertex is connected to exactly one of the three vertices,,3. In this case, fourth vertex can be connected to the i th vertex, (i=,,3) by the 9 possible edges given above. are isomorphic. For each possibility, the remaining edge can be connected by any of the above mentioned 9 edges. Hence. in this case, the associated connected Dynkin diagrams are x 9 = 6 Case iii) Fourth vertex is connected to all the 3 vertices. In this case, there are 9 3 connected Dynkin diagrams. or Thus, possible number connected Dynkin diagrams associated with QAC () in this case is x 9 = 8 (Note that, we are considering non isomorphic diagrams; Joining the th vertex with either st or 3 rd vertices will result in isomorphic diagrams) Totally there are =909 non isomorphic, connected Dynkin diagrams associated with QAC (). All these 909 Dynkin diagrams are drawn in the following table. Case ii) Fourth vertex is connected to exactly two of the 3 vertices; To get non isomorphic diagrams, these two 06, IRJET Impact Factor value:.5 ISO 900:008 Certified Journal Page 876

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13 International Research Journal of Engineering and Technology (IRJET) e-issn: Volume: 03 Issue: 0 Jan-06 p-issn: QAC () and identify the short and long real roots; Among the imaginary roots, we compute minimal imaginary and isotropic roots up to height 3 for these families; 0 0 Example : Consider the GCM A= 0, 0 a 0 0 b whose associated Dynkin diagram is given by (a,b) Proposition : The quasi affine Kac Moody algebra QAC () satisfies the purely imaginary root. Proof : From the characterization of Kac Moody algebras with the purely imaginary property (given in [0]), any indecomposable symmetrizable GCM up to order satisfies the purely imaginary property. Hence all imaginary roots are purely imaginary in QAC (). Proposition : Among the 909 Dynkin diagrams, 68 families are extended hyperbolic, 8 are hyperbolic type, 53 Dynkin diagrams in QAC () are indefinite, which are neither hyperbolic nor extended hyperbolic. Proof : Any Dynkin diagram that contains a bold faced edge can not be of extended hyperbolic type; 53 diagrams contain bold faced edges and hence they are neither hyperbolic nor of extended hyperbolic type; Still they are indefinite type of Dynkin diagrams. All other diagrams which do not contain a bold faced edge are either of extended hyperbolic or hyperbolic type, From the table it is clear that there are 8 diagrams of hyperbolic type, namely diagrams (89) (896), (90)-(903), since each proper connected sub diagram is of finite or affine type. Leaving out these 8+53=6 diagrams, the remaining 68 Dynkin diagrams are all of extended hyperbolic type, since each proper connected sub diagrams in these cases are of finite, affine or hyperbolic types. PROPERTIES OF ROOTS OF SPECIFIC CLASSES OF QAC () In this section, we consider three particular families of where ab>5; Note that the GCM is symmetrizable, A=DB = ; a b/ a 0 0 a a / b The non generate symmetric bilinear form (, ) defined on this algebra is given as follows: (α,α )=, (α,α )=-, (α,α )=, (α,α 3)=-, (α 3,α 3)=, (α 3,α )=-a, (α,α )=a/b All fundamental roots are real. Roots of height : (α +α, α +α ) is a real root. (α +α 3, α +α 3) is a real root. (α 3+α, α 3+α ) = +a/b+(-a) α 3+α is imaginary if +a/b-a 0 α 3+α is isotrophic if +a/b-a = 0 i.e. if a+b-ab = 0 a+b = ab if and only if a=b= But the GCM is not quasi affine indefinite type if a=b= Therefore, this possibility is excluded; To compute short and long real roots: If a=b, then α is the short root, and α, α 3, α are long. If a>b, then α is the short root and α is the long root. If a=b, then α, α are the short roots. If a/b>, then α, α 3 are the long roots. Roots of height 3: (α + α + α 3, α + α + α 3)= >0; α + α + α 3 can not be an imaginary root; (α + α 3+ α, α + α 3+ α )= +a/b-a α + α 3+ α is imaginary if +a/b-a 0 α + α 3+ α is not isotropic (For, If b=, a+=a, a= a=, b=, the GCM is not quasi affine indefinite type). Now,(α 3+α, α 3+α ) = +a/b-a α 3+α is Imaginary if +a/b-a < 0. 06, IRJET Impact Factor value:.5 ISO 900:008 Certified Journal Page 886

14 International Research Journal of Engineering and Technology (IRJET) e-issn: Volume: 03 Issue: 0 Jan-06 p-issn: Example : Consider another family in QAC () whose 0 0 GCM is c ; The symmertizable 0 a 0 d b decomposition is taken as A= c / a d / c 0 c / a c / d with cd,ab 0 and the condition for symmetrizability is ad=bc ; The non generate symmetric bilinear form values are: (α,α )=, (α,α )=-, (α,α )=, (α,α 3)=-, (α 3,α 3)=, (α 3,α )=-a, (α,α )=c/d, (α,α )=-c/ Roots of height : (α +α, α +α ) =>0; α +α is a real root. (α +α 3, α +α 3) =>0 α +α 3is a real root. (α 3+α, α 3+α ) = +(c/d)-a (α 3+α, α 3+α ) is imaginary iff d+c-a 0, i.e. iff c + d a, α 3+α is isotrophic if d+c-a 0, this implies c + d a (α +α, α +α )=+c/d-c; α +α is isotropic iff d+c-cd=0 ; Thus (α +α, α +α ) is isotropic iff d+c-cd=0 iff c=d= Now (α +α, α +α ) is imaginary if d+c-cd 0; Roots of height 3: (α + α + α 3, α + α + α 3)= >0 (α + α 3+ α, α + α 3+ α )= +c/d+-a-c The root α + α 3+ α is isotropic iff +c((/d)-)-a=0 and this root is imaginary iff +c/d-a-c 0 Consider (α +α, α +α ) = +c/d-c α +α is imaginary iff c-(c/d) α +α is isotropic iff = c-(c/d); Consider (α +α, α +α ) =+(c/d)-c α +α, α +α is isotropic iff =c-c/d α +α, α +α is imaginary iff c-(c/d) Next, consider (α 3+α, α 3+α ) = +c/d-a α +α is imaginary iff a-(c/d) α +α is isotropic iff = a-(c/d) Now consider (α 3+α, α +α ) =+(c/d)-a; α +α is isotropic iff =a-c/d α +α is imaginary iff < a-(c/d) Length of real roots respectively; : α i =,,, c/d for i=,,3, If c=d, then α and α are the short root If c/d< then α, α 3 are the long root If c/d= then α, α 3, α are the long roots If c/d > then α is the long root. Example 3: Consider the GCM a b c A= a 0,a,b,c are non negative integers b c 0 not all 0; It is easy to check that A is a indecomposable symmetrizable GCM and the bilinear form is given by: (α,α )=, (α,α )=-a, (α,α 3)=-b (α,α )=, (α,α 3)=-, (α 3,α 3)=, (α 3,α )=-, (α,α )=-c,(α,α )= Let k k k3 3 k ; (, ) k { k ak bk ck } 3 k { ak k k } k { bk k k k } k { ck k k }. 3 Case : k 3 = k = 0. Then (, ) k k akk If a > & k = k, k ( ) is an imaginary root. And, (, ) a a 0, if a If a =, ( ) (, ) 0 Also, if k + k ak k =0 is imaginary and ) is an isotropic root. ( ak a k k ak k a k k { a a } Any k & k satisfying the above condition will give an isotropic root k k. Also, the minimal imaginary root is if a >. k If a, if k k, k k is imaginary. k Case : k = k 3 = 0. Then (, ) ( k k ) ckk If a> & k = k, k ( ) is imaginary. k If a, if k k, k( ) is imaginary. k k For a, if k k, k is imaginary. 06, IRJET Impact Factor value:.5 ISO 900:008 Certified Journal Page 887

15 International Research Journal of Engineering and Technology (IRJET) e-issn: Volume: 03 Issue: 0 Jan-06 p-issn: (, ) ( c) c is minimal imaginary root if c. is isotropic if c =. If k + k ak k = 0 k ck c k k 0, if c k { c c } k k will be If k & k satisfy this relations then isotropic. Case 3: k = k = 0; If b > 3/ & k = k 3, Then k k3 3 is an imaginary root. 3k3 If b & k k3, is imaginary. k 3k If b & k3 k, is imaginary. k 3 ( 3, 3) 3 b; If b 3/, 3 is minimal imaginary root. 5. CONCLUSION In this work, classification of Dynkin diagrams and some properties of roots are obtained for the family QAC (). Further, using the representation theory, the dimensions of the root spaces and weight spaces can be computed. ACKNOWLEDGEMENT The author sincerely acknowledges University Grants Commission, India for granting Research Award and this work is part of the research work done under the UGC Research Award (RA-0--NEW-GE-TAM-). REFERENCES [] Benkart, G.M., Kang, S.J., and Misra, K.C. (993), Graded Lie algebras of Kac-Moody type, Adv. Math., Vol. 97, pp [] Bennett, C. (993), Imaginary roots of Kac Moody algebra whose reflections preserve root multiplicities, J. Algebra, Vol. 58, pp [3] Casperson, D. (99), Strictly imaginary roots of Kac Moody algebras, J. Algebra, Vol. 68, pp [] Feingold, A.J., and Frenkel, I.B. (983), A hyperbolic Kac-Moody algebra and the theory of siegal modular forms of genus, Math. Ann., Vol. 63, pp. 87- [5] Kac, V.G. (990), Infinite Dimensional Lie Algebra, 3rd ed. Cambridge : Cambridge University Press. [6] Kang, S.J. (993), Root multiplicities of the HA hyperbolic Kac-Moody Lie algebra, J. Algebra, Vol. 60, pp [7] Kang, S.J. (99a), Root multiplicities of the HA hyperbolic Kac-Moody Lie algebra, J. Algebra, Vol. 70, pp [8] Kang, S.J. (99), On the hyperbolic Kac-Moody Lie HA () algebra, Trans. Amer. Math. Soc., Vol. 3, pp , 99 [9] R.V. Moody. A new class of Lie algebras. J. Algebra, 0:-30, 968. [0] Sthanumoorthy, N., and Uma Maheswari, A. (996b), Purely imaginary roots of Kac-Moody algebras, Comm. Algebra, Vol., No., pp [] Sthanumoorthy, N., and Uma Maheswari, A. (996a), Root multiplicities of extended hyperbolic Kac- Moody algebras, Comm. Algebra, Vol., No., pp [] Sthanumoorthy, N., Lilly, P.L., and Uma Maheswari, A. (00), Root multiplicities of some classes of extended-hyperbolic Kac-Moody and extended - hyperbolic generalized Kac-Moody algebras, Contemporary Mathematics, AMS, Vol. 33, pp [3] Sthanumoorthy, N., Uma Maheswari, A., and Lilly, P.L., () Extended Hyperbolic Kac-Moody EHA Algebras structure and Root Multiplicities, Comm. Algebra, Vol. 3, No. 6, pp [] Sthanumoorthy, N., and Uma Maheswari, A. (0), Structure And Root Multiplicities for Two classes of Extended Hyberbolic Kac-Moody Algebras EHA () and EHA () for all cases, Communications in Algebra, Vol. 0, pp [5] Uma Maheswari. A. (0), A Study on Dynkin diagrams of Quasi hyperbolic Kac Moody algebras, Proceedings of the International Conference on Mathematics and Computer Science, ISBN , pp [6] Uma Maheswari. A. (0), Imaginary Roots and Dynkin Diagrams of Quasi Hyberbolic Kac-Moody Algebras, International Journal of Mathematics and Computer Applications Research,Vol., Issue, pp [7] Uma Maheswari. A,(0), A Study on the Structure of Indefinite Quasi-Affine Kac-Moody Algebras QAC(), International Mathematical Forum,Vol. 9, no. 3, [8] Uma Maheswari. A,(0), Quasi-Affine Kac-Moody Algebras: Classification of Dynkin Diagrams & Realization for a Class of QAG(),International Journal of Mathematical Sciences,Volume 3, ,Issue. [9] Uma Maheswari. A,(05), Structure Of Indefinite Quasi-Affine Kac-Moody Algebras QAD3(), () () n 06, IRJET Impact Factor value:.5 ISO 900:008 Certified Journal Page 888

16 International Research Journal of Engineering and Technology (IRJET) e-issn: Volume: 03 Issue: 0 Jan-06 p-issn: International Journal of Pure and Applied Mathematics,Volume 0 No., [0] Uma Maheswari, A., and Krishnaveni, S. (0), A Study on the Structure of a class of Indefinite Non- Hyperbolic Kac-Moody Algebras QHG, Trans Stellar, Vol., Issue, [] Uma Maheswari, A., and Krishnaveni, S. (0), On the Structure of Indefinite Quasi-Hyperbolic Kac- Moody Algebras QHA (), International Journal of Algebra. [] A.Uma Maheswari and S. Krishnaveni, Structure of the Indefinite Quasi-Hyperbolic Kac-Moody Algebra QHA (), International Mathematical Forum, HIKARI Ltd, 9(3):95 507, 0. [3] A.Uma Maheswari and S. Krishnaveni. A Study on the Structure of a class of Indefinite Non-Hyperbolic Kac- Moody Algebras QHA () 7, International ournal of Mathematical Sciences, 3():639-68,05. [] A.Uma Maheswari and S. Krishnaveni. A Study on the Quasi-Hyperbolic Algebra QHA () 5, International Journal of Pure and Applied Mathematics, 0(0), 3-38, 05 [5] Wan Zhe-Xian (99), Introduction to Kac-Moody Algebra, Singapore: World Scientific Publishing Co. Pvt. Ltd. 06, IRJET Impact Factor value:.5 ISO 900:008 Certified Journal Page 889