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Structure of Indefinite Quasi-Hyperbolic Kac-Moody

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Structure of Indefinite Quasi-Hyperbolic Kac-Moody Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1821–1833 © Research India Publications http://www.ripublication.com/gjpam.htm Structure of Indefinite Quasi-Hyperbolic (3) Kac-Moody Algebras QH2 A. Uma Maheswari Associate Professor & Head, Department of Mathematics, Quaid-E-Millath Government College for Women (Autonomous), Chennai-600 002, Tamil Nadu, India. M. Priyanka Research Scholar, Department of Mathematics, Quaid-E-Millath Government College for Women (Autonomous), Chennai-600 002, Tamil Nadu, India. Abstract The paper discusess the indefinite type of quasi-hyperbolic Kac-Moody algebras (3) QH2 which are extended from the indefinite class of hyperbolic Kac-Moody al- (3) gebras H2 . These Quasi hyperbolic algebras are realized as graded Lie algebra of Kac-Moody type. Using the homological approach and Hochschild-Serre spectral sequences theory, the structure of the components of the maximal graded ideals upto level five is determined. AMS subject classification: 17B67. Keywords: Kac-Moody algebra, indefinite, Quasi- hyperbolic, homology modules, maximal graded ideal, spectral sequences. 1822 A. Uma Maheswari and M. Priyanka 1. Introduction Victor Kac and Robert Moody independently introduced Kac-Moody Lie algebras in 1968. Since then, Kac-Moody algebras have grown into an important field with appli- cations in physics and many areas of mathematics. The structure of the Kac-Moody algebras of finite type and affine type are well understood. However, determining the structure of indefinite type Kac-Moody algebras is an open problem. In [1], root multiplicities upto level 2 were computed for the hyperbolic Kac-Moody (1) Lie algebra HA1 . Using homological techniques of graded Lie algebras and spectral sequence theory, the root multiplicities for the hyperbolic type of Kac-Moody Lie algebra (1) (n) HA1 and HA1 upto level five have been computed in [2–5]. A new class called ex- tended hyperbolic Kac-Moody algebras were introduced in [6] and measured the structure and root multiplicities for a specific class of extended hyperbolic Kac-Moody algebras (1) (2) for the families EHA1 and EHA1 in [7–10]. In [11] another class of indefinite type of Kac-Moody algebras called Quasi Hyperbolic(QH) type were defined and some properties of roots were studied. In [12] another class called Quasi affine Kac-Moody algebras of indefinite type were introduced and computed the structure of the maximal ideal upto level 4 for the indef- inite non-hyperbolic Kac-Moody algebras QH G2. In [13–18] some specific families (1) of indefinite type of Quasi-affine and Quasi-hyperbolic Kac-Moody algebras QH A2 , (1) (2) (2) (2) (2) QAC2 , QAD3 , QH A4 , QH A5 and QH A7 were considered and realized as a graded Lie algebra of Kac-Moody type and the maximal ideal upto level 4 were com- (1) puted. In [19] the root Multiplicities for the Indefinite Kac⣓Moody Algebra HDn were determined. In this paper, we consider the indefinite type of Quasi-hyperbolic Kac-Moody algebra (3) QH2 associated with the indecomposable generalized Cartan matrix is given by ⎛ ⎞ 2 −1 −2 −a ⎜ ⎟ ⎜−12−1 −b⎟ ⎝−2 −12−c⎠ −p −q −r 2 + where a, b, c, p, q, r ∈ Z . We start by realizing the Kac-Moody Lie algebra as the graded Lie algebra and then, the homology modules are determined using the Kostant formula. To compute the homology modules for higher levels, we invoke the technique of Hochschild-Serre spectral sequence. 2. Preliminaries In this section we give the basic definitions and results needed for our study. For detailed information one can refer [20, 21]. Definition 2.1. Let (H,,v) be a realization of A, where Structure of Indefinite Quasi-Hyperbolic Kac-Moody Algebras 1823 H =2n − l dimensional complex vector space (where l-rank of A), ∗ ={α1,...,αn}⊂H to be the set of simple roots and v ={ v v}⊂ α1,...,αn H to be the set of simple coroots respectively, v = = satisfying αj (αi ) aij for i, j 1,...,n. = n Definition 2.2. Let A (aij )i,j=1 be a symmetrizable GCM. The Kac-Moody algebra g(A) is the Lie algebra over C generated by the elements ei, fi, i = 1, 2,...,nand h satisfying the following relations: [h, h ]=0, h, h ∈ H ; [ ]= v ei,fj δij αi ; [h, ej ]=αj (h)ej [h, fj ]=−αj (h)fj , i, j ∈ N; 1−aij (adei) ej = 0; 1−aij (adfi) fj = 0, ∀ i = j,i,j ∈ N Root space Decomposition: g(A) =⊕gα(A), α∈Q where gα(A) ={x ∈ g(A)/[h, x]=α(h)x, for all h ∈ H}. An element α, α = 0inQ is called a root if gα = 0. Let (= (A)) denote the set of all roots of g(A) and , the set of all positive roots of g(A).Wehave− =−+ and = + ∪ −. Definition 2.3. [21] To every GCM A is associated a Dynkin diagram S(A) defined as follows: S(A) has n vertices and vertices i and j are connected by max {|aij |, |aji|} number of lines if aij .aji ≤ 4 and there is an arrow pointing towards i if |aij | > 1. If aij .aji > 4, i and j are connected by a bold faced edge, equipped with the ordered pair (|aij |, |aji|) of integers. Definition 2.4. [21] A Kac-Moody algebra g(A) is said to be of finite, affine or indefinite type if the associated GCM A is of finite, affine or indefinite type respectively. = n Definition 2.5. [11] Let A (aij )i,j=1 be an indecomposable GCM of indefinite type. We define the associated Dynkin diagram S(A) to be of Quasi-hyperbolic (QH) type if S(A) has a proper connected sub diagram of hyperbolic types with n − 1 vertices. The GCM A is of Quasi-hyperbolic type if S(A) is of Quasi-hyperbolic type. We then say the Kac-Moody algebra g(A) is of Quasi-hyperbolic type. 2.1. General construction of graded Lie algebra Let G – a Lie algebra over a field of characteristic zero. V – basic representation of highest weght module over G V – contragradiant of V ψ : V ⊗ V → G a G-module homomorphism. 1824 A. Uma Maheswari and M. Priyanka The space V ⊕ G ⊕ V has a local Lie algebra structure as in [22]. By the extension ∞ of the bracket operation, G = Gn becomes a graded Lie algebra structure with the n=−∞ local part. Defining the subspaces for n ≥ 1, we get I± ={x ∈ G±/[y1, [...[yn−1,x]] ...]=0 for all y1,...,yn−1 ∈ G∓}. Set I+ = In, I− = I−n, where I+ and I− is the maximal graded ideal of G trivially n>1 n>1 intersecting G−1+G0+G1. For n>1, let L±n = G±n/I±n. Let L = L(G,V,V ,ψ)= G−/I− ⊕ G0 ⊕ G+/I+ =···⊕L−2 ⊕ L−1 ⊕ L0 ⊕ L1 ⊕ L2 ⊕ ..., where L0 = G0, L1 = G1, L−1 = G−1. Thus ∃ graded Lie algebras G and L with the local part V ⊕ G ⊕ V . ∞ Thus L = G/I , where G = Gn is the maximal graded Lie algebra and n=−∞ ∞ I = In is the maximal graded ideal of G intersecting the local part trivially. n=−∞ ∼ Theorem 2.6. [2] There is an isomorphism of g(A)-modules H (L, J ) = H + (L), ∼ j j 2 for j ≥ 1. In particular Im+1 = (G1 ⊗ Im)/H3(L)m+1. (j) Let j ≥ m. Then I = In is an ideal of G generated by the subspace Ij .Now, n≥j − by considering the quotient algebra L(j) = G/I (j), we obtain N (j) = I (j)/I (j 1) and (m) ∼ L = L . The following relation is very useful in the actual computation: Ij+1 = (j) (G1 ⊗ Ij )/H3(L )j+1. Theorem 2.7. [Kostant’s formula] [23] − ˜ ∼ Hj (n (S), V (λ)) = ⊕ V(w(λ+ ρ) − ρ). w∈W(S) l(w)=j One can refer [2, 22, 23] for detailed explanation of graded Lie algebra construction, homology of Lie algebra and the Hochschild-Serre spectral sequence. (3) 3. Realization for QH2 (3) Let us consider QH2 , the class of Quasi-Hyperbolic Kac-Moody algebras whose as- sociated GCM is ⎛ ⎞ 2 −1 −2 −a ⎜−12−1 −b⎟ ⎜ ⎟ , ⎝−2 −12−c⎠ −p −q −r 2 Structure of Indefinite Quasi-Hyperbolic Kac-Moody Algebras 1825 + where a, b, c, p, q, r ∈ Z .(ie.), the class of all4x4GCMofQuasi-hyperbolic type (3) obtained from the subalgebra of H2 associated with the GCM A is given by ⎛ ⎞ 2 −1 −2 ⎝−12−1⎠ . −2 −12 (3) Note that the GCM associated with the Quasi-hyperbolic Kac-Moody algebras QH2 is symmetrizable. (3) The Dynkin diagram associated with QH2 is 12 (a,p) 4 (b,q) (c,r) 3 ∨ ={ } ∨ ={ ∨ ∨ ∨} Let (h, , ) be the realization of A with α1,α2,α3 and α1 ,α2 ,α3 . Let (. , .) be the symmetric non-degenerate invariant bilinear form defined in g(A). Then (α1,α1) = 2, (α1,α2) =−1, (α1,α3) =−2, (α2,α1) =−1, (α2,α2) = 2, (α2,α3) =−1, (α3,α1) =−2, (α3,α2) =−1, (α3,α3) = 2 ∗ ∨ = ∨ = ∨ = Let α4 be the element in h such that α4(α1 ) 0, α4(α2 ) 0, α4(α3 ) 1. { + + + + + + } =− Define λ =1/3 (2a b)α1 (a 2b)α2 (5a 4b 3c)α4 . Set α4 λ. Now, form (α ,α ) = 4 2 i j = the matrix C (< αi,αj >)i,j=1, where <αi,αj > = ,i,j 1, 2, 3, 4. (αi,αi) Then ⎛ ⎞ 2 −1 −2 −a ⎜−12−1 −b⎟ C = ⎜ ⎟ ⎝−2 −12−c⎠ −p −q −r 2 + where a, b, c, p, q, r ∈ Z with b/q = c/r = a/p is symmetrizable GCM of Quasi- (3) hyperbolic type, represented as QH2 .
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