Structure of Indefinite Quasi-Hyperbolic Kac-Moody

Home , G2 (mathematics), Robert Moody, Victor Kac

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 extended 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 indefinite 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 KacâŁ“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 associated 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

(a,p) 4 (b,q)

(c,r)

∨ ={ } ∨ ={ ∨ ∨ ∨} 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 . Let ⎛ ⎞ 2 −1 −2 ⎝−12−1⎠ −2 −12 1826 A. Uma Maheswari and M. Priyanka

(3) e ∗ be a GCM of hyperbolic type H2 . From the graded Lie algebra, we have L(G ,V,V ,ψ), L =∼ g(C), where L is a symmetrizable quasi-hyperbolic Kac-Moody algebra associated with the GCM C. Now, let us calculate the homology modules for the corresponding quasi-hyperbolic (3) = Kac-Moody algebra QH2 . Here it is enough to consider, only the negative part L− G−/I− by the symmetry of the involutive automorphism. Computation of Homology Modules

={ }⊂ ={ } (3) Let S 1, 2, 3 N 1, 2, 3, 4 . Let gs be the Kac-Moody Lie algebra H2 , + ={ + + + + = } (S) k1α1 k2α2 k3α3 k4α4 /k4 0 . s be the root system of gs.

Reﬂection of length 1 in W(S): r4. r (ρ) = ρ − α ; r (ρ) − ρ =−α 4 ∼ 4 4 4 ∴ H1(L−) = V(−α4).

Reﬂections of length 2 in W(S): r4r1, r4r2, r4r3. r4r1(ρ) − ρ =−(p + 1)α4 − α1; r4r2(ρ) − ρ =−(q + 1)α4 − α2; r4r3(ρ) − ρ =−(r + 1)α4 − α3.

Using Kostant’s formula, ∼ H2(L−) = {V(−(p + 1)α4 − α1) ⊕ V(−(q + 1)α4 − α2) ⊕(−(r + 1)α4 − α3)}.

Reﬂections of length 3 in W(S): r4r1r2,r4r1r3, r4r1r4, r4r2r1, r4r2r3, r4r2r4, r4r3r1, r4r3r2, r4r3r4. r4r1r2(ρ) − ρ =−(2p + q + 1)α4 − 2α1 − α2; r4r1r3(ρ) − ρ =−(3p + r + 1)α4 − 3α1 − α3; r4r1r4(ρ) − ρ =−(p + ap)α4 − (a + 1)α1; r4r2r1(ρ) − ρ =−(2q + p + 1)α4 − 2α2 − α1; r4r2r3(ρ) − ρ =−(3q + r + 1)α4 − 3α2 − α3; r4r2r4(ρ) − ρ =−(q + qb)α4 − (b + 1)α2; r4r3r1(ρ) − ρ =−(3r + p + 1)α4 − 3α3 − α1; r4r3r2(ρ) − ρ =−(2r + q + 1)α4 − 2α3 − α2; r4r3r4(ρ) − ρ =−(r + rc)α4 − (c + 1)α3. Hence, by Kostant formula, ∼ H3(L−) = V(−(2p + q + 1)α4 − 2α1 − α2)

⊕V(−(3p + r + 1)α4 − 3α1 − α3)

⊕V(−(p + ap)α4 − (a + 1)α1)

⊕V(−(2q + p + 1)α4 − 2α2 − α1)

⊕V(−(3q + r + 1)α4 − 3α2 − α3) Structure of Indeﬁnite Quasi-Hyperbolic Kac-Moody Algebras 1827

⊕V(−(q + qb)α4 − (b + 1)α2)

⊕V(−(3r + p + 1)α4 − 3α3 − α1)

⊕V(−(2r + q + 1)α4 − 2α3 − α2)

⊕V(−(r + rc)α4 − (c + 1)α3).

Continuing in the similar manner, the other homology modules H4(L−), H5(L−), H6(L−), etc. can be determined.

(3) 4. Structure of the Maximal Ideal in QH2

(3) We ﬁnd the structure of the components of the maximal ideals, upto level 5 for QH2 , using the homological techniques and Hochschild-Serre spectral sequences theory, as developed in [2]. We know that the homological subspace I−2 generates the idealI− of (2) (j) (n) G−. Hence we may write I− = I− . Similarly, for j ≥ 2, we write I− = I− , n≥j (j) (j) (j) (j) (j+1) L− = G/I− and N− = I− /I− .

Computation of I−2: Since G− is free, the subspace I− generates I− by the Hochschild-Serre ﬁve term exact ∼ 2 sequence and I−2 = H2(L−). Using Kostant formula, we get ∼ H2(L−) = V(wρ− ρ) w∈W(S) l(w)=2 ∼ = {V(−(p + 1)α4 − α1) ⊕ V(−(q + 1)α4 − α2) ⊕ (−(r + 1)α4 − α3)} ∼ ∴ I−2 = {V(−(p + 1)α4 − α1) ⊕ V(−(q + 1)α4 − α2) ⊕ V(−(r + 1)α4 − α3)}

Computation of I−3: ∼ (j) By Theorem 2.6, I−(j+1) = (V ⊗ I−j )/H3(L− )−(j+1) j ≥ 2. (2) − When j = 2, L− coincides with the subspace η (S) for S ={1, 2, 3, 4}. Using ∼ (2) Kostant’s formula we get I−3 = (V ⊗ I−2)/H3(L− )−3, where ⎧ ⎪ − − = = ⎪V( 3α4 3α1), p 1,a 2 ⎨ − − = = (2) ∼ V( 3α4 3α2), q 1,b 2 H3(L− )−3 = ⎪V(−3α − 3α ), r = 2,c = 2 ⎩⎪ 4 3 0 , otherwise

Computation of I−4: 1828 A. Uma Maheswari and M. Priyanka

→ (2) → (3) → (2) → { r } Let us consider the sequence 0 N− L− L− 0 and Ep,q be the (3) 2 =∼ (2) ⊗ corresponding spectral sequence converging to H∗(L− ) such that Ep,q Hp(L− ) q ∧ (I−2). (3) → 2 −→d2 To compute H3(L− )−4 we start with the short exact sequence 0 E2,0 2 → E0,1 0. (3) (3) =∼ ∞ ⊕ ∞ Since H∗(L− ) is the converging spectral sequence, we get, H1(L− ) E1,0 E0,1. (3) ∼ (3) (3) (3) ∼ Also, H1(L− ) = L− /[L− ,L− ] = L−1 = V . ∞ = 2 =∼ (2) =∼ (2) [ (2) (2)] =∼ = But E1,0 E1,0 H1(L− ) L− / L− ,L− L−1 V , which implies ∞ = 3 = E0,1 E0,1 0. ∴ d2 is surjective. 2 =∼ 2 = 2 = Since E2,0 I−2 and E2,0 E0,1 I−2, d2 must be an isomorphism. Thus ∞ = 3 = E2,0 E2,0 0. → 2 −→d2 2 → Next, consider the following sequence 0 E3,0 E1,1 0. From the previous equations we get, 2 =∼ =∼ − + + − − E3,0 H3(L−) V( (2p q 1)α4 2α1 α2)

⊕V(−(3p + r + 1)α4 − 3α1 − α3)

⊕V(−(p + ap)α4 − (a + 1)α1)

⊕V(−(2q + p + 1)α4 − 2α2 − α1)

⊕V(−(3q + r + 1)α4 − 3α2 − α3)

⊕V(−(q + qb)α4 − (b + 1)α2)

⊕V(−(3r + p + 1)α4 − 3α3 − α1)

⊕V(−(2r + q + 1)α4 − 2α3 − α2)

⊕V(−(r + rc)α4 − (c + 1)α3) and 2 =∼ (2) ⊗ =∼ ⊗ E1,1 H1(L− ) I−2 V I−2. 2 2 : 2 → 2 Thus, the levels of both the terms E3,0 and E1,1 are compared and we get d2 E3,0 E1,1 3 = 2 ∞ = 3 = 2 =∼ ⊗ is trivial. So E3,0 E3,0 and E1,1 E1,1 E1,1 V I−2. Since, I−3 generates (3) (3) ∼ I− ,wehaveH2(L− ) = I−3 = V ⊗ I−2. (3) =∼ ∞ ⊕ ∞ ⊕ ∞ ∞ = 4 = But H2(L− ) E2,0 E1,1 E0,2. It follows that E0,2 E0,2 0. ∴ : 3 → 3 The homomorphism d3 E3,0 E0,2 is surjective. 3 = : 3 → 3 : 2 → 2 Since E0,2 0. d3 E3,0 E0,2 is trivial and d2 E2,1 E0,2 is surjective in the → 2 −→d2 2 −→d2 2 → sequence 0 E4,0 E2,1 E0,2 0. Thus ∞ = 4 = : 3 → 3 : → 3 = 3 = 2 =∼ (2) E3,0 E3,0 Ker(d3 E3,0 E0,2)/Im(d3 0 E3,0) E3,0 E3,0 H3(L− ) Structure of Indeﬁnite Quasi-Hyperbolic Kac-Moody Algebras 1829

: 2 → 2 2 =∼ 2 By comparing levels, we see that d2 E4,0 E2,1 is trivial. Since E0,2 (I−2), 3 = 2 E4,0 E4,0 and

∞ = 3 = : 2 → 2 : 2 → 2 =∼ : 2 → 2 E2,1 E2,1 Ker(d2 E2,1 E0,2)/Im(d2 E4,0 E2,1) Ker(d2 E2,1 E0,2). : 2 → 2 Since d2 E2,1 E0,2 is surjective,

2 =∼ 2 =∼ 2 =∼ ⊗ (I−2) E0,2 E2,1/Ker d2 (I−2 I−2)/Ker d2

∴ =∼ 2 ∞ =∼ 2 Ker d2 S (I−2) and hence E2,1 S (I−2). 3 = : 3 → 3 3 = 2 Next let us consider E0,2 0 and d3 E3,0 E0,2 is surjective. Since E3,0 E3,0 : 3 → 3 ∞ = is irreducible, then there exists an isomorphism d3 E3,0 E0,2. Hence, E3,0 4 = E3,0 0 and

(2) =∼ 3 =∼ 3 =∼ 2 : 2 → 2 =∼ 2 : 2 → 2 H3(L− ) E3,0 E0,2 E0,2/Im(d2 E2,1 E0,2) (I−2)/Im(d2 E2,1 E0,2).

(3) : 2 → Since all the modules are completely reducible over H2 ,wehaveIm(d2 E2,1 2 =∼ 2 (2) : 2 → 2 E0,2) (I−2)/H3(L− ). Already, we have seen that d2 E4,0 E2,1 is trivial. Thus ∞ = 3 = : 2 → 2 : 2 → 2 = : 2 → 2 E2,1 E2,1 Ker(d2 E2,1 E0,2)/Im(d2 E4,0 E2,1) Ker(d2 E2,1 E0,2). Since

=∼ 2 (2) =∼ 2 =∼ ⊗ Im d2 (I−2)/H3(L− ) E2,1/Ker d2 (I−2 I−2)/Ker d2,

∼ 2 (2) we have, Ker d2 = S (I−2) ⊕ H3(L− ) → 2 → 2 → 2 → Now consider 0 E5,0 E3,1 E1,2 0. By comparing levels, we see that : 2 → 2 3 = 2 =∼ ⊗ 2 d2 E3,1 E1,2 is trivial. Thus E1,2 E1,2 V (I−2). Again by comparing → 3 −→d3 3 → the levels of the terms in the sequence 0 E4,0 E1,2 0, we conclude that = ∞ = 4 = 2 =∼ ⊗ 2 ∞ d3 0. Therefore E1,2 E1,2 E1,2 V (I−2). Finally, E0,3 is a sub module 2 =∼ 3 of E0,3 (I−2). ∼ (3) ∼ ∴ We obtain I−4 = (V ⊗ I−3)/H3(L− )−4 = V ⊗ I−3, where ⎧ 2 ⎪S ⊕ V(−4α4 − α2 − 2α1) ⊕ V(−4α4 − 2α2 − α1), p = 1,q = 1 ⎪ ⎪ − − = = ⎪V( 4α4 2α1), p 2,a 1 ⎨ − − = = (3) ∼ V( 4α4 2α2), q 2,b 1 H (L )− = 3 − 4 ⎪ 2 ⊕ − − − = = ⎪S V( 4α4 2α3 α1), q 1,r 1 ⎪ ⎪V(−4α − 2α ), r = 2,c = 1 ⎩⎪ 4 3 0, otherwise 1830 A. Uma Maheswari and M. Priyanka

Computation of I−5: (3) (4) (3) We start with the exact sequence, 0 → N− → L− → L− → 0 and the corresponding { r } (4) 2 =∼ (3) ⊗∧q spectral sequence Ep,q converging to H∗(L− ) such that Ep,q Hp(L− ) (I−3). (4) : 2 → 2 we shall compute H3(L− )−5 from this spectral sequence. Clearly d2 E2,0 E0,1 is ∞ = → 2 −→d2 2 → an isomorphism and E2,0 0. Consider the sequence 0 E3,0 E1,1 0. From previous equations, we get

2 =∼ (2) ⊕ 2 ⊕ ⊗∧2 ⊕ E3,0 H3(L− ) S (I−2) V (I−2) M and, 2 =∼ (3) × =∼ ⊗ E1,1 H1(L− ) I−3 V I−3. 2 =∼ ⊕ By elementary linear algebra, we have E3,0 Kerd2 Imd2. : 2 → 2 Consider d2 E3,0 E1,1 and by comparing the levels of the terms above, we get

∞ = 3 = 2 =∼ =∼ ⊗ E1,1 E1,1 E1,1/Imd2 I−4 V I−3/Imd2.

Hence we conclude that ∴ 3 =∼ 2 ∞ = 3 =∼ ∞ = 4 = E3,0 E3,0 and E1,1 E1,1 I−4. This implies that E0,2 E0,2 0. Thus the → 3 −→d3 3 → 3 homomorphism d3 is surjective in the sequence 0 E3,0 E0,2 0. E0,2 is a 2 =∼ ∧2 3 3 submodule of E0,2 (I−3). By comparing the levels of E3,0 and E0,2, we see that ⊗∧2 ∞ = 4 =∼ ⊗∧2 ⊕ Kerd3 must contain V (I−2). It follows that E3,0 E3,0 V (I−2) M (3) where M is a direct sum of level > 6 irreducible highest weight representation of H2 . ∞ = ∞ = ∞ = It is easy to see that (E2,1)−5 (E1,2)−5 (E0,3)−5 0. Therefore we have ⎧ ⎪ ⊗∧2 = = = ⎪V (I−2), p 1,q 1,r 1 ⎪ ⎪V(−5α − α − 3α ) ⊕ V(−5α − 3α − α ), p = 1,r = 1 ⎪ 4 3 1 4 3 1 ⎪ − − − = = ⎪V( 5α4 2α2 α1), p 2,q 1 ⎪ ⎪V(−5α4 − α2 − 2α1), p = 1,q = 2 ⎨⎪ V(−5α − 2α − α ), q = 2,r = 1 H (L(4)) =∼ 4 3 2 3 − −5 ⎪ − − − = = ⎪V( 5α4 α3 3α2), q 1,r 1 ⎪ ⎪V(−5α − 5α ), p = 1,a = 4 ⎪ 4 1 ⎪ − − = = ; ⎪V( 5α4 5α2), p 1,b 4 ⎪ ⎪V(−5α4 − 5α3), p = 1,c = 4 ⎩⎪ 0, otherwise

=∼ ⊗ (4) Hence I−5 V I−4/H−3(L− )−5. Thus by generalization of the above results, we state the following structure theorem. Structure of Indeﬁnite Quasi-Hyperbolic Kac-Moody Algebras 1831

=⊕ (3) Theorem 4.1. With the usual notations, let L Ln be the realization of QH2 ⎛ ⎞ n∈Z 2 −1 −2 −a ⎜ ⎟ −12−1 −b + associated with the GCM ⎜ ⎟ where a, b, c, p, q, r ∈ Z . Then we ⎝−2 −12−c⎠ −p −q −r 2 have following:

∼ i) I−2 = {V(−(p + 1)α4 − α1) ⊕ V(−(q + 1)α4 − α2) ⊕ V(−(r + 1)α4 − α3)}.

=∼ ⊗ (2) ii) I−3 V I−2⎧/H3(L− )−3 where ⎪ − − = = ⎪V( 3α4 3α1,p1,a 2 ⎨ − − = = (2) ∼ V( 3α4 3α2), q 1,b 2 H3(L− )−3 = ⎪V(−3α − 3α ), r = 2,c = 2 ⎩⎪ 4 3 0 , otherwise

∼ (3) iii) I−4 = V ⊗ I−3⎧/H3(L− )−4 where ⎪S2 ⊕ V(−4α − α − 2α )p= 1,q = 1 ⎪ 4 2 1 ⎪ ⊕ − − − ⎪ V( 4α4 2α2 α1), ⎪ ⎨⎪V(−4α4 − 2α1), p = 2,a = 1 (3) ∼ H3(L− )−4 = V(−4α − 2α ), q = 2,b = 1 ⎪ 4 2 ⎪ 2 ⊕ − − − = = ⎪S V( 4α4 2α3 α1), r 1,q 1 ⎪ ⎪V(−4α4 − 2α3,r= 2,c = 1 ⎩⎪ 0, otherwise

∼ (4) iv) I− = V ⊗ I− /H (L ) where 5 4⎧ 3 − −5 ⎪ ⊗∧2 = = = ⎪V (I−2), p 1,q 1,r 1 ⎪ ⎪V(−5α4 − α3 − 3α1)p= 1,r = 1 ⎪ ⎪ ⊕V(−5α − 3α − α ), ⎪ 4 3 1 ⎪ − − − = = ⎪V( 5α4 2α2 α1), p 2,q 1 ⎪ ⎨⎪V(−5α4 − α2 − 2α1), p = 1,q = 2 H (L(4)) =∼ − − − = = 3 − −5 ⎪V( 5α4 2α3 α2), q 2,r 1 ⎪ ⎪V(−5α4 − α3 − 3α2), q = 1,r = 1 ⎪ ⎪V(− α − α ), p = ,a = ⎪ 5 4 5 1 1 4 ⎪ − − = = ⎪V( 5α4 5α2), p 1,b 4 ⎪ ⎪V(−5α − 5α ), p = 1,c = 4 ⎩⎪ 4 3 0, otherwise 1832 A. Uma Maheswari and M. Priyanka

5. Conclusion

The work done in this paper, helps us to understand the structure of this Quasi-hyperbolic Kac-Moody algebra. Using the structure, we can further compute the multiplicities of roots.

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