• Title/Summary/Keyword: Kac-Moody algebra

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Complete Reducibility of some Modules for a Generalized Kac Moody Lie Algebra

  • Kim, Wansoon
    • Journal of the Chungcheong Mathematical Society
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    • v.5 no.1
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    • pp.195-201
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    • 1992
  • Let G(A) denote a generalized Kac Moody Lie algebra associated to a symmetrizable generalized Cartan matrix A. In this paper, we study on representations of G(A). Highest weight modules and the category O are described. In the main theorem we show that some G(A) modules from the category O are completely reducible. Also a criterion for irreducibility of highest weight modules is obtained. This was proved in [3] for the case of Kac Moody Lie algebras.

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GENERALIZED MCKAY QUIVERS, ROOT SYSTEM AND KAC-MOODY ALGEBRAS

  • Hou, Bo;Yang, Shilin
    • Journal of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.239-268
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    • 2015
  • Let Q be a finite quiver and $G{\subseteq}Aut(\mathbb{k}Q)$ a finite abelian group. Assume that $\hat{Q}$ and ${\Gamma}$ are the generalized Mckay quiver and the valued graph corresponding to (Q, G) respectively. In this paper we discuss the relationship between indecomposable $\hat{Q}$-representations and the root system of Kac-Moody algebra $g({\Gamma})$. Moreover, we may lift G to $\bar{G}{\subseteq}Aut(g(\hat{Q}))$ such that $g({\Gamma})$ embeds into the fixed point algebra $g(\hat{Q})^{\bar{G}}$ and $g(\hat{Q})^{\bar{G}}$ as a $g({\Gamma})$-module is integrable.

A NOTE ON THE RANK 2 SYMMETRIC HYPERBOLIC KAC-MOODY ALGEBRAS

  • Kim, Yeon-Ok
    • The Pure and Applied Mathematics
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    • v.17 no.1
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    • pp.107-113
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    • 2010
  • In this paper, we study the root system of rank 2 symmetric hyperbolic Kac-Moody algebras. We give the sufficient conditions for existence of imaginary roots of square length -2k ($k\;{\in}\;\mathbb{Z}$>0). We also give several relations between the roots on g(A).

A NOTE ON THE ROOT SPACES OF AFFINE LIE ALGEBRAS OF TYPE $D_{\iota}^{(1)}$

  • KIM YEONOK
    • The Pure and Applied Mathematics
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    • v.12 no.1
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    • pp.65-73
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    • 2005
  • Let g = g(A) = (equation omitted) + be a symmetrizable Kac-Moody Lie algebra of type D/sub l//sup (1) with W as its Weyl group. We construct a sequence of root spaces with certain conditions. We also find the number of terms of this sequence is less then or equal to the hight of θ, the highest root.

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ON SOME BEHAVIOR OF INTEGRAL POINTS ON A HYPERBOLA

  • Kim, Yeonok
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.4
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    • pp.1243-1259
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    • 2013
  • In this paper, we study the root system of rank 2 hyperbolic Kac-Moody algebras. We give some sufficient conditions for the existence of imaginary roots of square length $-2k(k{\in}\mathbb{Z}_{>0}$. We also give several relations between the integral points on the hyperbola $\mathfrak{h}$ to show that the value of the symmetric bilinear form of any two integral points depends only on the number of integral points between them. We also give some generalizations of Binet formula and Catalan's identity.

ON THE NILPOTENCY OF CERTAIN SUBALGEBRAS OF KAC-MOODY ALGEBRAS OF TYPE AN(r)

  • Kim, Yeon-Ok;Min, Seung-Kenu
    • Communications of the Korean Mathematical Society
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    • v.18 no.3
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    • pp.439-447
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    • 2003
  • Let (equation omitted) be a symmetrizable Kac-Moody algebra with the indecomposable generalized Cartan matrix A and W be its Weyl group. Let $\theta$ be the highest root of the corresponding finite dimensional simple Lie algebra ${\gg}$ of g. For the type ${A_N}^{(r)}$, we give an element $\omega_{o}\;\in\;W$ such that ${{\omega}_o}^{-1}({\{\Delta\Delta}_{+}})\;=\;{\{\Delta\Delta}_{-}}$. And then we prove that the degree of nilpotency of the subalgebra (equation omitted) is greater than or equal to $ht{\theta}+1$.

A NOTE ON THE INTEGRAL POINTS ON SOME HYPERBOLAS

  • Ko, Hansaem;Kim, Yeonok
    • The Pure and Applied Mathematics
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    • v.20 no.3
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    • pp.137-148
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    • 2013
  • In this paper, we study the Lie-generalized Fibonacci sequence and the root system of rank 2 symmetric hyperbolic Kac-Moody algebras. We derive several interesting properties of the Lie-Fibonacci sequence and relationship between them. We also give a couple of sufficient conditions for the existence of the integral points on the hyperbola $\mathfrak{h}^a:x^2-axy+y^2=1$ and $\mathfrak{h}_k:x^2-axy+y^2=-k$ ($k{\in}\mathbb{Z}_{>0}$). To list all the integral points on that hyperbola, we find the number of elements of ${\Omega}_k$.