• Bulletin of the Korean Mathematical Society
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• v.34 no.4
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• pp.519-524
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• 1997

We extent the notion of equivalent relation $\approx$ for Kac-Moody algebras to generalized Kac-Moody algebras and prove some analogues of results for Kac-Moody algebras.

• 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.

• 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 ${\Omega}_k$.

• 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 $ht{\theta}+1$.