• 대한수학회논문집
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• 제18권3호
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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 $ht{\theta}+1$.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제20권3호
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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 ${\Omega}_k$.