• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.635-645
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• 2002

Two examples of $\varepsilon$-famed manifolds are constructed. It is proved that an $\varepsilon$-framed structure on a manifold is not unique. Automorphism groups of r-framed manifolds are studied. Lastly we prove that a connected Lie group G admits a left invariant normal $\varepsilon$-framed structure if and only if the Lie algebra of all left invariant vector fields on G is an $\varepsilon$-framed Lie algebra.

• Bulletin of the Korean Mathematical Society
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• v.40 no.4
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• pp.545-551
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• 2003

We define subalgebras ${V_q}^{mZ{\times}nZ}\;of\;V_q\;where\;V_q$ are in the paper [4]. We show that the Lie algebra ${V_q}^{mZ{\times}nZ}$ is simple and maximally abelian decomposing. We may define a Lie algebra is maximally abelian decomposing, if it has a maximally abelian decomposition of it. The F-algebra automorphism group of the Laurent extension of the quantum plane is found in the paper [4], so we find the Lie automorphism group of ${V_q}^{mZ{\times}nZ}$ in this paper.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1407-1419
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• 2021

The Iwasawa decomposition NAK of the Lie group G = SO₀(2, 1) with a left invariant metric produces Riemannian submersions G → N\G, G → A\G, G → K\G, and G → NA\G. For each of these, we calculate the curvature of the base space and the lifting of a simple closed curve to the total space G. Especially in the first case, the base space has a constant curvature 0; the holonomy displacement along a (null-homotopic) simple closed curve in the base space is determined only by the Euclidean area of the region surrounded by the curve.

• Bulletin of the Korean Mathematical Society
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• v.41 no.2
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• pp.359-369
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• 2004

The purpose of this paper is to compare the radicals of a left symmetric algebra considered in 〔1〕 when the associated Lie algebra is nilpotent. In this case, we show that all the radicals considered there are equal. We also consider some other radicals and show they are also equal.

• Communications of the Korean Mathematical Society
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• v.16 no.1
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• pp.85-94
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• 2001

Let g=g(A) be an affine Lie algebra of type A(sub)2ι(sup)(2)(ι>1) and W be its Weyl group. In this paper, we find $\omega$$\in$W such that $\Delta$+U{1/2($\alpha$ι+$\delta$), 1/2($\alpha$ι+3$\delta$)}⊂$\Delta$+($\omega$).

• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.517-561
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• 2003

We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in $\mathbb{C}$. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise upper bound of the dimension of this Lie group for some specific systems of partial differential equations.

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.233-249
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• 2015

This paper is concerned with applications of representations of the Lie group of class $D_4$ to PDE. A realization of all irreducible finite-dimensional representations of $D_4$ is found and their application to a study of solutions of some systems of partial differential equations is given.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1193-1200
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• 2013

The center of the Lie group $SU(n)$ is isomorphic to $\mathbb{Z}_n$. If $d$ divides $n$, the quotient $SU(n)/\mathbb{Z}_d$ is also a Lie group. Such groups are locally isomorphic, and their Weyl groups $W(SU(n)/\mathbb{Z}_d)$ are the symmetric group ${\sum}_n$. However, the integral representations of the Weyl groups are not equivalent. Under the mod $p$ reductions, we consider the structure of invariant rings $H^*(BT^{n-1};\mathbb{F}_p)^W$ for $W=W(SU(n)/\mathbb{Z}_d)$. Particularly, we ask if each of them is a polynomial ring. Our results show some polynomial and non-polynomial cases.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.699-709
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• 2008

We study homology of the gauge group associated with the principal $G_2$ bundle over the four-sphere using the Eilenberg-Moore spectral sequence and the Serre spectral sequence with the aid of homology and cohomology operations.