• 대한수학회지
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• 제33권1호
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• pp.183-203
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• 1996

When dealing with a Lie group or, in general, with an analytic loop or quasigroup, its symmetry is broken by the election of the distinguished identity element.

• 대한수학회지
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• 제60권2호
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• pp.303-325
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• 2023

In this paper, we introduce relative Rota-Baxter systems on Leibniz algebras and give some characterizations and new constructions. Then we construct a graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter systems. This allows us to define a cohomology theory associated with a relative Rota-Baxter system. Finally, we study formal deformations and extendibility of finite order deformations of a relative Rota-Baxter system in terms of the cohomology theory.

• 호남수학학술지
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• 제21권1호
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• pp.75-95
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• 1999

Dimensions of irreducible $so_5(F)$-modules over an algebraically closed field F of characteristics p > 2 shall be obtained. It turns out that they should be coincident with $p^{m}$, where 2m is the dimension of coadjoint orbits of ${\chi}{\in}so_5(F)^*{\backslash}0$ as Premet asserted. But there is no subregular point for $g=sp_4(F)=so_5(F)$ over F.

• 대한수학회보
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• 제56권1호
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• pp.13-22
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• 2019

We compute the second cohomology of the affine Lie algebra aff(1) on the dimensional real space with coefficients in the space ${\mathcal{D}}^n_{{\underline{\lambda}},{\mu}}$ of n-ary linear differential operators acting on weighted densities where ${\underline{\lambda}}=({\lambda}_1,{\ldots},{\lambda}_n)$. We explicitly give 2-cocycles spanning these cohomology.

• 대한수학회논문집
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• 제20권3호
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• pp.443-456
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• 2005

Let G be a Lie group, let L(G) be its Lie algebra, and let exp : $L(G){\rightarrow}G$ denote the exponential mapping. For $S{\subseteq}G$, we define the tangent set of S by $L(S)\;=\;\{X\;{\in}\;L(G)\;:\;exp(tX)\;\in\;S\;for\;all\;t\;{\geq}\;0\}$. We say that a semigroup S is strictly infinitesimally generated if S is the same as the semigroup generated by exp(L(S)). We find a tangent set of the semigroup of all non-singular totally positive matrices and show that the semigroup is strictly infinitesimally generated by the tangent set of the semigroup. This generalizes the familiar relationships between connected Lie subgroups of G and their Lie algebras

• 대한수학회논문집
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• 제21권1호
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• pp.45-52
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• 2006

Every non-associative algebra L corresponds to its symmetric semi-Lie algebra $L_{[,]}$ with respect to its commutator. It is an interesting problem whether the equality $Aut{non}(L)=Aut_{semi-Lie}(L)$ holds or not [2], [13]. We find the non-associative algebra automorphism groups $Aut_{non}\; \frac\;{(WN_{0,0,1}_{[0,1,r_1...,r_p])}$ and $Aut_{non-Lie}\; \frac\;{(WN_{0,0,1}_{[0,1,r_1...,r_p])}$ where every automorphism of the automorphism groups is the composition of elementary maps [3], [4], [7], [8], [9], [10], [11]. The results of the paper show that the F-algebra automorphism groups of a polynomial ring and its Laurent extension make easy to find the automorphism groups of the algebras in the paper.

• 대한수학회지
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• 제51권2호
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• pp.427-442
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• 2014

A previous work gave a combinatorial description of the crystal B(${\infty}$), in terms of certain simple Young tableaux referred to as the marginally large tableaux, for finite dimensional simple Lie algebras. Using this result, we present an explicit description of the crystal B(${\lambda}$), in terms of the marginally large tableaux, for the $G_2$ Lie algebra type. We also provide a new description of B(${\lambda}$), in terms of Nakajima monomials, that is in natural correspondence with our tableau description.

• 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 ${\Omega}_k$.

• 호남수학학술지
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• 제31권3호
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• pp.407-419
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• 2009

The simple non-associative algebra $N(e^{A_S},q,n,t)_k$ and its simple sub-algebras are defined in the papers [1], [3], [4], [5], [6], [12]. We define the non-associative algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$ and its antisymmetrized algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$. We also prove that the algebras are simple in this work. There are various papers on finding all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [3], [5], [6], [9], [12], [14], [15]). We also find all the derivations $Der_{anti}(WN(e^{{\pm}x^r},0,2)_B^-)$ of te antisymmetrized algebra $WN(e^{{\pm}x^r}0,2)_B^-$ and every derivation of the algebra is outer in this paper.

• 한국수학사학회지
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• 제15권3호
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• pp.17-24
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• 2002

It will be described the discovery of fundamental algebras such as complex numbers and the quaternions. Cardano(1539) was the first to introduce special types of complex numbers such as 5$\pm$$\sqrt{-15}$. Girald called the number a$\pm$$\sqrt{-b}$ solutions impossible. The term imaginary numbers was introduced by Descartes(1629) in “Discours la methode, La geometrie.” Euler knew the geometrical representation of complex numbers by points in a plane. Geometrical definitions of the addition and multiplication of complex numbers conceiving as directed line segments in a plane were given by Gauss in 1831. The expression “complex numbers” seems to be Gauss. Hamilton(1843) defined the complex numbers as paire of real numbers subject to conventional rules of addition and multiplication. Cauchy(1874) interpreted the complex numbers as residue classes of polynomials in R[x] modulo $x^2$＋1. Sophus Lie(1880) introduced commutators [a, b] by the way expressing infinitesimal transformation as differential operations. In this paper, it will be studied general quaternion algebras to finding of algebraic structure in Algebras.