• Kyungpook Mathematical Journal
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• 제57권4호
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• pp.601-612
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• 2017

Let Q be the frame g-loop quiver, i.e. a generalized ADHM quiver obtained by replacing the two loops into g loops. The vector space M of representations of Q admits an involution ${\ast}$ if orthogonal and symplectic structures on the representation spaces are endowed. We prove equivalence between semisimplicity of representations of the ${\ast}-invariant$ subspace N of M and the orbit-closedness with respect to the natural adjoint action on N. We also explain this equivalence in terms of King's stability [8] and orthogonal decomposition of representations.

• Korean Journal of Mathematics
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• 제18권3호
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• pp.243-252
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• 2010

In this paper we extend the projective properties of representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left R-modules to the projective properties of representations of quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left $R[x]$-modules. We show that if P is a projective left R-module then $0{\rightarrow}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. And we show $0{\rightarrow}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module if and only if $0{\rightarrow}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. Then we show if P is a projective left R-module then $R[x]\longrightarrow^{id}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. We also show that if $L\longrightarrow^{id}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module, then $L[x]\longrightarrow^{id}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules.

• Korean Journal of Mathematics
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• 제20권3호
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• pp.343-352
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• 2012

In this paper we show that the projective properties of representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ as left $R[x]$-modules. We show that if P is a projective left R-module then $0{\longrightarrow}0{\longrightarrow}P[x]$ is a projective representation of a quiver Q as $R[x]$-modules, but $P[x]{\longrightarrow}0{\longrightarrow}0$ is not a projective representation of a quiver Q as $R[x]$-modules, if $P{\neq}0$. And we show a representation $0{\longrightarrow}P[x]\longrightarrow^{id}P[x]$ of a quiver Q is a projective representation, if P is a projective left R-module, but $P[x]\longrightarrow^{id}P[x]{\longrightarrow}0$ is not a projective representation of a quiver Q as $R[x]$-modules, if $P{\neq}0$. Then we show a representation $P[x]\longrightarrow^{id}P[x]\longrightarrow^{id}P[x]$ of a quiver Q is a projective representation, if P is a projective left R-module.

• 대한수학회지
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• 제57권6호
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• pp.1373-1388
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• 2020

In the paper, we give an explicit basis of the cyclotomic quiver Hecke algebra corresponding to a minuscule representation of finite type.

• 호남수학학술지
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• 제35권1호
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• pp.1-15
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• 2013

In this paper, we study a sheaf-theoretic analysis of the convolution algebra on quiver varieties. As by-products, we reinterpret the results of H. Nakajima. We also produce a refined form of the BBD decomposition theorem for quiver varieties. Finally, we study a construction of highest weight modules through constructible functions.

• Korean Journal of Mathematics
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• 제17권3호
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• pp.271-281
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• 2009

We define injective and projective representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$. Then we show that a representation $M_1\longrightarrow[50]^{f1}M_2\longrightarrow[50]^{f2}M_3$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ is projective if and only if each $M_1,\;M_2,\;M_3$ is projective left R-module and $f_1(M_1)$ is a summand of $M_2$ and $f_2(M_2)$ is a summand of $M_3$. And we show that a representation $M_1\longrightarrow[50]^{f1}M_2\longrightarrow[50]^{f2}M_3$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ is injective if and only if each $M_1,\;M_2,\;M_3$ is injective left R-module and $ker(f_1)$ is a summand of $M_1$ and $ker(f_2)$ is a summand of $M_2$.

• 대한수학회논문집
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• 제21권1호
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• pp.37-43
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• 2006

We prove that $M_1\longrightarrow^f\;M_2$ is an injective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ if and only if $M_1\;and\;M_2$ are injective left R-modules, $M_1\longrightarrow^f\;M_2$ is isomorphic to a direct sum of representation of the types $E_l{\rightarrow}0$ and $M_1\longrightarrow^{id}\;M_2$ where $E_l\;and\;E_2$ are injective left R-modules. Then, we generalize the result so that a representation$M_1\longrightarrow^{f_1}\;M_2\; \longrightarrow^{f_2}\;\cdots\;\longrightarrow^{f_{n-1}}\;M_n$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\cdots}{\rightarrow}{\bullet}$ is an injective representation if and only if each $M_i$ is an injective left R-module and the representation is a direct sum of injective representations.

• 헤리티지:역사와 과학
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• 제53권2호
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• pp.242-253
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• 2020

나주 복암리 정촌 고분 1호 석실에서 화살통 장식 6점이 출토되었다. 유기물로 만들어진 화살통은 매장 상태에서 부식되어 없어지고 금속으로 만들어진 화살통 장식물만 남게 된다. 정촌 고분 화살통 장식은 형태적으로 2점씩 쌍을 이루며, 출토 위치에 따라 화살통 2점을 장식한 것으로 추정된다. 화살통 장식은 화살의 방입부(方立部)를 꾸며주는 대륜상금구와 방입부와 허리띠를 연결하는 배판(背板)을 장식하는 판상금구로 나누어진다. 1호 석실 목관2에서 출토된 화살통 장식은 대륜상금구만 확인되었으며 1호 석실 동남쪽에서 확인된 화살통 장식은 허리띠에 사용된 추정 대구, 판상금구, 대륜상금구가 확인되었다. 화살통 장식의 분석 결과, 철제 판에 금동 판을 접합한 철지금동장식제(鐵地金銅裝飾製)이며 표면을 정(釘)으로 점을 찍어 선과 문양을 만든 것을 알 수 있다. 성분 분석 결과(XRF), 금동 표면은 24~40wt% Au, 50~93wt% Cu가 검출되어 금도금 표면에 청동 부식물이 형성되었음을 확인하였다. 금도금 층의 SEM-EDS 분석 결과 광택을 내기 위한 작업선이 확인되었다. 또한 7~9wt% Hg가 검출되고 도금 층에 아말감 덩어리가 확인되어 아말감 도금한 것을 알 수 있었다. CT와 FT-IR 분석 결과 대륜상금구는 철제 판 아래 견직물이 2중으로 겹쳐 있으며 그 아래 옻칠편도 붙어 있었다. 이는 대륜상금구를 방입부에 부착할 때 직물을 덧대어 밀착력과 장식성을 높였으며, 옻칠 된 방입부 표면이 함께 떨어진 것으로 추정된다. 반면, 판상금구는 철제 판 아래 유기물이 두껍게 붙어 있다. 재질을 확정하기 어려우나 배판의 잔재로 보인다. 이러한 나주 정촌 고분 출토 화살통 장식의 특징은 4세기 후반~5세기 후반의 백제, 신라, 가야 문화권과 유사한 형식을 보여주며 당시 수준 높은 고대 금속 공예 제작 기술을 확인할 수 있었다.

• 대한수학회지
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• 제52권2호
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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.

• 대한수학회보
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• 제50권2호
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• pp.389-398
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• 2013

Let R be a ring and $\mathcal{Q}$ be a quiver. In this paper we give another definition of purity in the category of quiver representations. Under such definition we prove that the class of all pure injective representations of $\mathcal{Q}$ by R-modules is preenveloping. In case $\mathcal{Q}$ is a left rooted semi-co-barren quiver and R is left Noetherian, we show that every cotorsion flat representation of $\mathcal{Q}$ is pure injective. If, furthermore, R is $n$-perfect and $\mathcal{F}$ is a flat representation $\mathcal{Q}$, then the pure injective dimension of $\mathcal{F}$ is at most $n$.