• Korean Journal of Mathematics
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• 제17권4호
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• pp.429-436
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• 2009

We define a projective representation $M_1{^{\rightarrow}_{\rightarrow}}M_2{\rightarrow}M_3$ of a quiver $Q={\bullet}{^{\rightarrow}_{\rightarrow}}{\bullet}{\rightarrow}{\bullet}$ and consider their properties. Then we show that any projective representation $M_1{^{\rightarrow}_{\rightarrow}}M_2{\rightarrow}M_3$ of a quiver $Q={\bullet}{^{\rightarrow}_{\rightarrow}}{\bullet}{\rightarrow}{\bullet}$ is isomorphic to the quotient of a direct sum of projective representations $0{^{\rightarrow}_{\rightarrow}}0{\rightarrow}P,\;0{^{\rightarrow}_{\rightarrow}}P{\rightarrow\limits^{id}}P$ and $P{^{\rightarrow}_{\rightarrow}}^{e1}_{e2}P{\oplus}P{\rightarrow\limits^{id_{P{\oplus}P}}}P{\oplus}P$, where $e_1(a)=(a,0)$ and $e_2(a)=(0,a)$.

• 호남수학학술지
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• 제33권3호
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• pp.347-353
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• 2011

In [1], Maffei proved a certain relationship between quiver varieties of type A and the geometry of partial flag varieties over the nilpotent cone. This relation was conjectured by Naka-jima, and Nakajima proved his conjecture for a simple case. In the Maffei's proof, the key step was a reduction of the general case of the conjecture to the simple case treated by Nakajima through a certain isomorphism. In this paper, we study properties of this isomorphism.

• 충청수학회지
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• 제24권3호
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• pp.427-436
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• 2011

In this paper, we first show that the pull-back morphism between two K-groups of the Steinberg varieties, obtained respectively from partial flag varieties and quiver varieties of type A, is a ring homomorphism with respect to the convolution product. Then, we prove that this ring homomorphism yields a property of compatibility between two certain convolution actions.

• 대한수학회논문집
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• 제34권3호
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• pp.701-718
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• 2019

In this paper we compute the U-projective resolution of kQ-modules where Q is quiver of type $A_n$ and ${\tilde{A}}_n$. The behavior of the sequence can be seen through its geometric representation.

• Korean Journal of Mathematics
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• 제16권3호
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• pp.323-334
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• 2008

We define injective and projective representations of quivers with two vertices with n arrows. In the representation of quivers we denote n edges between two vertices as ${\Rightarrow}$ and n maps as $f_1{\sim}f_n$, and $E{\oplus}E{\oplus}{\cdots}{\oplus}E$ (n times) as ${\oplus}_nE$. We show that if E is an injective left R-module, then $${\oplus}_nE{\Longrightarrow[50]^{p_1{\sim}p_n}}E$$ is an injective representation of $Q={\bullet}{\Rightarrow}{\bullet}$ where $p_i(a_1,a_2,{\cdots},a_n)=a_i,\;i{\in}\{1,2,{\cdots},n\}$. Dually we show that if $M_1{\Longrightarrow[50]^{f_1{\sim}f_n}}M_2$ is an injective representation of a quiver $Q={\bullet}{\Rightarrow}{\bullet}$ then $M_1$ and $M_2$ are injective left R-modules. We also show that if P is a projective left R-module, then $$P\Longrightarrow[50]^{i_1{\sim}i_n}{\oplus}_nP$$ is a projective representation of $Q={\bullet}{\Rightarrow}{\bullet}$ where $i_k$ is the kth injection. And if $M_1\Longrightarrow[50]^{f_1{\sim}f_n}M_2$ is an projective representation of a quiver $Q={\bullet}{\Rightarrow}{\bullet}$ then $M_1$ and $M_2$ are projective left R-modules.

• 대한수학회논문집
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• 제27권3호
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• pp.571-581
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• 2012

The moduli spaces of stable quasimaps unify various moduli appearing in the study of Gromov-Witten theory. This note is a survey article on the moduli of stable quasimaps, based on papers [9, 11, 18] as well as the author's talk at Kinosaki Algebraic Geometry Symposium 2010.

• Kyungpook Mathematical Journal
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• 제61권2호
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• pp.213-222
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• 2021

In this paper we investigate the Baer-Kaplansky theorem for module classes on algebras of finite representation types over a field. To do this we construct finite dimensional quiver algebras over any field.

• 대한수학회지
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• 제51권6호
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• pp.1177-1187
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• 2014

In the current paper we study absolutely pure representations of quivers. Then over some nice quivers including linear quivers some sufficient conditions guaranteeing a representation to be absolutely pure is characterized. Furthermore some relations between atness and absolute purity is investigated. Finally it is shown that the absolutely pure covering of representations of linear quivers (including $A^-_{\infty}$, $A^+_{\infty}$ and $A^{\infty}_{\infty}$) by R-modules whenever R is a coherent ring exists.

• 대한수학회지
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• 제56권2호
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• pp.353-385
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• 2019

In this paper, we introduce the notion of combinatorial Auslander-Reiten (AR) quivers for commutation classes [${\tilde{w}}]$ of w in a finite Weyl group. This combinatorial object is the Hasse diagram of the convex partial order ${\prec}_{[{\tilde{w}}]}$ on the subset ${\Phi}(w)$ of positive roots. By analyzing properties of the combinatorial AR-quivers with labelings and reflection functors, we can apply their properties to the representation theory of KLR algebras and dual PBW-basis associated to any commutation class [${\tilde{w}}_0$] of the longest element $w_0$ of any finite type.

• Asia pacific journal of information systems
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• 제11권2호
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• pp.121-139
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• 2001

VOQL* query language, recently proposed, is a visual language for object-oriented databases. It is based on Ven Diagram and graph, so that the underlying schema structure can be naturally implied in query expressions. In VOQL*, structural relationship among the objects used in a query expression is represented graphically and thus it has formal semantics that can be inductively defined, as well as it can be used with ease. In this paper, we proposed revised VOQL* and introduced its query processor, InQs(Intelligent Querying System). While retaining the merit of VOQL* that it allows the structural relationship among the objects to be represented visually, the revised VOQL* has another merit that users can formulate a query interactively using various forms supplied by InQs. As a query processor that translates queries in revised VOQL into those in ODMG OQL, InQs provides an environment in which users express queries in revised VOQL* and then the system automatically translates them into those in ODMG OQL. Translation algorithm of InQs is much simpler and intuitive than other algorithms used in QUIVER and other systems, since it reflects the formal semantics of VOQL*, which is defined inductively.