• 제목/요약/키워드: representations of a quiver

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On Semisimple Representations of the Framed g-loop Quiver

  • Choy, Jaeyoo
    • 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.

PROJECTIVE PROPERTIES OF REPRESENTATIONS OF A QUIVER Q = • → • AS R[x]-MODULES

  • Park, Sangwon;Kang, Junghee;Han, Juncheol
    • 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.

PURE INJECTIVE REPRESENTATIONS OF QUIVERS

  • Hosseini, Esmaeil
    • 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 $n$.

PROJECTIVE REPRESENTATIONS OF A QUIVER WITH THREE VERTICES AND TWO EDGES AS R[x]-MODULES

  • Han, Juncheol;Park, Sangwon
    • 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.

PROJECTIVE AND INJECTIVE PROPERTIES OF REPRESENTATIONS OF A QUIVER Q = • → • → •

  • Park, Sangwon;Han, Juncheol
    • 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 $M_2$.

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INJECTIVE REPRESENTATIONS OF QUIVERS

  • Park, Sang-Won;Shin, De-Ra
    • 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.

PROJECTIVE PROPERTIES OF REPRESENTATIONS OF A QUIVER OF THE FORM Q = • ⇉ • → •

  • Park, Sangwon;Han, Juncheol
    • 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 $e_2(a)=(0,a)$.

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ABSOLUTELY PURE REPRESENTATIONS OF QUIVERS

  • Aghasi, Mansour;Nemati, Hamidreza
    • 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.

GENERALIZED MCKAY QUIVERS, ROOT SYSTEM AND KAC-MOODY ALGEBRAS

  • Hou, Bo;Yang, Shilin
    • 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.