### Abstract

We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We exhibit several concrete examples and investigate duality theorem between reflection functors. We also show a complement of Gabriel's theorem. Let Γ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams over(A, ̃)_{n}(n ≥ 0), over(D, ̃)_{n}(n ≥ 4), over(E, ̃)_{6}, over(E, ̃)_{7} and over(E, ̃)_{8}, then there exists an indecomposable representation of Γ on separable infinite-dimensional Hilbert spaces.

Original language | English |
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Pages (from-to) | 959-991 |

Number of pages | 33 |

Journal | Journal of Functional Analysis |

Volume | 256 |

Issue number | 4 |

DOIs | |

Publication status | Published - Feb 15 2009 |

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### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

*Journal of Functional Analysis*,

*256*(4), 959-991. https://doi.org/10.1016/j.jfa.2008.12.011

**Indecomposable representations of quivers on infinite-dimensional Hilbert spaces.** / Enomoto, Masatoshi; Watatani, Yasuo.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 256, no. 4, pp. 959-991. https://doi.org/10.1016/j.jfa.2008.12.011

}

TY - JOUR

T1 - Indecomposable representations of quivers on infinite-dimensional Hilbert spaces

AU - Enomoto, Masatoshi

AU - Watatani, Yasuo

PY - 2009/2/15

Y1 - 2009/2/15

N2 - We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We exhibit several concrete examples and investigate duality theorem between reflection functors. We also show a complement of Gabriel's theorem. Let Γ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams over(A, ̃)n(n ≥ 0), over(D, ̃)n(n ≥ 4), over(E, ̃)6, over(E, ̃)7 and over(E, ̃)8, then there exists an indecomposable representation of Γ on separable infinite-dimensional Hilbert spaces.

AB - We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We exhibit several concrete examples and investigate duality theorem between reflection functors. We also show a complement of Gabriel's theorem. Let Γ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams over(A, ̃)n(n ≥ 0), over(D, ̃)n(n ≥ 4), over(E, ̃)6, over(E, ̃)7 and over(E, ̃)8, then there exists an indecomposable representation of Γ on separable infinite-dimensional Hilbert spaces.

UR - http://www.scopus.com/inward/record.url?scp=58149336757&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58149336757&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2008.12.011

DO - 10.1016/j.jfa.2008.12.011

M3 - Article

AN - SCOPUS:58149336757

VL - 256

SP - 959

EP - 991

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 4

ER -