Unbounded strongly irreducible operators and transitive representations of quivers on infinite-dimensional Hilbert spaces
Unbounded strongly irreducible operators and transitive representations of quivers on infinite-dimensional Hilbert spaces
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发表时间：
2016-03
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通讯作者：
M. Enomoto;Y. Watatani
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作者：
M. Enomoto;Y. Watatani
我们引入了无界强不可约算子和传递算子。这些算子与无限维Hilbert空间上箭图的一类不可分解的Hilbert表示有关。我们认为箭图的Hilbert表示理论是无界算子理论的推广。如果自同态代数是平凡的，则称箭图的非零Hilbert表示是传递的。如果箭图的希尔伯特表示是传递的，那么它是不可分解的。但事实并非如此。设$\Gamma$是箭图，其基础无向图是扩展的动态图。则存在$\Gamma$的无限维传递Hilbert表示当且仅当$\Gamma$不是定向循环箭图。
We introduce unbounded strongly irreducible operators and transitive operators. These operators are related to a certain class of indecomposable Hilbert representations of quivers on infinite-dimensional Hilbert spaces. We regard the theory of Hilbert representations of quivers is a generalization of the theory of unbounded operators. A non-zero Hilbert representation of a quiver is said to be transitive if the endomorphism algebra is trivial. If a Hilbert representation of a quiver is transitive, then it is indecomposable. But the converse is not true. Let $\Gamma$ be a quiver whose underlying undirected graph is an extended Dynkin diagram. Then there exists an infinite-dimensional transitive Hilbert representation of $\Gamma$ if and only if $\Gamma$ is not an oriented cyclic quiver.