Unbounded strongly irreducible operators and transitive representations of quivers on infinite-dimensional Hilbert spaces

2016/03/25 by Masatoshi Enomoto, Enomoto, Masatoshi, Yasuo Watatani +1
Mathematics · Physics and Astronomy · #16G20 (Secondary) #46C07 #47A15 #47A65 (Primary) #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Quantum many-body systems

paper · pdf · doi:10.48550/arxiv.1603.07836

openalex publication_date 2016/03/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

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 Γ be a quiver whose underlying undirected graph is an extended Dynkin diagram. Then there exists an infinite-dimensional transitive Hilbert representation of Γ if and only if Γ is not an oriented cyclic quiver.

Citations

• Locally-scalar representations of graphs in the category of Hilbert spaces

Related

• Strongly irreducible operators and indecomposable representations of quivers on infinite-dimensional Hilbert spaces
• Indecomposable representations of quivers on infinite-dimensional Hilbert spaces
• Gabriel's Theorem for Infinite Quivers
• The representations of quivers of type A_n. A fast approach
• On the indecomposability in the category of representations of a quiver and in its subcategory of orthoscalar representations