Indecomposable representations of quivers on infinite-dimensional Hilbert spaces

2007/07/06 by Masatoshi Enomoto, Enomoto, Masatoshi, Yasuo Watatani +1
Mathematics · #15A21 #16G20 #16G60 #46C07 #47A15 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Representation Theory (math.RT) #math.OA #math.RT #msc:15A21 #msc:16G20 #msc:16G60 #msc:46C07 #msc:47A15

paper · pdf · doi:10.48550/arxiv.0707.0966

34 pages

openalex publication_date 2007/07/06 · arxiv created 2007/07/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We consider a complement of Gabriel's theorem for these representations. Let Γ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams A_n (n ≥ 0), D_n (n ≥ 4), E₆,E₇ and E₈, then there exists an indecomposable representation of Γ on separable infinite-dimensional Hilbert spaces.

Citations

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

Related

• Gabriel's Theorem for Infinite Quivers
• Unbounded strongly irreducible operators and transitive representations of quivers on infinite-dimensional Hilbert spaces
• Gabriel's Theorem for Locally Finite-Dimensional Representations of Infinite Quivers
• Strongly irreducible operators and indecomposable representations of quivers on infinite-dimensional Hilbert spaces
• The representations of quivers of type A_n. A fast approach