Orthoscalar representations of quivers in the category of Hilbert spaces

As is known, finitely presented quivers correspond to Dynkin graphs (Gabriel, 1972) and tame quivers correspond to extended Dynkin graphs (Donovan and Freislich, Nazarova, 1973). In the article "Locally scalar representations of graphs in the category of Hilberts spaces" (Func. Anal. Apps., 2005), the authors showed a way for carrying over these results to Hilbert spaces, constructed Coxeter functors, and proved an analog of the Gabriel theorem for locally scalar representations (up to unitary equivalence). The category of locally scalar representations of a quiver can be regarded as a subcategory in the category of all representations (over the field ℂ). In the present paper, we study the relationship between the indecomposability of locally scalar representations in the subcategory and in the category of all representations (it is proved that for a class of quivers wide enough indecomposability in the subcategory implies indecomposability in the category). For a quiver corresponding to the extended Dynkin graph D̃4, locally scalar representations that cannot be obtained from the simplest ones by Coxeter functors (regular representations) are classified. Bibliography: 21 titles. © 2007 Springer Science+Business Media, Inc.