Orthoscalar representations of quivers in the category of Hilbert spaces

S. A. Kruglyak ; L. A. Nazarova ; A. V. Roiter

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 180-201

Résumé

As it is known, finitely presented quivers correspond to Dynkin graphs (Gabriel, 1972) and tame quivers – 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. and Apps., 2005) authors showed the way to transfere these results to Hilbert spaces, constructed Coxeter functors and proved an analogue of Gabriel theorem for locally scalar representations (up to the unitary equivalence). The category of locally scalar representations of a quiver can be considered as a subcategory in the category of all representations (over the field $\mathbb C$). In the present paper we study the connection between indecomposability of locally scalar representations in the subcategory and in the category of all representations (it is proved that for wide enough class of quivers indecomposability in the subcategory implies indecomposability in the category). For a quiver, corresponding to extended Dynkin graph $\widetilde D_4$ locally scalar representations which cannot be obtained from the simplest by Coxeter functors (regular representations) are classified.