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A. V. Roiter

Publications and source records attributed to A. V. Roiter.

9 recordsLinked to original sources
arXiv2009

Orthoscalar representations of quivers on the category of Hilberts spaces. II

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 reresentations of graphs in the category of Hilbers spaces" (Func. Anal. and Appl., 2005) authors showed the way to tranfer these results to Hilbert spaces, constructed Coxeter functors and proved an analogue of Gabriel theorem fol locally scalar (orthoscalar in the sequel) representations (up to the unitary equivalence). The category of orthoscalar representations of a quiver can be considered as a subcategory in the category of all representations (over a field $\mathbb C$). In the present paper we study the connection between indecomposable orthoscalar representations in the subcategory and in the category of all representations. For the quivers, corresponded to extended Dynkin graphs, orthoscalar representations which cannot be obtained from the simplest by Coxeter functors (regular representations) are classified.

math.RT↗
arXiv2006

Matrix Problems in Hilbert Spaces

We consider matrix problems in Hilbert spaces (orthoscalar representations of quivers and posets). A criterion of tameness of the problem of classification of indecomposable orthoscalar representations of a quiver is given.

math.RT↗
arXiv2005

Antimonotonous quadratic forms and partially ordered sets

Antimonotonous quadratic forms generalizing P-faithful posets defined by authors earlier are introduced. The criterion of antimonotonousness is given for posets with positive semidefinite quadratic forms. As consequence the new proofs of the criterions of finite representativity and tameness of posets are given.

math.RT↗
arXiv2003

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

In this paper authors consider representations of graphs in Hilbert spaces applying a restriction of local scalarity on them. It enables to obtain a theory, similar to the classical theory of representations of graphs in vector spaces. In particular, it is obtained a theorem analogous to the well-known Gabriel theorem: a connected finite graph (wood) is finitely representable in the category of Hilbert spaces if and only if it is a Dynkin graph.

math.RT↗
arXiv2003

Singular locally-scalar representations of quivers in Hilbert spaces and separating functions

A numeric function $ρ$: $ρ(k)=1+\frac{k-1}{k+1}, k \in N$ was considered in [1]. In its terms criterions of finite representability and tameness of marked quivers, posets with equivalence and dyadic posets can be obtained; Dynkin schemes and extended schemes also can be characterized. In this paper authors consider the connection of function $ρ$ with locally-scalar representations [2] of extended Dynkin graphs. Then a family of functions $ρ_n$ is defined -- a generalization of function $ρ$, which plays an analogous part for more wide class of graphs. Also some properties of functions $ρ$ and $ρ_k$ are proved. References [1] L.A. Nazarova, A.V. Roiter. {\it Norm of a relation, separating functions and representations of marked quivers.} Ukr. Math. Jour., 54(2002), No.6, p.808-840. [2] S.A. Kruglyak, A.V. Roiter. {\it Locally-scalar representations of graphs in the category of Hilbert spaces.} Prepr. Ukr. Math. Jour. (2003).

math.RT↗
arXiv2001

Representations of marked quivers

We introduce a generalization of representations of quivers that contains also representations of posets, vectorspace problems and other matrix problems. Many examples, some of which are given in the paper, show that the language of marked quivers is rather convenient and that the notion of their representations can probably be considered as basic for a general theory of matrix problems. Some historical comments from my personal point of view are added in the Appendix.

math.RT↗