Locally Scalar Graph Representations in the Category of Hilbert Spaces

Kruglyak, S. A.; Roiter, A. V.

doi:10.1007/s10688-005-0022-8

Cited by 32 publications

(82 citation statements)

References 15 publications

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“…Then indecomposability of π and lemma 3.5 [7] imply G π = {g}, i. e. π = Π g , (π, f ) is a simplest object.…”

Section: Let Repmentioning

confidence: 99%

“…In [7] for (arbitrary) graph G were considered Note that all listed categories, except Rep(G, H), are not additive. If vector f is such that f i > 0 when i = p + 1, n then in [7] it was constructed a functor…”

Section: Hypothesismentioning

confidence: 99%

“…Repeatedly attempts were made to generalize representations of quivers to metric and, in particular, Hilbert spaces, but at that mentioned connection was lost. In [7] a restriction of local scalarity was imposed on representations of graphs in Hilbert spaces, and after that it was managed to generalize the results of [1], [2] in a natural way by constructing, analogous to Coxeter functors, functors of even and add reflections, moreover it turned out that just these representations are of interest of functional analysis and in some particular cases in other terms in fact were considered in [8]- [9].In this paper we will consider a connection of locally-scalar representations of extended Dynkin graphs with function ρ considered in [10], and also functions ρ k playing analogous part for more wide class of graphs. …”

mentioning

confidence: 99%

“…We will widely use denotations and definitions of the article [7]. All considered graphs will be supposed finite, connected and acyclic (i. e. woods).…”

mentioning

confidence: 99%

“…All considered graphs will be supposed finite, connected and acyclic (i. e. woods). Multiplicity µ(g) of a vertex g ∈ G v is |M g | (M g = {g i ∈ G v | g i − g} [7]). …”

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confidence: 99%

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

Redchuk

Roiter

2004

Ukr Math J

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In the classical works [1], [2] representations of quivers (in the category of finite-dimensional vector spaces) were considered and their connection with root systems of corresponding graphs was established, which developing further in the works [3]- [6]. Repeatedly attempts were made to generalize representations of quivers to metric and, in particular, Hilbert spaces, but at that mentioned connection was lost. In [7] a restriction of local scalarity was imposed on representations of graphs in Hilbert spaces, and after that it was managed to generalize the results of [1], [2] in a natural way by constructing, analogous to Coxeter functors, functors of even and add reflections, moreover it turned out that just these representations are of interest of functional analysis and in some particular cases in other terms in fact were considered in [8]- [9].In this paper we will consider a connection of locally-scalar representations of extended Dynkin graphs with function ρ considered in [10], and also functions ρ k playing analogous part for more wide class of graphs.

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“…Then indecomposability of π and lemma 3.5 [7] imply G π = {g}, i. e. π = Π g , (π, f ) is a simplest object.…”

Section: Let Repmentioning

confidence: 99%

Section: Hypothesismentioning

confidence: 99%

mentioning

confidence: 99%

“…We will widely use denotations and definitions of the article [7]. All considered graphs will be supposed finite, connected and acyclic (i. e. woods).…”

mentioning

confidence: 99%

“…All considered graphs will be supposed finite, connected and acyclic (i. e. woods). Multiplicity µ(g) of a vertex g ∈ G v is |M g | (M g = {g i ∈ G v | g i − g} [7]). …”

mentioning

confidence: 99%

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

Redchuk

Roiter

2004

Ukr Math J

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On The Unitarization Of Linear Representations Of Primitive Partially Ordered Sets

Grushevoy

Yusenko

2009

Modern Analysis and Applications

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The representation theory of partially ordered sets (posets) in linear vector spaces has been studied extensively and found to be of great importance for studying indecomposable representations of group and algebras, Cohen-Macaulay modules and many others algebraical objects (see [2,3,6,7,16] and many others). A representation of a given poset P in some vector space V is a collection (V ; V i ), i ∈ P of vector subspaces V i ⊂ V such that V i ⊂ V j as soon as i ≺ j in P. Usually such representations are studied up to equivalence (which is given by linear bijections between two spaces that bijectively map the corresponded subspaces). M. Kleiner and L. Nazarova (see [6,11]) completely classified all posets into three classes: finite type posets (posets that have finite number of indecomposable nonequivalent representation), tame posets (posets that have at most one-parametric family of indecomposable representations in each dimension) and wild posets (the classification problem of their indecomposable representations contains as a subproblem a problem of classification up to conjugacy classes a pair of two matrices).It is also possible to develop a similar theory over Hilbert spaces. By representation we understand a collection (H; H i ) of Hilbert subspaces in some Hilbert space H such that H i ⊂ H j as soon as i ≺ j. The equivalence between two system of Hilbert subspaces is given by unitary operator which bijectively maps corresponding subspaces. It turns out that in this case the classification problem becomes much more harder: even the poset PThis work has been partially supported by the Scientific Program of National Academy of Sciences of Ukraine, Project No 0107U002333.2 Roman Grushevoi and Kostyantyn Yusenko poset (it is impossible to classify all representation of this poset in a reasonable way see [9]). We add an "extra" relation α 1 P 1 + . . . + α n P n = γI, (0.1) between the projections P i : H → H i on corresponding subspaces for some weight χ = (α 1 , . . . , α n ) ∈ R n + (this relation will be called orthoscalarity condition). When the system of subspaces is a so-called m-filtration, this relation plays an important role in different areas of mathematics (see [17,8] and references therein) and this is actually one of the original motivations to investigate such representations of posets in Hilbert space.The interconnection between linear and Hilbert representations of the posets is given by unitarization which asks whether for given linear representation (V, V i ) it is possible to provide a hermitian structure in V so that the linear relation (0.1) holds for some weight χ. In [4] for the case when P is a primitive poset we proved that a poset P is of finite orthoscalar type (has finitely many irreducible representations with orthoscalarity condition up to the unitary equivalence) if and only if it is of finite (linear) type. Also there were proved that each indecomposable representation of poset of finite (linear) type could be unitarized with some weight and for each representation we described...

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Strongly Irreducible Operators and Indecomposable Representations of Quivers on Infinite-Dimensional Hilbert Spaces

Enomoto

Watatani

2015

Integr. Equ. Oper. Theory

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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.

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Locally Scalar Graph Representations in the Category of Hilbert Spaces

Abstract: The condition of being locally scalar is imposed on graph (or quiver) representations in the category of Hilbert spaces. Under this condition, reflection and Coxeter functors are constructed in categories of such representations and used to prove an analog of the Gabriel theorem.

Cited by 32 publications

References 15 publications

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

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

On The Unitarization Of Linear Representations Of Primitive Partially Ordered Sets

Strongly Irreducible Operators and Indecomposable Representations of Quivers on Infinite-Dimensional Hilbert Spaces

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