On functions on graphs and representations of a certain class of ∗-algebras

Abstract

We study * -representations of certain algebras which can be described in terms of graphs and positive functions (weights) on the set of their vertices, continuing an earlier investigation of the case where the graph is a Dynkin graph or an extended Dynkin graph with a weight of a special kind.For the cases where the graph is one of the extended Dynkin graphsD 4 ,Ẽ 6 ,Ẽ 7 orẼ 8 , we prove that all irreducible * -representations of the corresponding algebras are finite-dimensional.In the case of a graph which properly contains an extended Dynkin graph, we study the evolution of weights under the action of the Coxeter functors, in particular, we show that there exist two linearly independent p-invariant weights. We also prove that there exists a weight which makes the corresponding algebra to have an infinite-dimensional irreducible * -representation.

Orthoscalar quiver representations corresponding to extended Dynkin graphs in the category of Hilbert spaces

“…We assign two N -dimensional G-vectors (N = m + n) to the representation T : the dimension d = {d(j)} j∈Qv where d(j) = dim T (j), and the character χ = {χ(j)} j∈Qv defined above (see (5), (6)). Two orthoscalar representations T and T are equivalent if there are exist unitary matrices U = diag{U i 1 , .…”

“…The case of odd n can be treated in a similar way by replacing the matrix [ 6 ; the corresponding split quiver Q has the form…”

“…The dimensions of indecomposable orthoscalar representations of quivers corresponding to extended Dynkin graphs are not uniformly bounded (see [2]). However, it follows from [4]- [6] that of all infinitely representable quivers only those corresponding to extended Dynkin graphs do not have infinite-dimensional indecomposable orthoscalar representations.…”

On simple n-tuples of subspaces of a Hilbert space

“…For example, the papers [30,1,37,31] are devoted to the investigation of orthoscalar n-tuples of subspaces of the Hilbert space H:…”

On reduction of block matrices in a Hilbert space

“…Q D n ; Q [7,2,6,4]). In the present paper, we introduce and study h-orthoscalar block matrices, i.e., block matrices B such that, for a certain h 2 N and positive real numbers In the present paper, we also introduce and study OE2; 4-orthoscalar block matrices, i.e., matrices that simultaneously satisfy the conditions of 2-and 4-orthoscalarity.…”