arXiv subjects
Publications and source records attributed to Kostyantyn Yusenko.
For a given poset, we consider its representations by systems of subspaces of a unitary space ordered by inclusion. We classify such systems for all posets for which an explicit classification is possible.
We investigate representations of *-algebras associated with posets. Unitarizable representations of the corresponding (bound) quivers (which are polystable representations for some appropriately chosen slope function) give rise to representations of these algebras. Considering posets which correspond to unbound quivers this leads to an ADE-classification which describes the unitarization behaviour of their representations. Considering posets which correspond to bound quivers, it is possible to construct unitarizable representations starting with polystable representations of related unbound quivers which can be glued together with a suitable direct sum of simple representations. Finally, we estimate the number of complex parameters parametrizing irreducible unitary non-equivalent representations of the corresponding algebras.
A subspace representation of a poset $\mathcal S=\{s_1,...,s_t\}$ is given by a system $(V;V_1,...,V_t)$ consisting of a vector space $V$ and its subspaces $V_i$ such that $V_i\subseteq V_j$ if $s_i \prec s_j$. For each real-valued vector $χ=(χ_1,...,χ_t)$ with positive components, we define a unitary $χ$-representation of $\mathcal S$ as a system $(U;U_1,...,U_t)$ that consists of a unitary space $U$ and its subspaces $U_i$ such that $U_i\subseteq U_j$ if $s_i\prec s_j$ and satisfies $χ_1 P_1+...+χ_t P_t= \mathbb 1$, in which $P_i$ is the orthogonal projection onto $U_i$. We prove that $\mathcal S$ has a finite number of unitarily nonequivalent indecomposable $χ$-representations for each weight $χ$ if and only if $\mathcal S$ has a finite number of nonequivalent indecomposable subspace representations; that is, if and only if $\mathcal S$ contains any of Kleiner's critical posets.
A number of recent papers treated the representation theory of partially ordered sets in unitary spaces with the so called orthoscalar relation. Such theory generalizes the classical theory which studies the representations of partially ordered sets in linear spaces. It happens that the results in the unitary case are well-correlated with those in the linear case. The purpose of this article is to shed light on this phenomena.
We prove that partially ordered set has finite number of finite-dimensional indecomposable nonequivalent Hilbert representations with orthoscalarity condition if and anly if it has finite number of indecomposable linear representations. We show that each indecomposable representation of the poset of finite type could be unitarized with some weight.
We describe all weights which are appropriated for the unitarization of linear representations of primitive partially ordered sets of finite type.