# Kostiantyn IusenkoUniversity of São Paulo | USP · Department of Mathematics (IME) (São Paulo)

Kostiantyn Iusenko

PhD

## About

19

Publications

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62

Citations

Introduction

Additional affiliations

March 2011 - August 2012

September 2009 - August 2010

## Publications

Publications (19)

For a finite poset P = {p(1),..., p(t)), we study systems (U-1,..., U-t)(U) of subspaces of a unitary space U such that U-i subset of U-j if p(i) < p(j). Two systems (U-1,..., U-t)(U) and (V-1,..., V-t)(V) are said to be isometric if there exists an isometry go : U -> V such that phi(U-i) = V-i. We classify such systems up to isometry if P is a sem...

In this expository article, we give a self-contained introduction to the wonderfully well-behaved class of pseudocompact algebras, focusing on the foundational classes of semisimple and separable algebras. We give characterizations of such algebras analogous to those for finite dimensional algebras. We give a self-contained proof of the Wedderburn-...

We consider a class of extensions of both abstract and pseudocompact algebras, which we refer to as "strongly proj-bounded extensions". We prove that the finiteness of the left global dimension and the support of the Hochschild homology is preserved by strongly proj-bounded extensions, generalizing results of Cibils, Lanzillota, Marcos and Solotar....

This short note describes some of the contributions to the Workshop GAAG 2019 held in Medellin, Colombia.

We define the path coalgebra and Gabriel quiver constructions as functors between the category of k-quivers and the category of pointed k-coalgebras, for k a field. We define a congruence relation on the coalgebra side, show that the functors above respect this relation, and prove that the induced Gabriel k-quiver functor is left adjoint to the cor...

We consider an intermediate category between the category of finite quivers and a certain category of pseudocompact associative algebras that contains all finite dimensional algebras. We define the completed path algebra and the Gabriel quiver as functors. We give an explicit quotient of the category of algebras on which these functors form an adjo...

We define the path coalgebra and Gabriel quiver constructions as functors between the category of $k$-quivers and the category of pointed $k$-coalgebras, for $k$ a field. We define a congruence relation on the coalgebra side, show that the functors above respect this relation, and prove that the induced Gabriel $k$-quiver functor is left adjoint to...

For a finite partially ordered set we calculate the dimension of the variety of its subspace representations having fixed dimension vector. The dimension is given in terms of the Euler quadratic form associated with a partially ordered set, which gives a geometric interpretation of this form.

The purpose of this paper is to study stable representations of partially ordered sets (posets) and compare it to the well known theory for quivers. In particular, we prove that every indecomposable representation of a poset of finite type is stable with respect to some weight and construct that weight explicitly in terms of the dimension vector. W...

These are the notes for a minicourse held in Odessa (2016) and Belo Horizonte (2017). My aim was to provide a short introduction to basic notions of category theory and representation theory of finite-dimensional algebras. We learnt the concept of adjoint functors and showed that the construction "quiver" <-> "algebra" can be interpreted as a pair...

In this paper we describe the twisted Hall algebra of bound quiver with small
homological dimension. The description is given in the terms of the quadratic
form associated with the corresponding bound quiver.

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...

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 $\chi=(\chi_1,...,\chi_t)$ with positive components, we define a unitary
$\chi$-representation of $\mathcal S$...

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-classif...

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.

We study conditions under which the images of irreducible quadruples of linearly connected projections give rise to all transitive systems of subspaces in a finite dimensional Hilbert space.

We describe the set of γ ∈ ℝ for which there exist quadruples of projectors P
i
for a fixed collection of numbers αi ℝ+, \(i = \overline {1,4} \), such that α1P
1 + α2P
2 + α3P
3 + α3P
4 = γI.

For -algebras associated with extended Dynkin graphs, we investigate a set of parameters for which there exist representations. We give structure properties of such sets and a complete description for the set related to the graph ˜ D4.