arXiv subjects

Mateusz Lowiel

Publications and source records attributed to Mateusz Lowiel.

3 recordsLinked to original sources

arXiv2026

The maximal dimensions of path and graph algebras

We consider the class of acyclic connected directed graphs with $N\geq 1$. In this paper we find the optimal upper bound for the number of paths amongst acyclic, connected graphs with $N$ edges. We prove that it is in fact optimal by finding an acyclic, connected graph with $N$ edges that realizes this bound. We then adapt these methods to find an optimal bound for Leavitt path algebras over a finite, acyclic, connected graph with $N$ edges.

math.CO↗

arXiv2026

The functoriality of moves on graphs and the extended covariant functoriality of graph algebras

Combinatorics of graphs is a very powerful tool to unravel various properties of graph algebras. In particular, isomorphisms between graph algebras are often implemented by moves between their graphs. In this paper, we make these combinatorial methods functorial, and show that collapsing an out-split graph to the original graph and transforming a graph to a shifted graph can be implemented by admissible graph homomorphisms and admissible path homomorphisms, respectively. To include the inverses of such isomorphisms, we introduce a new category of graphs where morphisms are given as regular homomorphisms of graph inverse semigroups. This new category admits a covariant functor to the category of C*-algebras and $*$-homomorphisms which extends the known covariant functor from the category of graphs and admissible path homomorphisms.

math.OA↗

arXiv2024

Quiver Grassmannians associated to nilpotent cyclic representations defined by single matrix

In the present paper we study the geometry of the closed Białynicki-Birula cells of the quiver Grassmannians associated to a nilpotent representation of a cyclic quiver defined by a single matrix. For the special case, where we choose subrepresentations of dimension $\mathbf{1}=(1,\dots,1)$, the main result of this paper is that the closed Białynicki-Birula cells are smooth. We also discuss the multiplicative structure of the cohomology ring of such spaces. Namely, we describe the so-called Knutson-Tao basis in context to the basis of equivariant cohomology that is dual to fundamental classes in equivariant homology.

math.RT↗