Narrow Results

We study finitary $2$-categories associated to dual projection functors for finite-dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type $A$), we show that the monoid generated by dual projection functors is the Hecke–Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.

Hecke–Kiselman monoids ${\mathrm{\text{HK}}}_{\mathrm{\Theta }}$ and their algebras $K[{\mathrm{\text{HK}}}_{\mathrm{\Theta }}]$, over a field $K$, associated to finite oriented graphs $\mathrm{\Theta }$ are studied. In the case $\mathrm{\Theta }$ is a cycle of length $n\ge 3$, a hierarchy of certain unexpected structures of matrix type is discovered within the monoid $C_n={\mathrm{\text{HK}}}_{\mathrm{\Theta }}$ and this hierarchy is used to describe the structure and the properties of the algebra $K[C_n]$. In particular, it is shown that $K[C_n]$ is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand–Kirillov dimension one. This is used to characterize all Noetherian algebras $K[{\mathrm{\text{HK}}}_{\mathrm{\Theta }}]$ in terms of the graphs $\mathrm{\Theta }$. The strategy of our approach is based on the crucial role played by submonoids of the form $C_n$ in combinatorics and structure of arbitrary Hecke–Kiselman monoids ${\mathrm{\text{HK}}}_{\mathrm{\Theta }}$.