arXiv subjects

David Reynoso-Mercado

Publications and source records attributed to David Reynoso-Mercado.

6 recordsLinked to original sources

arXiv2025

On the Number of Exceptional Pairs Over a Class of Nakayama Algebras

In this paper some combinatorial and homological tools are used to describe and give an explicit formula for the number of exceptional pairs (exceptional sequences of length two) for some classes of Nakayama Algebras and for the Auslander algebra of a radical square zero algebra of type $\mathbb{A}_n$. In addition, we explore how the number of exceptional pairs can be connected with some integer sequences in the OEIS (The On-line Encyclopedia of Integer Sequences).

math.CO↗

arXiv2025

Standard automorphisms of semisimple Lie algebras and their relations

Let $\mathfrak{g}$ be the simple Lie algebra of square matrices $(n+1)\times (n+1)$ with zero trace. There are certain relations concerning standard automorphisms that are considered ``folklore". One can find a complete proof of these in Millson and Toledano Laredo's work [Transformation Groups, Vol. 10, No. 2, 2005, pp. 217-254], but said proof is based on topological arguments which are non-trivial. The aim of this work is to verify these relations in an elementary way.

math.RA↗

arXiv2025

About UMP algebras and a special classification case

The class of UMP algebras arises in several classification problems in the context of derived categories of finite-dimensional algebras. In this paper we define the class of UMP algebras and develop algebraic combinatorics tools in order to present a characterization of this class of algebras which are locally monomial (see Definition 4.1) and special multiserial algebras. Among other things, we describe the ramifications graph of symmetric special biserial algebras and we classify which of them are UMP algebras in terms of their bound quivers and their associated Brauer graphs.

math.RT↗

arXiv2025

UMP Monomial Algebras: Combinatorial and Homological Consequences

In this paper, we apply the techniques developed in [5] to present several consequences of studying UMP algebras and the ramifications graph of a monomial bound quiver algebra. Specifically, we prove that every weakly connected component of the ramifications graph of a UMP monomial algebra is unilaterally connected. Furthermore, using the main result characterizing UMP algebras in the monomial context, we prove that the class of UMP algebras is equivalent to the class of special multiserial algebras when the algebra is a quadratic monomial algebra. Based on this equivalence and the classification of Chen-Shen-Zhou on Gorenstein projective modules in [6], we extend their results to the class of monomial special multiserial UMP algebras, where we use the analysis of homological properties on quadratic monomial algebras given by these authors.

math.RT↗

arXiv2024

A model for the canonical algebras of bimodules type (1, 4) over truncated polynomial rings

Let $k=\mathbb{C}(\!(ε)\!)$ be the field of complex Laurent series. We use Galois descent techniques to show that the simple regular representations of the species of type $(1,\, 4)$ over $k$ are naturally parametrized by the closed points of $\mathrm{Spec}(k[x])\dot{\cup}\{1,\,2\}$. Moreover we provide weak normal forms for those representations. We use our representatives of the simple regular representations to describe the canonical algebras associated to the species of type (1, 4) over k. This suggest a model of those algebras in the sense of the work of Geiss, Leclerc and Schröer [GLS17] and [GLS20].

math.RT↗

arXiv2023

A partial classification of simple regular representations of bimodules type $(2,\,2)$ over $\mathbb{C}(\!(\varepsilon)\!)$

In this paper, we use Galois descent techniques to find suitable representatives of the regular simple representations of the species of type $(2,2)$ over $k_n := k[\varepsilon^{1/n}]$, where $n$ is a positive integer and $k:=\mathbb{C}(\!(\varepsilon)\!)$ is the field of Laurent series over the complexes. These regular representations are essential for the definition of canonical algebras. Our work is inspired by the work done for species of type $(1,4)$ on $k$ in ``A model for the canonical algebras of bimodules type (1, 4) over truncated polynomial rings''. We presents all the regular simple representations on the $n$-crown quiver, and from these, we establish a partial classification of regular simple representations of bimodules type $(2,2)$.

math.RT↗