arXiv subjects

Marcelo Flores

Publications and source records attributed to Marcelo Flores.

8 recordsLinked to original sources

arXiv2025

On framed rook algebras

We introduce and study the framed rook algebra, a structure that unifies two significant generalizations of the Iwahori-Hecke algebra. The first one, introduced by Solomon, extends the Hecke algebra to the full matrix monoid, yielding the rook monoid algebra. The second one, developed by Yokonuma, replaces the Borel subgroup with the unipotent subgroup, resulting in the Yokonuma-Hecke algebra. Our concrete algebra is constructed from the double cosets of the unipotent subgroup within the full matrix monoid. We show that this double coset decomposition is indexed by the framed symmetric inverse monoid. We also define the Rook Yokonuma-Hecke algebra as an abstract structure using generators and relations. We then prove the main isomorphism theorem, which establishes that this abstract algebra is isomorphic to the framed rook algebra under a specific parameter specialization. To complete our characterization, we provide a faithful representation on a tensor space and establish a linear basis for the Rook Yokonuma-Hecke algebra.

math.RT↗

arXiv2024

Invariants for singular links via the two parameter bt-algebra

We construct a new invariant of singular links through representations of the singular braid monoid into the two parameters bt-algebra. Additionally, we recover this invariant by using the approach of Paris and Rabenda. Hence, we introduce the so called two parameter Singular bt-algebra. Finally, we provide the skein relations that define our invariant, and we prove that this invariant is more powerful than previous invariants of singular links in literature.

math.GT↗

arXiv2020

An Alexander type invariant for doodles

We construct an Alexander type invariant for oriented doodles from a deformation of the Tits representation of the twin group and from the Chebyshev polynomials of second kind. Similar to the Alexander polynomial, our invariant vanishes on unlinked doodles with more than one component. We also include values of our invariant on several doodles.

math.GT↗

arXiv2019

Framization of a Temperley-Lieb algebra of type $\mathtt{B}$

We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$ and prove that such an extension supports a unique linear Markov trace function. We then introduce the Framization of the Temperley-Lieb algebra of type $\mathtt{B}$ as a quotient of the Yokonuma-Hecke algebra of type $\mathtt{B}$. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type $\mathtt{B}$ to pass to the quotient algebra. Using the main theorem, we construct invariants for framed links and classical links inside the solid torus.

math.RA↗

arXiv2019

Tied links in the solid torus

We introduce the concept of tied links in the solid torus, which generalize naturally the concept of tied links in $S^3$ previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and subsequently we recover this invariant by using Jones' method over the bt-algebra of type $\mathtt{B}$ and the Markov trace defined on this.

math.RA↗

arXiv2017

A bt-algebra of type B

We introduce a bt-algebra of type B. We define this algebra doing the natural analogy with the original construction of the bt-algebra. Notably we find a basis for it, a faithful tensorial representation, and we prove that it supports a Markov trace, from which we derive invariants of classical links in the solid torus.

math.RA↗

arXiv2016

A Framization of the Hecke algebra of Type B

In this article we introduce a framization of the Hecke algebra of type B. For this framization we construct a faithful tensorial representation and two linear bases. We finally construct a Markov trace on these algebras and from this trace we derive isotopy invariants for framed and classical knots and links in the solid torus.

math.RA↗

arXiv2013

Representations of Generalized Almost-Jordan algebras

This paper deals with the variety of commutative algebras satisfying one of the four families of degree four identities found by Carini, Hentzel and Piacentini-Cattaneo in 1988. We give a characterization of representations and irreducible modules on these algebras. Our results require that the characteristic of the ground field be different from 2,3.

math.RA↗