Artículos (Álgebra)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/10804

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Constructions in R[x1,…,xn]: applications to K-theory
(2001-09-05) Gago Vargas, Manuel Jesús; Álgebra
A classical result in K-theory about polynomial rings like the Quillen–Suslin theorem admits an algorithmic approach when the ring of coefficients has some computational properties, associated with Gröbner bases. There are several algorithms when we work in , a field. In this paper we compute a free basis of a finitely generated projective module over R[x1,…,xn], R a principal ideal domain with additional properties, test the freeness for projective modules over D[x1,…,xn], with D a Dedekind domain like and for the one variable case compute a free basis if there exists any.
Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups
(De Gruyter Bill, 2024-03-09) Garnier, Owen; Álgebra
We consider a particular class of Garside groups, which we call circular groups. We mainly prove that roots are unique up to conjugacy in circular groups. This allows us to completely classify these groups up to isomorphism. As a consequence, we obtain the uniqueness of roots up to conjugacy in complex braid groups of rank 2. We also consider a generalization of circular groups, called hosohedral-type groups. These groups are defined using circular groups, and a procedure called the Δ-product, which we study in generality. We also study the uniqueness of roots up to conjugacy in hosohedral-type groups.
Enumerating regions of Shi arrangements per Weyl cone
(Elsevier, 2024-07-04) Dermenjian, Aran; Tzanaki, E.; Álgebra
Given a Shi arrangement , it is well-known that the total number of regions is counted by the parking number of type and the total number of regions in the dominant cone is given by the Catalan number of type . In the case of the latter, in Shi (1997), Shi gave a bijection between antichains in the root poset of and the regions in the dominant cone. This result was later extended by Armstrong, Reiner and Rhoades in Armstrong et al. (2015) where they gave a bijection between the number of regions contained in an arbitrary Weyl cone in and certain subposets of the root poset. In this article we expand on these results by giving a determinantal formula for the precise number of regions in using paths in certain digraphs related to Shi diagrams.
Thin MC left regular bands
(Springer, 2025-09-10) Dermenjian, Aram; Álgebra
We study left regular bands using the adjacency graph of chambers, labeled by facets. In particular, we define a particular family of left regular bands which we call thin MC left regular bands whose face poset is a meet-semilattice and whose adjacency graph is connected. We provide a criterion for when the face poset is a meet-semilattice using the multiplication of the semigroup and its associated support lattice. It turns out that the thin MC left regular bands have a particularly nice adjacency graph. In particular we prove that the adjacency graph of an arbitrary thin MC left regular band is such that each label appears precisely once and every simple cycle has an even number of edges whose associated facets have equal support, implying every simple cycle is of even length. Conversely, we define a set of graphs which we call thin LRB graphs which encode rank two thin MC left regular bands.
The OrientedMatroids package for SageMath
(msp, 2025-07-28) Dermenjian, Aram; Flight, Elizabeth; Tanasa, Tudor; Álgebra
Weintroduce the OrientedMatroids package for SageMath. It provides an interface for constructing oriented matroids using one of various cryptomorphic definitions. It also provides methods for transferring between the various definitions and implements some basic properties for oriented matroids.
Conjugacy Class Growth in Virtually Abelian Groups
(EPIsciences, 2025-02-24) Dermenjian, Aram; Evetts, Alex; Álgebra
We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius n in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually abelian groups, we prove that this function is always asymptotically equivalent to a polynomial. Furthermore, we show that in any affine Coxeter group, the degree of polynomial growth of a conjugacy class is equivalent to the reflection length of any element of that class.
The root extraction problem in braid group-based cryptography
(Springer, 2024-06-17) Cumplido Cabello, María; Kahrobaei, Delaram; Noce, Marialaura; Álgebra; FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
The root extraction problem in braid groups is the following: given a braid and a number , find such that . In the last decades, several cryptosystems based on the root extraction problem, including authentication schemes and digital signatures, have been proposed. In this survey paper, we describe these cryptosystems built around braid groups. Then we explain that, in general, these authentication schemes and digital signatures are not secure, and we present for each of them a possible attack.
On parabolic subgroups of Artin–Tits groups of spherical type
(Elsevier, 2019-06-18) Cumplido Cabello, María; Gebhardt, Volker; González Meneses López, Juan; Wiest, Bert; Álgebra; FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
We show that, in an Artin–Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin–Tits groups of spherical type a result that was previously known for braid groups. To obtain the above results, we show that every element in an Artin–Tits group of spherical type admits a unique minimal parabolic subgroup containing it, which we call its parabolic closure. We also show that the parabolic closure of an element coincides with the parabolic closure of any of its powers or roots. As a consequence, if an element belongs to a parabolic subgroup, all its roots belong to the same parabolic subgroup. We define the simplicial complex of irreducible parabolic subgroups, and we propose it as the analogue, in Artin–Tits groups of spherical type, of the celebrated complex of curves which is an important tool in braid groups, and more generally in mapping class groups. We conjecture that the complex of irreducible parabolic subgroups is δ-hyperbolic.
Intersection of parabolic subgroups in Euclidean braid groups: a short proof
(Académie des sciences, 2024-11-14) Cumplido Cabello, María; Gavazzi, Federica; Paris, Luis; Álgebra; FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
We give a short proof for the fact, already proven by Thomas Haettel, that the arbitrary intersection of parabolic subgroups in Euclidean Braid groups A[An] is again a parabolic subgroup. To that end, we use that the spherical-type Artin group A[Bn+1] is isomorphic to A[An]⋊Z .
Block mapping class groups and their finiteness properties
(Springer, 2025-02-18) Aramayona, J.; Aroca, Javier; Cumplido Cabello, María; Skipper, R.; Wu, Xi Xi; Álgebra; FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
Given , let denote the closed surface of genus g with a Cantor set removed, if ; or the blooming Cantor tree, when . We construct a family of subgroups of whose elements preserve a block decomposition of , and eventually like act like an element of H, where H is a prescribed subgroup of the mapping class group of the block. The group surjects onto an appropriate symmetric Thompson group of Farley–Hughes; in particular, it answers positively. Our main result asserts that is of type if and only if H is. As a consequence, for every and every , we construct a subgroup that is of type but not of type , and which contains the mapping class group of every compact surface of genus and with non-empty boundary.
Effective computation of the Gelfand-Kirillov dimension
(Edinburgh Mathematical Society, 2008-01-16) Bueso, José L.; Castro Jiménez, Francisco Jesús; Jara, Pascual; Álgebra; FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
In this note we propose an effective method based on the computation of a Gröbner basis of a left ideal to calculate the Gelfand-Kirillov dimension of modules.
Special issue on effective methods in rings of differential operators: Foreword by the guest editors
(Elsevier, 2001-12) Castro Jiménez, Francisco Jesús; Sendra Pons, Juan Rafael; Álgebra; FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
Singularities of the hypergeometric system associated with a monomial curve
(American Matematical Society, 2003-05-29) Castro Jiménez, Francisco Jesús; Takayama, Nobuki; Álgebra; FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
We compute, using D-module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems.
Logarithmic Comparison Theorem and some Euler homogeneous free divisors
(American Mathematical Society, 2004-11-01) Castro Jiménez, Francisco Jesús; Ucha Enríquez, José María; Álgebra; FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
Let D,x be a free divisor germ in a complex manifold X of dimension n>2. It is an open problem to find out which are the properties required for D,x to satisfy the so-called Logarithmic Comparison Theorem (LCT),that is, when the complex of logarithmic differential forms computes the cohomology of the complement of D,x. We give a family of Euler homogeneous free divisors which, somewhat unexpectedly, does not satisfy the LCT.
On the b-function of hypergeometric ideals associated with some space curves
(Real Academia de Ciencias, 2018) Castro Jiménez, Francisco Jesús; Cobo Pablos, M. Helena; Álgebra; FQM333: Algebra Computacional en Anillos no Conmutativos y Aplicaciones
In this paper we study the b-function with respect to a weight vector, associated to a hypergeometric ideal HA( ), with A of the form (1, p, q) and any complex number.
Tame Galois realizations of GL(2) (F-l) over Q
(Elsevier, 2009-05) Arias de Reyna Domínguez, Sara; Vila, Nuria; Álgebra; FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
This paper concerns the tame Galois inverse problem. For each prime number ℓ we construct infinitely many semistable elliptic curves over with good supersingular reduction at ℓ. The Galois action on the ℓ-torsion points of these elliptic curves provides tame Galois realizations of over .
Toward differentiation and integration between Hopf algebroids and Lie algebroids
(Universitat Autònoma de Barcelona, 2023-01-16) Ardizzoni, Alessandro; El Kaoutit, Laiachi; Saracco, Paolo; Álgebra
In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie–Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie–Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors from the category of Lie–Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented.
Determinant of the distance matrix of a tree
(University of Vienna, 2024) Briand, Emmanuel; Esquivias Quintero, Luis; Gutierrez, Alvaro; Lillo, Adrian; Rosas Celis, Mercedes Helena; Matemática Aplicada I (ETSII); Álgebra; MICIU/AEI/10.13039/501100011033; Junta de Andalucía, FEDER, PAIDI2020
We present a combinatorial proof of the Graham–Pollak formula for the determinant of the distance matrix of a tree, via sign-reversing involutions and the Lindström–Gessel–Viennot lemma.
Explicit models for perverse sheaves, II
(Springer, 2008) Gudiel Rodríguez, Félix; Narváez Macarro, Luis; Matemática Aplicada I (ETSII); Álgebra
In this paper we show that any p-perverse sheaf on an arbitrary stratified topological space ( p is a perversity function) is functorially determined by a system of usual sheaves on the open sets Ur (r ≥ 0) and certain gluing data, where Ur is the union of strata of perversity ≤ r.
On ν-quasiordinary surface singularities and their resolution
(Springer, 2024-08-05) Aroca, F.; Tornero Sánchez, José María; Álgebra; FQM218: Singularidades, Geometría Algebraica Aritmética, Grupos y Homotopía
Quasiordinary power series were introduced by Jung at the beginning of the 20th century, and were not paid much attention until the work of Lipman and, later on, Gao. They have been thoroughly studied since, as they form a very interesting family of singular varieties, whose properties (or at least many of them) can be encoded in a discrete set of integers, much as what happens with curves. Hironaka proposed a generalization of this concept, the so-called ν-quasiordinary power series, which has not been examined in the literature in such detailed way. This paper explores the behavior of these series under the resolution process in the surface case.

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