Papers by Dolores Martín Barquero
Journal of Algebraic Combinatorics, Jan 23, 2023
We will study evolution algebras A that are free modules of dimension two over domains. We start ... more We will study evolution algebras A that are free modules of dimension two over domains. We start by making some general considerations about algebras over domains: They are sandwiched between a certain essential D-submodule and its scalar extension over the field of fractions of the domain. We introduce the notion of quasiperfect algebras and we characterize the perfect and quasiperfect evolution algebras in terms of the determinant of its structure matrix. We classify the two-dimensional perfect evolution algebras over domains parametrizing the isomorphism classes by a convenient moduli set.
arXiv (Cornell University), Jan 23, 2007
In this paper we characterize the minimal left ideals of a Leavitt path algebra as those ones whi... more In this paper we characterize the minimal left ideals of a Leavitt path algebra as those ones which are isomorphic to principal left ideals generated by line point vertices, that is, by vertices whose trees do not contain neither bifurcations nor closed paths. Moreover, we show that the socle of a Leavitt path algebra is the two-sided ideal generated by these line point vertices. This characterization allows us to compute the socle of some algebras that arise as the Leavitt path algebra of some row-finite graphs. A complete description of the socle of a Leavitt path algebra is given: it is a locally matricial algebra.
arXiv (Cornell University), Aug 25, 2020
In this work we approach three-dimensional evolution algebras from certain constructions performe... more In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from two-dimensional algebras. Also we introduce two parameters, the annihilator stabilizing index and the socle stabilizing index, which are useful tools in the classification theory of these algebras. Finally, we use moduli sets as a convenient way to describe isomorphism classes of algebras.
arXiv (Cornell University), Aug 10, 2017
We classify row-finite Leavitt path algebras associated to graphs with no more than two vertices.... more We classify row-finite Leavitt path algebras associated to graphs with no more than two vertices. For the discussion we use the following invariants: decomposability, the K 0 group, detpN 1 E q (included in the Franks invariants), the type, as well as the socle, the ideal generated by the vertices in cycles with no exits and the ideal generated by vertices in extreme cycles. The starting point is a simple linear algebraic result that determines when a Leavitt path algebra is IBN. An interesting result that we have found is that the ideal generated by extreme cycles is invariant under any isomorphism (for Leavitt path algebras whose associated graph is finite). We also give a more specific proof of the fact that the shift move produces an isomorphism when applied to any row-finite graph, independently of the field we are considering.
arXiv (Cornell University), Nov 11, 2021
The starting point of this work is that the class of evolution algebras over a fixed field is clo... more The starting point of this work is that the class of evolution algebras over a fixed field is closed under tensor product. This arises questions about the inheritance of properties from the tensor product to the factors and conversely. For instance nondegeneracy, irreducibility, perfectness and simplicity are investigated. The four-dimensional case is illustrative and useful to contrast conjectures so we achieve a complete classification of four-dimensional perfect evolution algebras arising as tensor product of two-dimensional ones. We find that there are 4-dimensional evolution algebras which are the tensor product of two nonevolution algebras.
Epsilon: Revista de la Sociedad Andaluza de Educación Matemática "Thales", 1989
arXiv (Cornell University), Mar 8, 2023
In this article, we introduce a relation including ideals of an evolution algebra and hereditary ... more In this article, we introduce a relation including ideals of an evolution algebra and hereditary subsets of vertices of its associated graph and establish some properties among them. This relation allows us to determine maximal ideals and ideals having the absorption property of an evolution algebra in terms of its associated graph. We also define a couple of order-preserving maps, one from the sets of ideals of an evolution algebra to that of hereditary subsets of the corresponding graph, and the other in the reverse direction. Conveniently restricted to the set of absorption ideals and to the set of hereditary saturated subsets, this is a monotone Galois connection. According to the graph, we characterize arbitrary dimensional finitely-generated (as algebras) evolution algebras under certain restrictions of its graph. Furthermore, the simplicity of finitely-generated perfect evolution algebras is described on the basis of the simplicity of the graph.
Linear Algebra and its Applications, Aug 1, 2021
In this work we approach three-dimensional evolution algebras from certain constructions performe... more In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from two-dimensional algebras. Also we introduce two parameters, the annihilator stabilizing index and the socle stabilizing index, which are useful tools in the classification theory of these algebras. Finally, we use moduli sets as a convenient way to describe isomorphism classes of algebras.
arXiv (Cornell University), 2023
The notion of conservative algebras appeared in a paper of Kantor in 1972. Later, he defined the ... more The notion of conservative algebras appeared in a paper of Kantor in 1972. Later, he defined the conservative algebra W (n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. If n > 1, then the algebra W (n) does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). It looks like that W (n) in the theory of conservative algebras plays a similar role with the role of gl n in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra W (n) for some n ∈ N. The present paper is a part of a series of papers, which dedicated to the study of the algebra W (2) and its principal subalgebras.
Kyoto Journal of Mathematics, Jun 1, 2015
We introduce a revised notion of gauge action in relation with Leavitt path algebras. This notion... more We introduce a revised notion of gauge action in relation with Leavitt path algebras. This notion is based on group schemes and captures the full information of the grading on the algebra as it is the case of the gauge action of the graph C *-algebra of the graph.
arXiv (Cornell University), Apr 18, 2022
We introduce certain functors from the category of commutative rings (and related categories) to ... more We introduce certain functors from the category of commutative rings (and related categories) to that of Z-algebras (not necessarily associative or commutative). One of the motivating examples is the Leavitt path algebra functor R → L R (E) for a given graph E. Our goal is to find "descending" isomorphism results of the type: if F , G are algebra functors and K ⊂ K ′ a field extension, under what conditions an isomorphism F (K ′) ∼ = G (K ′) of K ′-algebras implies the existence of an isomorphism F (K) ∼ = G (K) of Kalgebras? We find some positive answers to that problem for the so-called "extension invariant functors" which include the functors associated to Leavitt path algebras, Steinberg algebras, path algebras, group algebras, evolution algebras and others. For our purposes, we employ an extension of the Hilbert's Nullstellensatz Theorem for polynomials in possibly infinitely many variables, as one of our main tools. We also remark that for extension invariant functors F , G , an isomorphism F (H) ∼ = G (H), for some Hopf K-algebra H, implies the existence of an isomorphism F (S) ∼ = G (S) for any commutative and unital Kalgebra S.
arXiv (Cornell University), Apr 18, 2022
We will study evolution algebras A which are free modules of dimension 2 over domains. Furthermor... more We will study evolution algebras A which are free modules of dimension 2 over domains. Furthermore, we will assume that these algebras are perfect, that is A 2 = A. We start by making some general considerations about algebras over domains: they are sandwiched between a certain essential D-submodule and its scalar extension over the field of fractions of the domain. We introduce the notion of quasiperfect algebras and modify slightly the procedure to associate a graph to an evolution algebra over a field given in [7]. Essentially, we introduce color in the connecting arrows, depending on a suitable criterion related to the squares of the natural basis elements. Then we classify the algebras under scope parametrizing the isomorphic classes by convenient moduli.
... Métodos matemáticos: álgebra lineal y geometría. Información General. Autores: Pablo Gabriel ... more ... Métodos matemáticos: álgebra lineal y geometría. Información General. Autores: Pablo Gabriel Alberca Bjrregaard, Dolores Martín Barquero; Editores: Archidona (Málaga) : Aljibe, 2001; Año de publicación: 2001; País: España; Idioma: Español; ISBN : 84-9700-027-7. ...
arXiv (Cornell University), Dec 21, 2022
The Gelfand-Kirillov dimension is a well established quantity to classify the growth of infinite ... more The Gelfand-Kirillov dimension is a well established quantity to classify the growth of infinite dimensional algebras. In this article we introduce the algebraic entropy for path algebras. For the path algebras, Leavitt path algebras and the path algebra of the extended (double) graph, we compare the Gelfand-Kirillov dimension and the entropy. We give a complete classification of path algebras over finite graphs by dimension, Gelfand-Kirillov dimension and algebraic entropy. We show indeed how these three quantities are dependent on cycles inside the graph. Moreover we show that the algebraic entropy is conserved under Morita equivalence. In addition we give several examples of the entropy in path algebras and Leavitt path algebras.
Epsilon: Revista de la Sociedad Andaluza de Educación Matemática "Thales", 1987
arXiv (Cornell University), Jul 19, 2021
For an arbitrary field K and a family of inner products in a Kvector space V of arbitrary dimensi... more For an arbitrary field K and a family of inner products in a Kvector space V of arbitrary dimension, we study necessary and sufficient conditions in order to have an orthogonal basis relative to all the inner products. If the family contains a nondegenerate element plus a compatibility condition, then under mild hypotheses the simultaneous orthogonalization can be achieved. So we investigate several constructions whose purpose is to add a nondegenerate element to a degenerate family and we study under what conditions the enlarged family is nondegenerate.
arXiv (Cornell University), Mar 2, 2021
Evolution algebras with one dimensional square are classified using the theory of inner product s... more Evolution algebras with one dimensional square are classified using the theory of inner product spaces. More precisely, for A an evolution algebra with dim(A 2) = 1 and a a generator of A 2 , the product of A is given by xy = x, y a. Three broad classes of algebras are obtained: (1) a ∈ Ann(A); (2) a / ∈ Ann(A) and a is isotropic relative to •, • ; (3) a / ∈ Ann(A) and a is anisotropic relative to •, • .
arXiv (Cornell University), Jun 23, 2020
It is known that the ideals of a Leavitt path algebra L K (E) generated by P l (E), by P c (E) or... more It is known that the ideals of a Leavitt path algebra L K (E) generated by P l (E), by P c (E) or by P ec (E) are invariant under isomorphism. Though the ideal generated by P b ∞ (E) is not invariant we find its "natural" replacement (which is indeed invariant): the one generated by the vertices of P b ∞ p (vertices with pure infinite bifurcations). We also give some procedures to construct invariant ideals from previous known invariant ideals. One of these procedures involves topology, so we introduce the DCC topology and relate it to annihilators in the algebraic counterpart of the work. To be more explicit: if H is a hereditary saturated subset of vertices providing an invariant ideal, its exterior ext(H) in the DCC topology of E 0 generates a new invariant ideal. The other constructor of invariant ideals is more categorical in nature. Some hereditary sets can be seen as functors from graphs to sets (for instance P l , etc). Thus a second method emerges from the possibility of applying the induced functor to the quotient graph. The easiest example is the known socle chain Soc (1) () ⊂ Soc (2) () ⊂ • • • all of which are proved to be invariant. We generalize this idea to any hereditary and saturated invariant functor. Finally we investigate a kind of composition of hereditary and saturated functors which is associative.
arXiv (Cornell University), Sep 19, 2012
This paper is devoted to the study of the center of several types of path algebras associated to ... more This paper is devoted to the study of the center of several types of path algebras associated to a graph E over a field K. In a first step we consider the path algebra KE and prove that if the number of vertices is infinite then the center is zero; otherwise, it is K, except when the graph E is a cycle in which case the center is K[x], the polynomial algebra in one indeterminate. Then we compute the centers of prime Cohn and Leavitt path algebras. A lower and an upper bound for the center of a Leavitt path algebra are given by introducing the graded Baer radical for graded algebras.
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Papers by Dolores Martín Barquero