Evolution algebras, automorphisms, and graphs

Elduque, Alberto; Labra, Antonieta

doi:10.1080/03081087.2019.1598931

Cited by 18 publications

(24 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For all i ∈ N there exist k ∈ N such that t ki = 0. Then, by (13) we have that f (e i ) = 0 for all i ∈ N as desired. In this case we can prove that f is injective.…”

Section: Preliminary Results For the Proof Of Theorem 33mentioning

confidence: 79%

“…Let i ∈ N. We want to prove that i ∈ Ω ℓ , that is, t il = 0, for some ℓ ∈ N. To fix notation let us rewrite (13) as…”

Section: Preliminary Results For the Proof Of Theorem 33mentioning

confidence: 99%

“…The theory was developed further on in the seminal work [30], where the author show many interesting correspondences between these algebras and subjects like discrete-time Markov chains, graph theory, group theory, and others. We refer the reader to [1][2][3][4][5][6][7][8][9][10][11][12][13][14]25,29,31] and references therein for an overview of recent theoretical results, applications to other fields, and interesting open problems. We point out that, such defined, these algebras are a special class of the genetic algebras introduced by [26].…”

mentioning

confidence: 99%

“…The purpose of our work is to advance in the study of Hilbert evolution algebras by extending results coming from the interplay between evolution algebras and graphs. Such connection has been a subject of current research, see for instance [3,5,12,13], and references therein. One of the open problems related to the subject is to understand the relationship between the evolution algebra determined by a graph and the evolution algebra induced by the symmetric random walk on the same graph.…”

mentioning

confidence: 99%

See 3 more Smart Citations

On Hilbert evolution algebras of a graph

Vidal¹,

Cadavid²,

Rodríguez³

2021

Preprint

View full text Add to dashboard Cite

Evolution algebras are a special class of non-associative algebras exhibiting connections with different fields of Mathematics. Hilbert evolution algebras generalize the concept through a framework of Hilbert spaces. This allows to deal with a wide class of infinitedimensional spaces. In this work we study Hilbert evolution algebras associated to a graph. Inspired in definitions of evolution algebras we define the Hilbert evolution algebra associated to a given graph and the Hilbert evolution algebra associated to the symmetric random walk on a graph. For a given graph, we provide conditions under which these structures are or are not isomorphic. Our definitions and results extend to graphs with infinitely many vertices a similar theory developed for evolution algebras associated to finite graphs.

show abstract

“…For all i ∈ N there exist k ∈ N such that t ki = 0. Then, by (13) we have that f (e i ) = 0 for all i ∈ N as desired. In this case we can prove that f is injective.…”

Section: Preliminary Results For the Proof Of Theorem 33mentioning

confidence: 79%

“…Let i ∈ N. We want to prove that i ∈ Ω ℓ , that is, t il = 0, for some ℓ ∈ N. To fix notation let us rewrite (13) as…”

Section: Preliminary Results For the Proof Of Theorem 33mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

On Hilbert evolution algebras of a graph

Vidal¹,

Cadavid²,

Rodríguez³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In the particular case of an evolution algebra, a complete characterization of such space is still an open question. We recommend to the reader the references [1,6,8,9,15,21] for a good idea of the existing results about this topic.…”

Section: Introductionmentioning

confidence: 99%

On the characterization of the space of derivations in evolution algebras

Casado

Cadavid

Montoya

et al. 2020

Annali di Matematica

View full text Add to dashboard Cite

We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph we prove that the space of derivations is zero. For the remaining families of evolution algebras we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish this task by identifying the null entries of the respective derivation matrix. Our results suggest how strongly the associated graph's structure impacts in the characterization of derivations for a given evolution algebra. Therefore our approach constitutes an alternative to the recent developments in the research of this subject. As an illustration of the applicability of our results we provide some examples and we exhibit the classification of the derivations for non-degenerate irreducible 3-dimensional evolution algebras.

show abstract

Connecting ideals in evolution algebras with hereditary subsets of its associated graph

Casado,

Barquero,

González

et al. 2024

Collect. Math.

View full text Add to dashboard Cite

In this article, we introduce a relation including ideals of an evolution algebra A 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. In particular, the maximal ideals can be determined through maximal hereditary subsets of vertices except for those containing $$A^2$$ A 2 . 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.

show abstract

Evolution algebras, automorphisms, and graphs

Cited by 18 publications

References 8 publications

On Hilbert evolution algebras of a graph

On Hilbert evolution algebras of a graph

On the characterization of the space of derivations in evolution algebras

Connecting ideals in evolution algebras with hereditary subsets of its associated graph

Contact Info

Product

Resources

About