arXiv subjects

Dolores Martín Barquero

Publications and source records attributed to Dolores Martín Barquero.

28 records · Page 2Linked to original sources

arXiv2020

Chains in evolution algebras

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.

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arXiv2020

Invariants ideals in Leavitt Path algebras

It is known that the ideals of a Leavitt path algebra $L_K(E)$ generated by $\Pl(E)$, by $\Pc(E)$ or by $\Pec(E)$ are invariant under isomorphism. Though the ideal generated by $\Pb(E)$ is not invariant we find its \lq\lq natural\rq\rq\ replacement (which is indeed invariant): the one generated by the vertices of $\Pbp$ (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 $\tops$ 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 $\tops$ 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 $\Pl$, 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 $\soc^{(1)}(\ )\subset\soc^{(2)}(\ )\subset\cdots$ 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.

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arXiv2020

On the centroid of a Leavitt path algebra

We describe the centroid of some Leavitt path algebras. More precisely, we show that for Leavitt path algebras over a field $K$ that are simple its centroid is isomorphic to $K$, and for prime Leavitt path algebras its centroid is isomorphic to $K$ except if the graph is a row-finite comet, in which case the centroid is isomorphic to $K[x,x^{-1}]$.

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arXiv2019

Ternary mappings of triangular algebras

We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of triangular algebras are also introduced and studied.

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arXiv2018

Squares and associative representations of two dimensional evolution algebras

We associate an square to any two dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behaviour of the algebra. We determine the identities of degrees at most four, as well as derivations and automorphisms. We look at the group of automorphisms as an algebraic group, getting in this form a new algebraic invariant. The study of associative representations of evolution algebras is also started and we get faithful representations for most two-dimensional evolution algebras. In some cases we prove that faithful commutative and associative representations do not exist, giving raise to the class of what could be termed as "exceptional" evolution algebras (in the sense of not admitting a monomorphism to an associative algebra with deformed product).

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arXiv2017

Classification of Leavitt path algebras with two vertices

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, $\det(N'_E)$ (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.

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arXiv2016

Using the Steinberg algebra model to determine the center of any Leavitt path algebra

Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient is a characterization of compact open invariant subsets of the unit space of the graph groupoid in terms of the underlying graph: an open invariant subset is compact if and only if its associated hereditary and saturated set of vertices satisfies Condition (F). We also give a basis of the center. Its cardinality depends on the number of minimal compact open invariant subsets of the unit space.

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arXiv2013

Centers of path algebras, Cohn and Leavitt path algebras

We study the center of several types of path algebras. We start with the path algebra $KE$ and prove that if the number of vertices is infinite then the center is zero. Otherwise, it coincides with the field $K$ 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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arXiv2013

On the gauge action of a Leavitt path algebra

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.

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arXiv2011

Atlas of Leavitt Path Algebras of small graphs

The aim of this work is the description of the isomorphism classes of all Leavitt path algebras coming from graphs satisfying Condition (Sing) with up to three vertices. In particular, this classification recovers the one achieved by Abrams et al. in the case of graphs whose Leavitt path algebras are purely infinite simple. The description of the isomorphism classes is given in terms of a series of invariants including the K_0 group, the socle, the number of loops with no exits and the number of hereditary and saturated subsets of the graph.

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