Dolores Martín Barquero

Socle theory for Leavitt path algebras of arbitrary graphs

The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate the socle of a Leavit t path algebra, extending so the results for row-finite graphs in the previous paper (12) ( but with different methods). A concrete description of the socle of a Leavitt path algebra i s obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field. New proofs of the Graded Uniqueness and of the Cuntz-Krieger Uniqueness Theorems are given, shorthening significantly the original ones.

Using Steinberg algebras to study decomposability of Leavitt path algebras

Abstract Given an arbitrary graph E we investigate the relationship between E and the groupoid G E {G_{E}} . We show that there is a lattice isomorphism between the lattice of pairs ( H , S ) {(H,S)} , where H is a hereditary and saturated set of vertices and S is a set of breaking vertices associated to H, onto the lattice of open invariant subsets of G E ( 0 ) {G_{E}^{(0)}} . We use this lattice isomorphism to characterise the decomposability of the Leavitt path algebra L K ⁢ ( E ) {L_{K}(E)} , where K is a field. First we find a graph condition to characterise when an open invariant subset of G E ( 0 ) {G_{E}^{(0)}} is closed. Then we give both a graph condition and a groupoid condition each of which is equivalent to L K ⁢ ( E ) {L_{K}(E)} being decomposable in the sense that it can be written as a direct sum of two nonzero ideals. We end by relating decomposability of a Leavitt path algebra with the existence of nontrivial central idempotents. In fact, all the nontrivial central idempotents can be described.

Extreme cycles. The center of a Leavitt path algebra

There are fourteen ne gradings on the exceptional Lie algebra e6 over an algebraically closed eld of zero characteristic. We provide their descriptions and a proof that any ne grading is equivalent to one of them. 2010 Mathematics Subject Classication: 17B25, 17B70.There are fourteenfine gradings on the exceptional Lie algebra e6 over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that any fine grading is equivalent to one of them.In this paper we prove mixed norm estimates for Riesz transforms on the group SU(2). From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the key ingredients in order to determine the center of a Leavitt path algebra. Our work will rely on our previous approach to the center of a prime Leavitt path algebra.In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.We show that the product BMO space can be characterized by iterated commutators of a large class of Calderon-Zygmund operators. This result followsfrom a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hytonens representation theorem of Calderon-Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.In this paper we survey some results on the Dirichlet problem for nonlocal operators of the form. We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators L in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the field.

Centers of Path Algebras, Cohn and Leavitt Path Algebras

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. First 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. In the final section we describe the center of a prime graph C

On the gauge action of a Leavitt path algebra

^*

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

∗-algebra for a row-finite graph.

ATLAS OF LEAVITT PATH ALGEBRAS OF SMALL GRAPHS

We start with the Lorentz algebra \(L = \mathfrak{o}_{\mathbb{R}}(1,3)\) over the reals and find a suitable basis B relative to which the structure constants are integers. Thus, we consider the \(\mathbb{Z}\)-algebra \(L_{\mathbb{Z}}\) which is free as a \(\mathbb{Z}\)-module and its \(\mathbb{Z}\)-basis is B. This allows us to define the Lorentz type algebra \(L_{R}:= L_{\mathbb{Z}} \otimes _{\mathbb{Z}}R\) over any (unital and commutative) ring R.It is well known that the real Lorentz algebra is simple, however its complexification is not. We study under what conditions on the ground ring of scalars the Lorentz type algebra is simple (for a suitable notion of simplicity).In this paper we study the ideal structure of Lorentz type algebras over rings. We use the notion of basic ideal and that of free ideal to study conditions under which the Lorentz type algebras are basically simple and freely simple. If the ground ring of scalars is a field, both notions of simplicity agree with the usual one.

Computing the socle 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