Cándido Martín González

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.

On the Cartan-Jacobson Theorem

Abstract The well-known Cartan–Jacobson theorem claims that the Lie algebra of derivations of a Cayley algebra is central simple if the characteristic is not 2 or 3. In this paper we have studied these two cases, with the following results: if the characteristic is 2, the theorem is also true, but, if the characteristic is 3, the derivation algebra is not simple. We have also proved that in this last case, there is a unique nonzero proper seven-dimensional ideal, which is a central simple Lie algebra of type A2, and the quotient of the derivation algebra modulo this ideal turns out to be isomorphic, as a Lie algebra, to the ideal itself. The original motivation of this work was a series of computer-aided calculations which proved the simplicity of derivation algebras of Cayley algebras in the case of characteristic not 3. These computations also proved the existence of a unique nonzero proper ideal (which turns out to be seven-dimensional) in the algebra of derivations of split Cayley algebras in characteristic 3.

FINE GRADINGS ON EXCEPTIONAL SIMPLE LIE SUPERALGEBRAS

The fine abelian group gradings on the simple exceptional classical Lie superalgebras over algebraically closed fields of characteristic 0 are determined up to equivalence.

Strongly prime alternative pairs with minimal inner ideals

In this paper, we use the known classification of the finite capacity simple alternative pairs and the version of the Litoff Theorem for Jordan pairs to describe all the strongly prime alternative pairs with nonzero socle. We study the inheritance of some properties (primeness, nondegenerancy,…) when passing from the original alternative pair to the symmetrized pair. Thus, we can apply Jordan theoretical results to the alternative case.

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.

Associative Dialgebras from a Structural Viewpoint

In this note we study associative dialgebras proving that the most interesting such structures arise precisely when the algebra is not semiprime. In fact the presence of some “perfection” property (simpleness, primitiveness, primeness, or semiprimeness) imply that the dialgebra comes from an associative algebra with both products ⊣ and ⊢ identified. We also describe the class of zero-cubed algebras and apply its study to that of dialgebras. Finally, we describe two-dimensional associative dialgebras.

Special jordan H*-triple systems

This work, jointly with [9], completes the structure theory and classification of the Jordan H *- triple systems. The problem of describing the Jordan H *-triple systems is reduced in [5] to that of describing the topologically simple ones. Ruling out the finite-dimensional case, we have that any of these H *-triples has an underlying triple system structure of quadratic type (and these can be fully described), or it is the H *-triple system associated to the odd part of a topologically simple Z2-graded Jordan H *-algebra, whose classification is given in [13].

Prime Alternative Triple Systems

In [8] Loos obtains a structure theorem for finite dimensional simple alternative triple systems over an algebraically closed field of characteristic ≠ 2. This theorem has been generalized in [10] (resp. [9]) for simple alternative triple systems (resp. alternative pairs) containing a maximal tripotent (resp. a maximal idempotent). Other classes of alternative triple systems without maximal tripotent over a Hilbert space (real or complex) have been characterized in [2] and [3] where the simplicity is replaced by the topological simplicity and other hypothesis involving an involution and the inner product. In this paper, we obtain a description of nondegenerate prime alternative triple systems containing a maximal tripotent in terms of its central closure. A socle theory for prime nondegenerate alternative triple systems with maximal tripotent is established in a forthcoming paper [4].

Jordan H*-triple systems

This work, jointly with [7], gives a complete classification of Jordan H*-triple systems. There, the infinite—dimensional topologically simple special nonquadratic Jordan H*-triple systems are fully described in terms of the odd part of a ℤ2—graded H*—algebra. Here we complete the structure theory endowing to any simple finite—dimensional real Jordan triple system, of an H*—structure, essentialy unique, and determining the ones of quadratic type.