Towards a goldie theory for jordan pairs

A Goldie theory for Jordan pairs is started in this paper. We introduce a notion of order in linear Jordan pairs and study orders in nondegenerate linear Jordan pairs with descending chain condition on principal inner ideals. This work has been supported by DGICYT Grant PB93-0990 and by the “Convenio Marco de Cooperación Hispano-Marroquí”     

Local and Subquotient Inheritance of Simplicity in Jordan Systems

In this paper we prove that the local algebras of a simple Jordan pair are simple. Jordan pairs all of which local algebras are simple are also studied, showing that they have a nonzero simple heart, which is described in terms of powers of the original pair. Similar results are given for Jordan triple systems and algebras. Finally, we characterize the inner ideals of a simple pair which determine simple subquotients, answering the question posed by O. Loos and E. Neher (1994, J. Algebra166, 255–295).     

Wedderburnzerlegung für Jordan-Paare

In the first section we summarize some properties of Jordan pairs. Then we state some results about some groups defined by Jordan pairs.In the next section we construct a Lie algebra to a Jordan pair. This construction is a generalization of the wellknown Koecher-Tits-construction. We calculate the radical of this Lie algebra in terms of the given Jordan pair.In the last section we prove a Wedderburn decomposition theorem for Jordan pairs in the characteristic zero case. Some special cases in arbitrary characteristic have been shown by R. Carlson (see [5]). Also we show that any two such decompositions are conjugate under a certain group of automorphism. Analogous theorems will be shown for Jordan Triples.     

Special jordan pairs

There are some important Jordan pairs contained in the free associative pair, e.g. the free special Jordan pair and the Jordan pair of all symmetric elements under the reversal involution. We study the connection between these two Jordan pairs in the case when the ring of scalars contains ½.     

Elementary groups and stability for Jordan pairs

A stability condition for Jordan algebras and, more generally, for Jordan pairs is introduced which extends the first stable range condition for associative and alternative rings. It is formulated in terms of the projective elementary group of a Jordan pair and has most of the properties expected from the associative case. The proofs involve a new kind of fourfold quasi-inverse.     

Additive Derivations on Jordan Banach Pairs

Abstract

We show the invariance of “almost all” primitive ideals under additive derivations on a Jordan Banach pair and we extend the well known result of Johnson and Sinclair to the Jordan Banach pairs framework.     

Jordan centroids

Central simple triples are important for the classification of prime Jordan triples of Clifford type in arbitrary characterstics. For triples and pairs (or even for unital Jordan algebras of characteristic 2), there is no workable notion of center, and the concept of “central simple” system must be understood as “centroid-simple”. The centroid of a Jordan system (algebra, triple, or pair) consists of the “natural” scalars for that system: the largest unital, commutative ring Γ such that the system can be considered as a quadratic Jordan system over Γ. In this paper we will characterize the centroids of the basic simple Jordan algebras, triples, and pairs. (Consideration of the tangled ample outer ideals in Jordan algebras of quadratic forms will be left to a separate paper.) A powerful tool is the Eigenvalue Lemma, that a centroidal transformation on a prime system over φ which has an eigenvalue α in φ must actually be scalar multiplication by α. An important consequence is that a prime system over φ with reduced elements P_xJ = φx (or which grows reduced elements under controlled conditions) must already be central, Γ = φ.     

Integrable and non-integrable structures in Einstein-Maxwell equations with Abelian isometry group <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

Using the fact that absolute zero divisors in Jordan pairs become Lie sandwiches of the corresponding Tits–Kantor–Koecher Lie algebras, we prove local nilpotency of the McCrimmon radical of a Jordan system (algebra, triple system, or pair) over an arbitrary ring of scalars. As an application, we show that simple Jordan systems are always nondegenerate.     

Zeros of Gegenbauer-Sobolev Orthogonal Polynomials: Beyond Coherent Pairs

Iserles et al. (J. Approx. Theory 65:151–175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt (J. Approx. Theory 114:115–140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt, when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair. This research is supported by grants from CNPq and FAPESP.     

Dynkin diagrams and short Peirce gradings of Kantor pairs

In a recent article with Oleg Smirnov, we defined short Peirce (SP) graded Kantor pairs. For any such pair P, we defined a family, parameterized by the Weyl group of type BC₂, consisting of SP-graded Kantor pairs called Weyl images of P. In this article, we classify finite dimensional simple SP-graded Kantor pairs over an algebraically closed field of characteristic 0 in terms of marked Dynkin diagrams, and we show how to compute Weyl images using these diagrams. The theory is particularly attractive for close-to-Jordan Kantor pairs (which are variations of Freudenthal triple systems), and we construct the reflections of such pairs (with nontrivial gradings) starting from Jordan pairs of matrices.     

On the convergence of the Weiszfeld algorithm

In this work we analyze the paper “Brimberg, J. (1995): The Fermat-Weber location problem revisited. Mathematical Programming 71, 71–76” which claims to close the question on the conjecture posed by Chandrasekaran and Tamir in 1989 on the convergence of the Weiszfeld algorithm. Some counterexamples are shown to the proofs showed in Brimberg’s paper. Received: January 1999 / Accepted: December 2001?Published online April 12, 2002 RID="*" ID="*"Partially supported by PB/11/FS/97 of Fundación Séneca of the Comunidad Autónoma de la Región de Murcia RID="**" ID="**"Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica (I+I+D), project TIC2000-1750-C06-06 RID="*" RID="**"     

On Iteration Semiring-Semimodule Pairs

Conway semiring-module pairs and iteration semiring-semimodule pairs were shown to provide an axiomatic basis to automata on ω -words in [Bloom, Esik: Iteration Theories, Springer, 1993]. In this paper, we show that two natural classes of semiring-semimodule pairs, the complete and the bi-inductive semiring-semimodule pairs both give rise to iteration semiring-semimodule pairs. Complete semiring-semimodule pairs are defined by infinite sums and products, while a bi-inductive semiring-semimodule pair is an ordered semiring-semimodule pair possessing enough least pre-fixed points and greatest post-fixed points to solve linear inequations. Moreover, we show that when V is idempotent, then a semiring-semimodule pair equipped with a star and an omega operation satisfies the Conway equations (iteration semiring-semimodule pair equations, respectively) if and only if the quemiring associated with (S,V) embeds in a Conway semiring (iteration semiring, respectively).     

Some new closures on orders

For each of the relations “less than or equal to”, “less than”, “covered by”, and “covered by or equal to”, we characterize finite orders (also called posets) with the property that the pair of Galois closure operators induced by the relation in question coincides with the pair of closure operators introduced and applied in our previous paper in 2007. We also consider the “less than or equal to” relation between the set of join-irreducible elements and the set of meet-irreducible elements, and we show that the above-mentioned pairs of closure operators coincide for finite modular lattices.     

Derived Rees Matrix Semigroups as Semigroups of Transformations

An ordered pair (e,f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. Previously [1] we have established that there are four distinct types of skew pairs of idempotents. We have also described (as quotient semigroups of certain regular Rees matrix semigroups [2]) the structure of the smallest regular semigroups that contain precisely one skew pair of each of the four types, there being to within isomorphism ten such semigroups. These we call the derived Rees matrix semigroups. In the particular case of full transformation semigroups we proved in [3] that T_X contains all four skew pairs of idempotents if and only if |X| ≥ 6. Here we prove that T_X contains all ten derived Rees matrix semigroups if and only if |X| ≥ 7.     

Automatic continuity of homomorphisms and fixed points on metric compacta

Let G₂(ℝ) × Sp₆(ℝ) and G₂(ℝ) × F₄(ℝ) be split dual pairs in split E₇(ℝ) and E₈(ℝ), respectively. It is known that the exceptional correspondences for these dual pairs are functorial on the level of infinitesimal characters. In this paper we show that these dual pair correspondences are functorial for the minimal K-types of principal series representations. Gordan Savin’s research is partially supported be NSF Grant no. DMS-0551846     

Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

It is shown that any generalized Jordan (triple-)derivation on a 2–torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2–torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.     

A counterexample to the bold conjecture

A pair of vertices (x,y) of a graph G is an o-critical pair if o(G + xy) > o(G), where G + xy denotes the graph obtained by adding the edge xy to G and o(H) is the clique number of H. The o-critical pairs are never edges in G. A maximal stable set S of G is called a forced color class of G if S meets every o-clique of G, and o-critical pairs within S form a connected graph. In 1993, G. Bacsó raised the following conjecture which implies the famous Strong Perfect Graph Conjecture: If G is a uniquely o-colorable perfect graph, then G has at least one forced color class. This conjecture is called the Bold Conjecture. Here we show a simple counterexample to it. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 165–168, 1997     

Coherent pairs of linear functionals on the unit circle

In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if (μ₀,μ₁) is a coherent pair of measures on the unit circle, then μ₀ is a semi-classical measure. Moreover, we obtain that the linear functional associated with μ₁ is a specific rational transformation of the linear functional corresponding to μ₀. Some examples are given.     

Extension of primal-dual interior point algorithms to symmetric cones

 In this paper we show that the so-called commutative class of primal-dual interior point algorithms which were designed by Monteiro and Zhang for semidefinite programming extends word-for-word to optimization problems over all symmetric cones. The machinery of Euclidean Jordan algebras is used to carry out this extension. Unlike some non-commutative algorithms such as the XS+SX method, this class of extensions does not use concepts outside of the Euclidean Jordan algebras. In particular no assumption is made about representability of the underlying Jordan algebra. As a special case, we prove polynomial iteration complexities for variants of the short-, semi-long-, and long-step path-following algorithms using the Nesterov-Todd, XS, or SX directions. Received: April 2000 / Accepted: May 2002 Published online: March 28, 2003 RID="⋆" ID="⋆" Part of this research was conducted when the first author was a postdoctoral associate at Center for Computational Optimization at Columbia University. RID="⋆⋆" ID="⋆⋆" Research supported in part by the U.S. National Science Foundation grant CCR-9901991 and Office of Naval Research contract number N00014-96-1-0704.