Polynomial Identities and Nonidentities of Split Jordan Pairs

We show that split Jordan pairs over rings without 2-torsion can be distinguished by polynomial identities with integer coefficients. In particular, this holds for simple finite-dimensional Jordan pairs over algebraically closed fields of characteristic not 2. We also generalize results of Drensky and Racine and of Rached and Racine on polynomial identities of, respectively, Jordan algebras and Jordan triple systems.     

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”     

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.     

On the Stability of Jordan *-Derivation Pairs

In this article, the Hyers–Ulam stability of Jordan *-derivation pairs for the Cauchy additive functional equation and the Cauchy additive functional inequality is proved. A fixed point method to establish of the stability and the superstability for Jordan *-derivation pairs is also employed.     

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.     

Homotope polynomial identities in prime Jordan systems

We prove an analogue of the Posner-Rowen theorem for strongly prime Jordan pairs and triple systems: the central closure of a strongly prime Jordan system satisfying a homotope polynomial identity is simple with finite capacity. We also prove that if a Jordan system satisfies a homotope polynomial identity it also satisfies a strict homotope polynomial identity.     

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.     

Track formulas and derivations in simple jordan pairs

In finite dimensional Lie algebras, Jordan algebras, and other algebraic structures the study of derivations has been facilitated by having a nontrivial trace formula on hand (see for example [?]) . Tuere is no common pattern in proving these trace formulas, they all depend on the underlying structures. In this note we derive such a trace formula for finite dimensional central simple Jordan pairs. We use it to determine all derivations the Killing form and the dimensions of the derivation algebras of the Jordan pairs. Dur primary tool is a Trace Reduction Formula.     

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.     

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. This work has been partially supported by 1) the “Plan Andaluz de Investigación y Desarrollo Tecnológico” with project no. 1027 and 2) the DGICYT with project no. PB93-0990     

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 forms for mutually annihilating nilpotent pairs

In this paper we completely characterize all possible pairs of Jordan canonical forms for mutually annihilating nilpotent pairs, i.e. pairs (A,B) of nilpotent matrices such that AB=BA=0.     

Outer inheritance in jordan systems 1

We show that outer ideals of Jordan algebras, pairs and triple systems inherit nondegeneracy, strong primeness and primitivity.     

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).     

A geometric construction of Moufang planes

An algebraic construction of degree 3 Jordan algebras (including the exceptional one) as trace 0 elements in a degree 4 Jordan algebra is translated to give a geometric construction of Barbilian planes coordinatized by composition algebras (including the Moufang plane) as skew polar line pairs and points on the quadratic surface determined by a polarity of projective 3-space over a smaller composition algebra.     

Inversion of regular analytic matrix functions: Local Smith form and subspace duality

This paper analyzes the relation between the local rank-structure of a regular analytic matrix function and the one of its inverse function. The local rank factorization (lrf) of a matrix function is introduced, which characterizes extended canonical systems of root functions and the local Smith form. An interpretation of the local rank factorization in terms of Jordan chains and Jordan pairs is provided. Duality results are shown to hold between the subspaces associated with the lrf of the matrix function and the one of its reduced adjoint.     

On Polynomial Identities in Associative and Jordan Pairs

We prove that a Jordan system satisfies a polynomial identity if and only if it satisfies a homotope polynomial identity. In the obtention of that result, we also prove an analogue for associative pairs with involution of Amitsur’s theorem on associative algebras satisfying a polynomial identity with involution.     

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.