Ottmar Loos

An elemental characterization of strong primeness in Jordan systems

Abstract We give elemental characterizations of strong primeness for Jordan algebras, pairs and triple systems. We use our characterization to study the transfer of strong primeness between a Jordan system and its local algebras and subquotients.

Diagonalization in Jordan pairs

As every student of linear algebra knows, a rectangular matrix over a division ring can be diagonalized by elementary row and column operations. Similarly, there are normal forms for alternating and hermitian matrices which can be achieved by simultaneous row and column operations. Since matrices of this kind form the main examples of Jordan pairs, it is natural to ask whether similar results hold in the Jordan setting. We show that this is in fact the case, study the obstruction to diagonalizability, the defect, and also prove that nondegenerate Jordan pairs admit a rank function sharing many properties with classical matrix rank. Let V= (V+, V) be a Jordan pair and S = {e,, . . . . e,} a set of orthogonal idempotents. The S-diagonal elements are those in xi= 1 V,(e,). Now suppose that V is nondegenerate and satisfies dcc on principal inner ideals. An element x E V” (a = i) is called diagonalizabfe if x is S-diagonal for some S consisting of division idempotents. To see that, for matrices, this is the same as the usual notion of diagonalizability, suppose that V has in addition act on principal inner ideals. Then V contains a frame, that is, a finite set F of orthogonal division idempotents such that V,,(F) = 0. For rectangular or hermitian matrices over division rings, F can be taken to consist of the diagonal matrix units (e,? = Eii) whereas for alternating matrices over a field, e,? = Eli1,2iE,,,zi-, . The Jordan analogues of elementary row and column operations are the inner automorphisms B( V:(e), V;(e)) and /?( V:(e), V,(e)) (where e E F) which generate the group of F-elementary automorphisms of V [6]. Any set of orthogonal division idempotents can be transformed (up to association) into F by an F-elementary automorphism [6, Th. 21. It follows that x is diagonalizable if and only if cp +(x) is F-diagonal for some elementary automorphism cp of V. This shows the equivalence with the usual definition.

Reflection systems and partial root systems

Abstract We develop a general theory of reflection systems and, more specifically, partial root systems which provide a unifying framework for finite root systems, Kac–Moody root systems, extended affine root systems and various generalizations thereof. Nilpotent and prenilpotent subsets are studied in this setting, based on commutator sets and the descending central series. We show that our notion of a prenilpotent pair coincides, for Kac–Moody root systems, with the one defined by Tits in terms of positive systems and the Weyl group.

Steinberg groups for Jordan pairs

Abstract We announce results on projective elementary groups and on Steinberg groups associated to Jordan pairs V with a grading by a locally finite 3-graded root system Φ: The projective elementary group PE ( V ) of V is a group with Φ-commutator relations with respect to appropriately defined root subgroups. Under some mild additional conditions, the Steinberg group associated to PE ( V ) uniquely covers all central extensions of PE ( V ) and is the universal central extension of PE ( V ) if Φ is irreducible and has infinite rank.

Associative and Jordan shift algebras

Let R be the shift algebra, i.e., the associative algebra presented by generators u, v and the relation uv = 1. As N. Jacobson showed, R contains an infinite family of matrix units. In this paper, we describe the Jordan algebra R + and its unital special universal envelope by generators and relations. Moreover, we give a presentation for the Jordan triple system defined on R by P x y = xy*x where * is the involution on R with u* = v. As a consequence, we prove the existence of an infinite rectangular grid in a Jordan triple system V containing tripotents c and d with V 2 (c) = V 2 (d) ○+ (V 2 (c) ∩ V 1 (d)) and V 2 (c) ∩ V 1 (d) ¬= 0