In this paper we study Jordan algebras having nonzero local algebras that satisfy the property of being Lesieur-Croisot (i.e., being orders in nondegenerate Jordan algebras of finite capacity). We will prove that the set of the elements of a nondegenerate Jordan algebra at which the local algebra is Lesieur-Croisot is an ideal.

Abstract. In this paper we prove that the Kostrikin radical of an extended affine Lie algebra of reduced type coincides with the center of its core, and use this characterization to get a type-free description of the core of such algebras. As a consequence we get that the core of an extended affine Lie algebra of reduced type is invariant under the automorphisms of the algebra.

We study the sets of elements of Jordan pairs whose local Jordan algebras are Lesieur-Croisot algebras, that is, classical orders in nondegenerate Jordan algebras with finite capacity. It is then proved that, if the Jordan pair is nondegenerate, the set of its Lesieur-Croisot elements is an ideal of the Jordan pair.

We classify two classes of B2-graded Lie algebras which have a second compatible grading by an abelian group Λ: (a) Λ-gradedsimple, Λ torsion-free and (b) division-Λ-graded. Our results describe the centreless cores of a class of affine reflection Lie algebras, hence apply in particular to the centreless cores of extended affine Lie algebras, the so-called Lie tori, for which we recover results of Allison-Gao and Faulkner. Our classification (b) extends a recent result of Benkart-Yoshii.Both classifications are consequences of a new description of Jordan algebras covered by a triangle, which correspond to these Lie algebras via the Tits-Kantor-Koecher construction. The Jordan algebra classifications follow from our results on graded-triangulated Jordan triple systems. They generalize work of McCrimmon and the first author as well as the Osborn-McCrimmon-Capacity-2-Theorem in the ungraded case.

No abstract