Lie tori of type B_2 and graded-simple Jordan structures covered by a triangle

TL;DR

This paper classifies certain B_2-graded Lie algebras with additional abelian group gradings and describes related Jordan algebra structures, extending previous classifications and connecting Lie and Jordan algebra theories.

Contribution

It provides a new classification of B_2-graded Lie algebras with compatible gradings and describes Jordan algebras covered by a triangle, extending prior work.

Findings

Classification of graded-simple Lie algebras for torsion-free A

Description of division-A-graded Lie algebras

Generalization of Jordan algebra classifications

Abstract

We classify two classes of B_2-graded Lie algebras which have a second compatible grading by an abelian group A: (a) graded-simple Lie algebras for A torsion-free and (b) division-A-graded Lie algebras. 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.