Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids

Abstract

Let R be an irreducible root system. A Lie Algebra L is called graded by R if L is graded with grading group the root lattice of R such that the only nonzero homogeneous subspaces of L have degree 0 or a root in R , the grading is induced by the adjoint action of a split Cartan subalgebra of a finite-dimensional simple Lie subalgebra of L with root system R , and L is generated by the homogeneous subspaces of nonzero degree. This class of Lie algebras was introduced and studied by S. Berman and R. Moody in Invent. Math. 108 (1992), where, in particular, a classification up to central equivalence is given in the simply-laced case. The doubly-laced cases have recently been classified by G. Benkart and E. Zelmanov. Let R be a 3-graded root system, i.e., R is not of type E_8, F_4 or G_2. In this paper, Lie algebras graded by R are described in a unified way, without case-by-case considerations. Namely, it is shown that a Lie algebra L is 3-graded if and only if L is a central extension of the Tits-Kantor-Koecher algebra of a Jordan pair covered by a grid whose associated 3-graded root system is isomorphic to R . This result is then used to classify Lie algebras graded by R : we give the classification of Jordan pairs covered by a grid and describe their Tits-Kantor-Koecher algebras. One of the advantages of this approach is that it works over rings containing 1/2 and 1/3, and also for infinite root systems. Another application is the description of Slodowy's intersection matrix algebras arising from multiply-affinized Cartan matrices.