LIE-ALGEBRAS GRADED BY 3-GRADED ROOT SYSTEMS AND JORDAN PAIRS COVEREDBY GRIDS

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 i n R, the grading is induced by the adjoint action of a split Cartan su balgebra of a finite-dimensional simple Lie subalgebra of L with root system R, and L is generated by the homogeneous subspaces of nonzero d egree. This class of Lie algebras was introduced and studied by S. Ber man 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. Benka rt 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 des cribed 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 c entral extension of the Tits-Kantor-Koecher algebra of a Jordan pair c overed by a grid whose associated 3-graded root system is isomorphic t o 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 describ e their Tits-Kantor-Koecher algebras. One of the advantages of this ap proach is that it works over rings containing 1/2 and 1/3, and also fo r infinite root systems. Another application is the description of Slo dowy's intersection matrix algebras arising from multiply-affinized Ca rtan matrices.