LIE SUPERALGEBRAS GRADED BY THE ROOT SYSTEM B(m,n) Georgia Benkart 1 Alberto Elduque 2 September 3, 2001 We determine the Lie superalgebras that are graded by the root system B(m, n) of the orthosymplectic Lie superalgebra (2m +1, 2n). §1. Introduction The notion of a Lie algebra graded by a finite root system has proved to be a very useful one for studying many important classes of Lie algebras such as the affine and toroidal Lie algebras and various generalizations of them: for example, the extended affine Lie algebras of [AABGP] or the intersection matrix Lie algebras of [Sl], which arise in the study of singularities. Any finite-dimensional simple Lie algebra of characteristic zero containing an ad-nilpotent element (or equivalently a copy of sl 2 ) is a Lie algebra graded by a finite root system (see [S]). Thus, this concept encompasses a diverse array of Lie algebras under one unifying theme - they all exhibit a grading by a finite (possibly nonreduced) root system. Our focus here and in [BE2] is on Lie superalgebras graded by the root systems of the finite-dimensional basic classical simple Lie superalgebras A(m, n), B(m, n), C(n), D(m, n), D(2, 1; α), (α =0, −1), G(3), and F(4) (a standard reference for re- sults on simple superalgebras is Kac’s ground-breaking paper [K1]). In this work, we investigate Lie superalgebras graded by the root system B(m, n) of the orthosym- plectic Lie superalgebra osp(2m +1, 2n). These superalgebras differ from the others by their complexity and most closely resemble the Lie algebras graded by the nonre- duced root systems BC r . The others have some similarities with the Lie algebras graded by the reduced root systems, but there are many striking differences as seen in [BE2]. 2000 Mathematics Subject Classification. Primary 17A70. 1 Support from National Science Foundation Grant #DMS–9970119 is gratefully acknowledged. 2 Supported by the Spanish DGES (Pb 97-1291-C03-03) and by a grant from the Spanish Direcci´on General de Ense˜ nanza Superior e Investigaci´ on Cient´ ıfica (Programa de Estancias de Investigadores Espa˜ noles en Centros de Investigaci´ on Extranjeros), while visiting the University of Wisconsin at Madison. Typeset by A M S-T E X 1