PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 4, April 2009, Pages 1245–1254 S 0002-9939(08)09634-2 Article electronically published on October 20, 2008 GROUP GRADINGS ON SIMPLE LIE ALGEBRAS IN POSITIVE CHARACTERISTIC YURI BAHTURIN, MIKHAIL KOCHETOV, AND SUSAN MONTGOMERY (Communicated by Birge Huisgen-Zimmermann) Abstract. In this paper we describe all gradings by a finite abelian group G on the following Lie algebras over an algebraically closed field F of character- istic p = 2: sl n (F )(n not divisible by p), so n (F )(n ≥ 5, n = 8) and sp n (F ) (n ≥ 6, n even). 1. Introduction We are interested in gradings on finite-dimensional simple Lie algebras over an algebraically closed field F . If a simple Lie algebra L is graded by a group G, then the support of the G-grading on L, Supp L := {g ∈ G | L g =0}, generates an abelian subgroup of G [6, Lemma 2.1]. Thus it is sufficient to consider the case when G is abelian. If L is finite-dimensional, one can also assume that G is finitely generated. Gradings by the groups G = Z n correspond to actions by tori. They have been extensively studied and find numerous applications (see e.g. [11]). We will restrict ourselves to the case of finite groups. Gradings by finite groups arise in the study of generalized symmetric spaces in differential geometry (see e.g. [12] and many more references in [1]), in the theory of Kac–Moody algebras [13], and also in the classification of infinite-dimensional simple Lie algebras endowed with a finite grading by a torsion-free group [15]. For some applications it is desirable to know all possible gradings on a given Lie algebra — e.g. in the context of symmetric spaces [1] and for the classification of simple Lie coloralgebras via the coloration–discoloration process. In the case char F = 0, all gradings on the classical simple Lie algebras (except of type D 4 ) have been described in [3, 6, 4]. Here we will focus on the case char F = p> 0. We will assume that p = 2. Our main results are Theorems 5.1 and 5.5, where we prove, respectively, that any grading on sl n (F ), n ≥ 2, in characteristic p = 2, with p ∤ n, and any grading on so n (F ), n ≥ 5, n = 8, and on sp n (F ), n even, n ≥ 6, in characteristic p = 2, Received by the editors July 5, 2007, and, in revised form, February 8, 2008, and, April 21, 2008. 2000 Mathematics Subject Classification. Primary 16W10, 16W50, 17B50, 17B70. The first author was partially supported by NSERC grant # 227060-04 and by a URP grant, Memorial University of Newfoundland. The second author was supported by a Start-up Grant, Memorial University of Newfoundland. The third author was supported by NSF grant DMS 0401399. c 2008 American Mathematical Society 1245 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use