
Journal of Lie Theory 28 (2018), No. 3, 843864 Copyright Heldermann Verlag 2018 Lie Algebras Attached to Clifford Modules and Simple Graded Lie Algebras Kenro Furutani Dept. of Mathematics, Science University of Tokyo, Japan furutani_kenro@ma.noda.tus.ac.jp Mauricio Godoy Molina Dep. de Matemática y Estadística, Universidad de la Frontera, Chile mauricio.godoy@ufrontera.cl Irina Markina Dept. of Mathematics, University of Bergen, Norway irina.markina@uib.no Tohru Morimoto Seki Kowa Inst. of Mathematics, Yokkaichi University, and: Inst. K. Oka de Mathématiques, Nara Women's University, Japan morimoto@cc.narawu.ac.jp Alexander Vasil'ev Dept. of Mathematics, University of Bergen, Norway We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo Htype Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type B_{n} with 2grading do not contain nonHeisenberg pseudo Htype Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types A_{n}, C_{n} and D_{n} provide such a possibility. Among exceptional algebras only F_{4} and E_{6} contain nonHeisenberg pseudo Htype Lie algebras as their negative part of 2grading. An analogous question addressed to real simple graded Lie algebras is more delicate, and we give results revealing the main differences with the complex situation. Keywords: Simple Lie algebras, root system, Dynkin diagram, graded Lie algebras, parabolic subalgebras, Htype algebra, Clifford algebra, nondegenerate bilinear form. MSC: 17B10, 17B22, 17B25, 22E46. [ Fulltextpdf (200 KB)] for subscribers only. 