Journal of Lie Theory 28 (2018), No. 3, 843--864
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.nara-wu.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 H-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type Bn with |2|-grading do not contain non-Heisenberg pseudo H-type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types An, Cn and Dn provide such a possibility. Among exceptional algebras only F4 and E6 contain non-Heisenberg pseudo H-type Lie algebras as their negative part of |2|-grading. 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, H-type algebra, Clifford algebra, non-degenerate bilinear form.
MSC: 17B10, 17B22, 17B25, 22E46.
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