Lie Algebras Attached to Clifford Modules and Simple Graded Lie Algebras

摘要

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo ￡f-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 A_n, C_n and D_n provide such a possibility. Among exceptional algebras only F_4 and E_6 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.