Lie algebras attached to Clifford modules and simple graded Lie algebras

الأصل · EN

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 difficult, and we give results revealing the main differences with the complex situation.