Lie algebras attached to Clifford modules and simple graded Lie algebras

TL;DR

This paper classifies complex simple graded Lie algebras of depth 2 that can be constructed from Clifford algebra representations, revealing which types contain pseudo H-type Lie algebras in their negative part.

Contribution

It provides a classification of complex simple graded Lie algebras of depth 2 related to Clifford modules and identifies which types include pseudo H-type Lie algebras.

Findings

Type B_n algebras do not contain pseudo H-type negative parts.

Types A_n, C_n, D_n can contain such negative parts.

Only F_4 and E_6 among exceptional algebras contain these structures.

Abstract

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