Lie algebras attached to Clifford modules and simple graded Lie algebras
Kenro Furutani, Mauricio Godoy Molina, Irina Markina, Tohru Morimoto,, Alexander Vasil'ev

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 -type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type with -grading do not contain non-Heisenberg pseudo -type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types , and provide such a possibility. Among exceptional algebras only and contain non-Heisenberg pseudo -type Lie algebras as their negative part of -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.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic structures and combinatorial models · Advanced Topics in Algebra
