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arXiv · 1712.08890

Lie algebras attached to Clifford modules and simple graded Lie algebras

Kenro Furutani·Mauricio Godoy Molina·Irina Markina·Tohru Morimoto·Alexander Vasil'ev
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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.

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Kenro Furutani, Mauricio Godoy Molina, Irina Markina, Tohru Morimoto, Alexander Vasil'ev. 2017-12-24. Lie algebras attached to Clifford modules and simple graded Lie algebras. https://arxiv.org/abs/1712.08890

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