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EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS $\mathfrak {f}_4$ AND $\mathfrak {e}_6$ VIA CONTACTIFICATIONS

Part of: Classical differential geometry Low-dimensional topology Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  03 July 2024

Paweł Nurowski Open the ORCID record for Paweł Nurowski [Opens in a new window]
Paweł Nurowski*
Affiliation:
Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
*
[email protected]
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Abstract

In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$. Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$. In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$.

The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$.

Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$.

Keywords

contactificationsexceptional simple Lie algebrasreal forms

MSC classification

Primary: 17B05: Structure theory
Secondary: 17B10: Representations, algebraic theory (weights) 17B60: Lie (super)algebras associated with other structures (associative, Jordan, etc.) 17B70: Graded Lie (super)algebras 53A35: Non-Euclidean differential geometry 53A55: Differential invariants (local theory), geometric objects 57M50: Geometric structures on low-dimensional manifolds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 24 , Issue 1 , January 2025 , pp. 157 - 201
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Footnotes

The research was funded from the Norwegian Financial Mechanism 2014-2021 with project registration number 2019/34/H/ST1/00636

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