Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 076, 20 pages      arXiv:2501.02789      https://doi.org/10.3842/SIGMA.2025.076

Prolongation of $(8,15)$-Distribution of Type $F_4$ by Singular Curves

Goo Ishikawa a and Yoshinori Machida b
a) Department of Mathematics, Hokkaido University, Kita 10 Nishi 8, Kita-ku, Sapporo 060-0810, Japan
b) Department of Mathematics, Faculty of Science, Shizuoka University, 836, Ohya, Suruga-ku, Shizuoka 422-8529, Japan

Received January 30, 2025, in final form September 12, 2025; Published online September 18, 2025

Abstract
Cartan gives the model of $(8, 15)$-distribution with the exceptional simple Lie algebra $F_4$ as its symmetry algebra in his paper (1893), which is published one year before his thesis. In the present paper, we study abnormal extremals (singular curves) of Cartan's model from viewpoints of sub-Riemannian geometry and geometric control theory.Then we construct the prolongation of Cartan's model based on the data related to its singular curves, and obtain the nilpotent graded Lie algebra which is isomorphic to the negative part of the graded Lie algebra $F_4$.

Key words: exceptional Lie algebra; singular curve; constrained Hamiltonian equation.

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