Goo Ishikawa, Yoshinori Machida, “Prolongation of <nobr>$(8,15)$</nobr>-Distribution of Type <nobr>$F_4$</nobr> by Singular Curves”, SIGMA, 2025, Volume 21,076, 20 pp.

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

Goo Ishikawa^a, 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

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 $F_4$.

Keywords: exceptional Lie algebra, singular curve, constrained Hamiltonian equation.

MSC: 53C17, 58A30, 17B25, 34H05, 37J37, 49K15, 53D25

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

Language: English

DOI: 10.3842/SIGMA.2025.076