Naturally graded Lie algebras of slow growth

D. V. Millionshchikov

Sbornik. Mathematics, Tome 210 (2019) no. 6, pp. 862-909

Résumé

A pro-nilpotent Lie algebra $\mathfrak g$ is said to be naturally graded if it is isomorphic to its associated graded Lie algebra $\operatorname{gr}\mathfrak g$ with respect to the filtration by the ideals in the lower central series. Finite-dimensional naturally graded Lie algebras are known in sub-Riemannian geometry and geometric control theory, where they are called Carnot algebras. We classify the finite-dimensional and infinite-dimensional naturally graded Lie algebras $\mathfrak g=\bigoplus_{i=1}^{+\infty}\mathfrak g_i$ with the property $$ \dim\mathfrak g_i+\dim\mathfrak g_{i+1}\leqslant3,\qquad i\geqslant1. $$ An arbitrary Lie algebra $\mathfrak g=\bigoplus_{i=1}^{+\infty}\mathfrak g_i$ of this class is generated by the two-dimensional subspace $\mathfrak g_1$, and the corresponding growth function $F_\mathfrak g^\mathrm{gr}(n)$ satisfies the bound $F_\mathfrak g^\mathrm{gr}(n)\leqslant3n/2+1$. Bibliography: 32 titles.

Détail

Keywords: graded Lie algebra, Carnot algebra, Kac-Moody algebras, central extension
Mots-clés : automorphism.