Naturally graded Lie algebras of slow growth

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 $$\dim\mathfrak g_i+\dim\mathfrak g_{i+1}\le3,\qquad i\ge1.$$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 $F_\mathfrak g^\mathrm{gr}(n)\le3n/2+1$. Bibliography: 32 titles.

graded Lie algebra, Carnot algebra, Kac-Moody algebras, central extension, automorphism