Naturally graded Lie algebras (Carnot algebras) of slow growth

Abstract

A nilpotent Lie algebra $\mathfrak{g}$ is said to be naturally graded if it is isomorphic to its associated graded Lie algebra $\mathrm{gr}\mathfrak{g}$ with respect to filtration by ideals of the lower central series. This concept is equivalent to the concept of the Carnot algebra arising in sub-Riemannian geometry and the geometric control theory. We classify finite-dimensional and infinite-dimensional naturally graded Lie algebras (Carnot algebras) $\mathfrak{g}={\oplus }_{i=1}^{+\infty }{\mathfrak{g}}_i$ with properties $$[{\mathfrak{g}}_1,{\mathfrak{g}}_i]={\mathfrak{g}}_{i+1},\dim {\mathfrak{g}}_i+\dim {\mathfrak{g}}_{i+1}\le 3,i\ge 1.$$ For growth functions of such Lie algebras, we have the estimate $F(n)\le \frac{3}{2}n+1$.

corrected version

Cited by in corpus (4)

• Geometry of central extensions of nilpotent Lie algebras
• On pro-solvable Lie algebras with maximal pro-nilpotent ideals ${\mathfrak{m}}_0$ and ${\mathfrak{m}}_2$
• Lie algebras of slow growth and Klein-Gordon equation
• On derivations and low-dimensional (co)homology groups of pro-sovable Lie algebras associated with ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$