Naturally graded Lie algebras (Carnot algebras) of slow growth

2017/05/21 by Dmitry V. Millionschikov, Millionschikov, Dmitry V. · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #math.RA #msc:17B30

paper · pdf · doi:10.48550/arxiv.1705.07494

corrected version

arxiv created 2017/12/25 · arxiv updated 2017/12/27

Abstract

A nilpotent Lie algebra \mathfrak g is said to be naturally graded if it is isomorphic to its associated graded Lie algebra \rm gr \mathfrakg 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=⊕_i=1^+∞\mathfrak g_i with properties [\mathfrak g₁, \mathfrak g_i]=\mathfrak g_i+1, dim\mathfrak g_i+dim\mathfrak g_i+1 ≤ 3, i ≥ 1. For growth functions of such Lie algebras, we have the estimate F(n) ≤ (3)/(2)n+1.

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• Lie algebras of slow growth and Klein-Gordon equation
• On pro-solvable Lie algebras with maximal pro-nilpotent ideals \mathfrakm₀ and \mathfrakm₂

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