title: A Cornucopia of Carnot groups in Low Dimensions
authors: Enrico Le Donne Francesca Tripaldi
text: 
  
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  	E. Le Donne and F. Tripaldi]Enrico Le Donne and Francesca Tripaldi
  
  
  Francesca Tripaldi: 
  University of Bologna, Department of Mathematics, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
  francesca.tripaldi2@unibo.it
  Enrico Le Donne: 
  Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy 
  
  & 
  
  University of Jyväskylä, Department of Mathematics and Statistics, P.O. Box (MaD), FI-40014, Finland
  enrico.ledonne@unipi.it
   
  
  
   
   E.L.D. and F.T were partially supported by the Academy of Finland (grant
  288501
  `Geometry of subRiemannian groups' and by grant
  322898
  `Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory')
  and by the European Research Council
   (ERC Starting Grant 713998 GeoMeG `Geometry of Metric Groups'). F.T. was also partially supported by the University of Bologna, funds for selected research topics, and by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 777822 GHAIA (`Geometric and Harmonic Analysis with Interdisciplinary Applications').
  
  
  
  
   Stratified groups  are those  simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating.
   When a stratified  group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space  is known as  Carnot group. 
   Carnot groups appear in several mathematical contexts.
   To understand their algebraic structure, it is useful to study some examples explicitly.
   In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invariant 1-forms, a basis of right-invariant vector fields, and some other properties.  
   We exhibit all stratified  groups in dimension up to 7 and also study some free-nilpotent groups in dimension up to 14.
  
  
  
  []
  53C17, 
  
  
  43A80, 
  
  
  
  
  
  
  
  
  
  
  22E25, 
   22F30, 
  14M17. 
  
  
  
  
  
  
   
  
  
   
    A cornucopia of Carnot groups in low dimensions
      [
      December 30, 2023
  ===============================================
  
  
  
  
  
  
  
  
  
  
  
    
  
  
  
  
  
  
  
  
  
  
  
  
  § INTRODUCTION
  
  Introduzione/Introduction.tex
  
  
   §.§ Notations and differential objects considered
  
  Introduzione/Notations.tex
  
  
  
  
  
  
  
  § 1D–4D NILPOTENT LIE ALGEBRAS
  
    
  The nilpotent Lie groups of dimension up to 4 are well known, and they are all stratifiable.  The list of the stratifiable Lie algebras of dimension less than or equal to 4 is: ℝ, ℝ^2, ℝ^3, N_3,2, ℝ^4, N_3,2×ℝ, and N_4,2, where here ℝ^n denotes the n-dimensional  abelian Lie algebra, N_3,2 is the first Heisenberg Lie algebra, and N_4,2 is the Engel Lie algebra.
  
  We shall now recall the non-zero brackets of these last two Lie algebras in order to help the reader get familiar with our diagram notation.
  
  The algebra N_3,2 and its stratification are represented as
  
  
      [end anchor=north]
      X_1 [dr, no head]       X_2[dl, no head]
  
         X_3        .
      
  
  
  Whereas the algebra N_4,2 and its stratification are represented as
  
  
      [end anchor=north]
      X_1[dr, no head][ddr, no head]        X_2[dl, no head]
  
         X_3[d, no head]     
  
          X_4       .
      
  
  
  
  In the case of the first Heisenberg group and the Engel group, it is sometimes convenient to work in exponential coordinates of the first kind, and some other times in exponential coordinates of the second kind. 
  
  low-dim/Heis.tex
  low-dim/Engel.tex 
  
  
  § 5D INDECOMPOSABLE NILPOTENT LIE ALGEBRAS
  
    
    Among all the indecomposable nilpotent Lie algebras of dimension 5, there are 4 Carnot algebras and 2 more nilpotent Lie algebras, which are gradable. Moreover, there are other 3 decomposable nilpotent algebras, which are stratifiable: the abelian ℝ^5, the direct products of the first Heisenberg group times ℝ^2, and the Engel group times ℝ.
    
  5-dim/5-1.tex
  
   5-dim/5-2-1.tex
    5-dim/5-2-2.tex
     5-dim/5-2-3.tex
    5-dim/5-3-1.tex
   5-dim/5-3-2.tex
      
      
      
        
  
  § 6D INDECOMPOSABLE    NILPOTENT LIE ALGEBRAS
  
    Among all the indecomposable nilpotent Lie algebras in dimension 6, there are 13 Carnot algebras and 11 more nilpotent Lie algebras, which are gradable. 
   
    
    6-dim/6-1-1.tex
     6-dim/6-1-2.tex
    6-dim/6-1-3.tex
    6-dim/6-1-4.tex  
    6-dim/6-2-1.tex
     6-dim/6-2-2.tex
    6-dim/6-2-3.tex
    6-dim/6-2-4.tex
    6-dim/6-2-5.tex
     6-dim/6-2-5a.tex
    6-dim/6-2-6.tex
     6-dim/6-2-7.tex
     6-dim/6-2-8.tex
     6-dim/6-2-9.tex
     6-dim/6-2-9a.tex
     6-dim/6-2-10.tex
     6-dim/6-3-1.tex
     6-dim/6-3-1a.tex 
     6-dim/6-3-2.tex
     6-dim/6-3-3.tex
     6-dim/6-3-4.tex
     6-dim/6-3-5.tex
     6-dim/6-3-6.tex
     6-dim/6-4-4a.tex
    
    
  
  § 7D INDECOMPOSABLE CARNOT NILPOTENT LIE ALGEBRAS
  
    The  indecomposable Carnot Lie algebras in dimension 7 are uncountable. They can be subdivided into 45 examples plus two families whose expressions depend on a real parameter λ. These two families are denoted as (147E) and (147E_1).
    In dimension 7 there are also uncountable non-stratifiable indecomposable nilpotent Lie algebras, and not all of them are gradable. For the complete list we refer to <cit.>.
    
    
   7-dim/37A.tex
   7-dim/37B.tex
   7-dim/37B1.tex
   7-dim/37C.tex
   7-dim/37D.tex
   7-dim/37D1.tex
   7-dim/357A.tex
   7-dim/357B.tex
   7-dim/27A.tex
   7-dim/27B.tex
   7-dim/257B.tex
   7-dim/247A.tex
   7-dim/247B.tex
   7-dim/247C.tex
   7-dim/247D.tex
   7-dim/247E.tex
   7-dim/247E1.tex
   7-dim/247F.tex
   7-dim/247F1.tex
   7-dim/247G.tex
   7-dim/247H.tex
   7-dim/247H1.tex
   7-dim/247I.tex
   7-dim/247J.tex
   7-dim/247K.tex
   7-dim/247N.tex
   7-dim/247P.tex
   7-dim/247P1.tex
   7-dim/2457A.tex
   7-dim/2457B.tex
   7-dim/2457L.tex
   7-dim/2457L1.tex
   7-dim/2457M.tex
  7-dim/23457A.tex
  7-dim/23457B.tex
  7-dim/23457C.tex
  7-dim/17.tex
  7-dim/147D.tex
  7-dim/147E.tex
  7-dim/147E1.tex
  7-dim/137A.tex
  7-dim/137A1.tex
  7-dim/137C.tex
  7-dim/12457H.tex
  7-dim/12457L.tex
  7-dim/12457L1.tex
  7-dim/123457A.tex
    
  
  § SOME FREE-NILPOTENT GROUPS IN LOW DIMENSION
  
    In this final section we analyze free-nilpotent groups of low dimension.
    We shall denote with 𝔽_rs the simply connected Lie group whose Lie algebra has rank r and is free up to nilpotency step s. In the specific, we shall study 
    𝔽_23,
    𝔽_24,
    𝔽_33,
    𝔽_25, respectively.
  
  FreeGroups/Rank2Step3.tex
  FreeGroups/Rank2Step4.tex
  FreeGroups/Rank3Step3.tex
  FreeGroups/Rank2Step5.tex
  
  
  
  
    
    
  
        
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
     
    
  
  
   
  
   
  
  
   
  
   
  
  
  
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