A Cornucopia of Carnot groups in Low Dimensions
aa r X i v : . [ m a t h . DG ] A ug A CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS
ENRICO LE DONNE AND FRANCESCA TRIPALDI
Abstract.
Stratified groups are those simply connected Lie groups whose Lie alge-bras 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 homo-geneous with respect to the automorphisms induced by the derivation, this metricspace is known as Carnot group. Carnot groups appear in several mathematicalcontexts. To understand their algebraic structure, it is useful to study some exam-ples 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, togetherwith 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 7and also study some free-nilpotent groups in dimension up to 14.
Contents
1. Introduction 41.1. Notations and differential objects considered 62. 1D–4D nilpotent Lie algebras 10 N , N , N , non-stratifiable 14 N , , N , , non-stratifiable 17 N , , N , , Date : August 31, 2020.2010
Mathematics Subject Classification.
Geometry ofsubRiemannian groups ’ and by grant 322898 ‘
Sub-Riemannian Geometry via Metric-geometry andLie-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, fundsfor selected research topics, and by the European Union’s Horizon 2020 research and innovationprogramme under the Marie Sk lodowska-Curie grant agreement No 777822 GHAIA (‘
Geometric andHarmonic Analysis with Interdisciplinary Applications ’). N , , N , , non-stratifiable 23 N , , non-stratifiable 25 N , , non-stratifiable 27 N , , non-stratifiable 30 N , , N , , N , , non-stratifiable 36 N , , non-stratifiable 38 N , , N , , N , , N , , N , , non-stratifiable 46 N , , non-stratifiable 48 N , , non-stratifiable 50 N , , non-stratifiable 52 N , , N , , N , , non-stratifiable 57 N , , N , , N , , N , , N , , A ) 67(37 B ) 68(37 B ) 69(37 C ) 71(37 D ) 72(37 D ) 74(357 A ) 75(357 B ) 77 CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 3 (27 A ) 78(27 B ) 80(257 B ) 81(247 A ) 83(247 B ) 84(247 C ) 86(247 D ) 87(247 E ) 89(247 E ) 91(247 F ) 92(247 F ) 94(247 G ) 95(247 H ) 97(247 H ) 99(247 I ) 100(247 J ) 102(247 K ) 104(247 N ) 105(247 P ) 107(247 P ) 108(2457 A ) 110(2457 B ) 111(2457 L ) 113(2457 L ) 115(2457 M ) 116(23457 A ) 118(23457 B ) 120(23457 C ) 122(17) 124(147 D ) 125(147 E ) 127(147 E ) 129(137 A ) 131(137 A ) 132(137 C ) 134 E. LE DONNE AND F. TRIPALDI (12457 H ) 136(12457 L ) 138(12457 L ) 140(123457 A ) 1436. Some free-nilpotent groups in low dimension 145 F . 145 F . 146 F . 148 F . 153References 1611. Introduction
Stratified groups, equipped with their homogeneous metrics, appear in severalmathematical contexts. Such metric groups arise in harmonic analysis, in the study ofhypoelliptic differential operators, and as boundaries of strictly pseudo-convex com-plex domains, see the books [Ste93, CDPT07] as initial references. When equippedwith Carnot-Carath´eodory metrics, stratified groups are also known as Carnot groupsand they appear in geometric group theory as asymptotic cones of nilpotent finitelygenerated groups, see [Gro96, Pan89]. Sub-Riemannian stratified groups are limits ofRiemannian manifolds and are metric tangents of sub-Riemannian manifolds. Sub-Riemannian geometries arise in many areas of pure and applied mathematics (suchas algebra, geometry, analysis, mechanics, control theory, mathematical physics), aswell as in applications (e.g., robotics), for references see the book [Mon02]. The lit-erature on geometry and analysis on stratified groups is plentiful. In addition to theprevious references, we also cite few more: [RS76, FS82, NSW85, KR85, VSCC92,Hei95, Mag02, Vit08, BLU07, Jea14, Rif14, ABB19].Stratified groups are simply connected Lie groups for which the Lie algebra admitsa special grading, called a stratification, and is equipped with one such a stratifica-tion. Namely, a grading is a stratification if the degree-one layer of the grading is Liegenerating. The presence of a positive grading implies that the Lie algebra is nilpo-tent. Not all nilpotent Lie algebras admit a stratification; however, free-nilpotentLie algebras do. Each positive grading induces a one-parameter family of automor-phisms, giving rise to the consideration of homogeneous distances, which are uniqueup to biLipschitz equivalence, see [LD17] for an introduction. Hence, stratified groupshave many analogies with (finite-dimensional) vector spaces. However, their possiblenon-commutativity provides some crucial differences.
CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 5
In this paper, we provide the list of all stratifiable Lie algebras of dimension up to7, as well as the free-nilpotent Lie algebras of dimension up to 14. Moreover, up todimension 6, we also consider those nilpotent Lie algebras that are not stratifiable,and since they all happen to be positively gradable, we provide one such grading forthem.Let us recall the terminology for stratifiability, stratifications, and gradings. Wefirst stress that all Lie algebras considered here are over R and finite-dimensional.Also, given two subspaces V, W of a Lie algebra, we set [
V, W ] := Span { [ X, Y ]; X ∈ V, Y ∈ W } . A stratification of step s of a Lie algebra g is a direct-sum decomposition g = V ⊕ V ⊕ · · · ⊕ V s , for some integer s ≥
1, where V s = { } and [ V , V j ] = V j +1 for all integers j ∈{ , . . . , s } and where we set V s +1 = { } . We say that a Lie algebra is stratifiable ifthere exists a stratification of it. Equivalently, as pointed out in [Cor19], stratifiablealgebras are those nilpotent Lie algebras that possess a derivation inducing the iden-tity map modulo the derived subalgebra. Stratifiable algebras are also called Carnotalgebras . We say that a Lie algebra is stratified when it is stratifiable and endowedwith a stratification. We should stress that in the case of a stratifiable algebra, thechoice of a stratification is essentially unique: every two stratifications of g differ bya Lie algebra automorphism of g , see [LD17, Proposition 2.17] for a reference.A stratification is a particular example of grading. Indeed, it is a grading wherethe layer of degree one is Lie generating. A positive grading of a Lie algebra g is afamily ( V t ) t ∈ (0 , + ∞ ) of linear subspaces of g , where all but finitely many of the V t ’s are { } , such that g is their direct sum g = M t ∈ (0 , + ∞ ) V t and where [ V t , V u ] ⊂ V t + u , for all t, u > . We say that a Lie algebra is positively gradable if there exists a positive grading of it.We say that a Lie algebra is graded (or positively graded , to be more precise) when itis positively gradable and endowed with a positive grading. More considerations onthis subject can be found in [LR19].As usual, we only consider those Lie algebras that are indecomposable , i.e., thosethat are not the direct sum of two nontrivial Lie algebras. The classification of strati-fied algebras that we provide in this paper will give rise to the following consequence:
Proposition 1.1.
There are 4 indecomposable stratified algebras in dimension 5, and13 in dimension 6. All nilpotent Lie algebras of dimension less than or equal to 6 arepositively gradable; but 2 in dimension 5 and 11 in dimension 6 are not stratifiable.In dimension 7, there are two one-parameter families of indecomposable stratifiedalgebras, plus 45 more single examples.
E. LE DONNE AND F. TRIPALDI
For the list of nilpotent Lie algebras of dimension less than or equal to 7, we followGong’s classification from his thesis [Gon98]. However, in our paper we shall also pointout for each Lie algebra what the corresponding name/number is in the classificationspresent in de Graaf [dG07], Magnin [Mag], and Del Barco [dB15], respectively. Thiswill provide the reader with a database to navigate between the different notations.One should stress that the list in Magnin’s paper [Mag] consists of indecomposablenilpotent Lie algebras with complex structural constants (1.2), and is therefore shorterthan the other ones. This is due to the fact that some Lie algebras are isomorphicover algebraically closed fields, but are not isomorphic over R (we refer to Chapter 7in [Gon98] for a more thorough explanation of this fact). Regarding the free nilpotentalgebras of dimension greater than 7, we shall use Hall’s construction from [Hal50].For each group in our list, we exhibit a basis of its Lie algebra following the presen-tation in [Gon98] and we study the differential structure in exponential coordinates.More precisely, we calculate the group law, the vector fields that are left-invariant ex-tensions of the given basis, and the respective left-invariant 1-forms. We also providethe expressions for the right-invariant vector fields. One should stress that nilpotentgroups that are not stratifiable, do not have a canonical sub-Riemannian structure.For these groups, we shall also add a subsection to present a possible grading anddiscuss which polarizations give rise to a maximal Hausdorff dimension for their re-spective sub-Riemannian distance, and calculate the tangent space. We also computethe asymptotic cones of all non-stratifiable nilpotent Lie groups of dimension lessthan or equal to 6.1.1. Notations and differential objects considered.
For the groups discussed inthis paper, we shall use the following notation for describing their differential structurein exponential coordinates.We present each Lie algebra with a choice of a basis denoted by X , . . . , X n , andprovide the list of non-zero bracket relations in the form(1.2) [ X i , X j ] = n X k =1 c kij X k , where c kij ∈ R are called structural constants , and of course we only present those forwhich i < j .Since for such low-dimensional Lie algebras the list of equations (1.2) is rather short,we aim to provide a visualization of the grading of the Lie algebra through a diagramas follows. For Carnot groups up to dimension 7, the sum in (1.2) is of at most oneaddend and most of the time the only non-zero coefficient is 1 or −
1. Hence, if forgiven i, j there exists k such that [ X i , X j ] = X k , we will then visualize this relationas CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 7 X i X j X k . In other words, the bracket relation expressed in the diagram should always be read from left to right . If the diagram should be read differently, we shall use the followingnotation. Namely, if the bracket relation is [ X i , X j ] = − X k , we will draw the diagramas X i X j X k . If instead the bracket relation is [ X i , X j ] = cX k for some c ∈ R , we will then write X i X j X k . c In the case of Carnot algebras where the given basis can be adapted to a stratifi-cation, we will also draw the diagram by rows according to the different strata. Forexample, a diagram of the form X X X X means that X , X is a basis of the first stratum V , X is a basis of the second stratum V , X is a basis of the third stratum V , and the bracket relations are [ X , X ] = X ,[ X , X ] = X , and [ X , X ] = X .Moreover, other information is also readily available from simply looking at thediagram, such as the rank of the Lie algebra, which is equal to the number of vectorsin the first row, and the nilpotency step, which is given by the number of rows in thediagram.Given a basis X , . . . , X n of a nilpotent Lie algebra g , there exists a unique (up toisomorphism) simply connected Lie group G with g as Lie algebra. Moreover, theexponential map exp : g → G is a diffeomorphism, see [CG90] for a reference. We E. LE DONNE AND F. TRIPALDI shall then parametrize G via the exponential map and our choice of basis. Namely,we will use the identification R n ←→ G ( x , . . . , x n ) exp Å n X i =1 x i X i ã . (1.3)Since in nilpotent groups the Baker-Campbell-Hausdorff formula converges globally,the identification above allows us to write the group product. In fact, for all x , y ∈ R n ,there exists a unique z ∈ R n such that(1.4) exp Å n X i =1 x i X i ã exp Å n X i =1 y i X i ã = exp Å n X i =1 z i X i ã . Via the identification (1.3), one can write the group law in (1.4) as(1.5) x ∗ y = z . Hence, we have a group law ∗ on R n that makes ( R n , ∗ ) a simply connected Liegroup with Lie algebra g , whose identity element is .The basis X , . . . , X n of g induces left-invariant vector fields on R n via the formula(1.6) x d Ä L x ä e i , i = 1 , . . . , n , where L x ( y ) := x ∗ y is the left translation , and by e i we denote the i -th vector of thestandard basis of R n . We will still denote by X i the left-invariant vector fields givenby equation (1.6). One should stress that each vector field X i is represented by the i -th column of the matrix d Ä L x ä , for i = 1 , . . . , n . For better readability, however,in our paper we will provide both d Ä L x ä as a matrix, as well as the vector fields X , . . . , X n written as derivations.We also provide the explicit expression in coordinates of the basis θ , . . . , θ n of left-invariant 1-forms that is dual to the basis of left-invariant vector fields X , . . . , X n ,that is(1.7) θ i ( X j ) = δ ij , for i, j = 1 , . . . , n . Likewise, one can repeat the same procedure for right translations. For shortness,we will only provide the differential at of right translations R x ( y ) := y ∗ x . Thereader can then deduce the right-invariant vector fields (and subsequently the right-invariant 1-forms) from the columns of the matrix d Ä R x ä .In the case of non-stratifiable nilpotent groups, we will insert an extra subsectionto discuss the possible Carnot-Carath´eodory structures that maximize the Hausdorffdimension. In order to do so, we now recall the notion of polarization and the methodto calculate the dimension of the metric space that it defines (up to biLipschitzequivalence). Given g a nilpotent Lie algebra, we denote by G the simply connectedLie group that has g as Lie algebra. By applying left-translations, we have that any CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 9 subspace V of g induces a left-invariant subbundle ∆ of T G . Following Gromov’sterminology, we call the pair ( G , ∆) a polarization of G . We only focus on thecase where V is Lie generating, which is equivalent to saying that ∆ is a bracket-generating distribution. One can check that, since g is nilpotent, a subspace V ⊂ g is Lie generating if and only if(1.8) V + [ g , g ] = g . Once we fix a left-invariant norm k · k on ∆, we get that the triple ( G , ∆ , k · k ) is anexample of a Carnot-Carath´eodory space. We refer to Gromov seminal paper [Gro96]for the theory of Carnot-Carath´eodory spaces, also called CC-spaces . Every CC-spaceis a metric space when equipped with the control distance. The Hausdorff dimensionwith respect to this distance can be expressed as(1.9) X i i Ä rank ∆ i − rank ∆ i − ä , where rank ∆ = 0, rank ∆ = dim V , andrank ∆ i = dim Ä V + [ V, V ] + · · · + [ V, [ V, · · · , [ V, V ]]] | {z } i − ä . Let us point out that if g is stratifiable, then a Lie generating subspace V ⊂ g maxi-mizes the Hausdorff dimension if and only if V is the first layer of a stratification. Onthe other hand, this characterization is not present for non-stratifiable Lie algebras.For this reason in our paper, when presenting a non-stratifiable nilpotent Lie algebra g , we will add an extra subsection to discuss for which choice of polarization ( G , ∆)we obtain maximal Hausdorff dimension. In low dimension, except for a few cases,such polarizations are unique up to automorphism. Namely, in dimension up to 5, allpossible ∆ with (1.8) differ by an automorphism. In dimension 6, except for N , , and N , , , polarizations of maximal dimension are unique up to automorphism. Forthe considered polarizations we calculate the tangent cone, which is also known assymbol, and by a theorem of Mitchell is a very easily computable Carnot group, see[Mit85].By a theorem of Pansu, the asymptotic cone of every nilpotent Lie group equippedwith a CC-metric is a Carnot group, which does not depend on the choice of metric.Thus, for every non-stratifiable nilpotent Lie algebra of dimension 5 or 6 we shall alsodetermine its asymptotic cone. The calculation is done via the associated Carnot-graded Lie algebra , for which we refer to [Pan83].Finally, after the list of Carnot groups of dimension 7, we analyze free-Carnot groupsof low dimension. Regarding the step-2 case, one can easily write down the productlaw and the left-invariant vector fields in arbitrary dimension (this calculation is notoriginal and can also be found in [LR19, LS16, BLU07]). Regarding the step-3 case,the free-Carnot groups of rank 2 is 5-dimensional, so it is already included in Section4 (see Lie algebra N , , on page 18). In addition, we will present the rank-3 step-3 free-Carnot group, which has dimen-sion 14, and the rank-2 free-Carnot groups o step at most 5, which have dimensions5, 8, and 14, respectively. We will not discuss the rank-4 step-3 case, which hasdimension 30, nor the one with rank 3 and step 4, which has dimension 32.Of the free-Carnot groups above, we will also provide exponential coordinates ofthe second type, together with the change of variables with respect to the ones offirst type. We shall add an s , as an exponent, to those differential objects that areexpressed in exponential coordinates of the second type.2. The nilpotent Lie groups of dimension up to 4 are well known, and they are allstratifiable. The list of the stratifiable Lie algebras of dimension less than or equal to4 is: R , R , R , N , , R , N , × R , and N , , where here R n denotes the n -dimensionalabelian Lie algebra, N , is the first Heisenberg Lie algebra, and N , is the Engel Liealgebra.We shall now recall the non-zero brackets of these last two Lie algebras in order tohelp the reader get familiar with our diagram notation.The algebra N , and its stratification are represented as X X X . Whereas the algebra N , and its stratification are represented as X X X X . In the case of the first Heisenberg group and the Engel group, it is sometimesconvenient to work in exponential coordinates of the first kind, and some other timesin exponential coordinates of the second kind. N , . The following Lie algebra is denoted as N , by Gong in [Gon98], as L , by deGraaf in [dG07], as h by Del Barco in [dB15], and as G by Magnin in [Mag].The only non-trivial bracket is the following:[ X , X ] = X . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 11
This is a nilpotent Lie algebra of rank 2 and step 2 that is stratifiable, also known asthe first Heisenberg algebra. The Lie brackets can be pictured with the diagram: X X X . The composition law (1.4) of N , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ).Since d( L x ) = − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx . Finally, we have d( R x ) = x − x . One can also consider the exponential coordinates of the second kind. In this case,we obtain the following expression for the left-invariant vector fields: • X s = ∂ x ; • X s = ∂ x + x ∂ x ; • X s = ∂ x . N , . The following Lie algebra is denoted as N , by Gong in [Gon98], as L , by deGraaf in [dG07], as (2) by Del Barco in [dB15], and as G by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 3 that is stratifiable, also knownas the filiform Lie algebra of dimension 4. The Lie brackets can be pictured with thediagram: X X X X . The composition law (1.4) of N , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 13
Finally, we have d( R x ) = x − x x − x x x − x . One can also consider the exponential coordinates of the second kind. In this case,we obtain the following expression for the left-invariant vector fields: • X s = ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x ; • X s = ∂ x + x ∂ x ; • X s = ∂ x .
5D indecomposable nilpotent Lie algebras
Among all the indecomposable nilpotent Lie algebras of dimension 5, there are 4Carnot algebras and 2 more nilpotent Lie algebras, which are gradable. Moreover,there are other 3 decomposable nilpotent algebras, which are stratifiable: the abelian R , the direct products of the first Heisenberg group times R , and the Engel grouptimes R . N , non-stratifiable. The following Lie algebra is denoted as N , by Gong in[Gon98], as L , by de Graaf in [dG07], as (1) by Del Barco in [dB15], and as G , byMagnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 4 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X . The composition law (1.4) of N , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − x y ( x y − x y ). CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 15
Since d( L x ) = − x x − x x − x x x − x x + x − x x x − x x + x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x + x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x − x ä dx + Ä x x + x − x ä dx + Ä x − x x + x + x x ä dx . Finally, we haved( R x ) = x − x x − x x x − x x − x x + x x x + x x − x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , is not stratifiable, but it is gradable as V i = span { X i } , i = 1 , . . . , . We claim that in this Lie algebra every two complementary subspaces to the derivedsubalgebra, as in (1.8), differ by an automorphism. Indeed, when u ji varies, the matrix u u u u u u u u − u u is a Lie algebra automorphism and sends the subspace span { X , X } to span { X + u X + u X + u X , X + u X + u X + u X } , which is an arbitrary subspace as in(1.8). In particular, every ∆ as in (1.8) gives maximal dimension. The tangent coneof each of such polarizations has Lie algebra isomorphic to N , , , see page 19.The asymptotic cone of the Lie group with Lie algebra N , has Lie algebra isomor-phic to N , , , which is the filiform algebra of step 4. N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , byde Graaf in [dG07], as (2) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 4 that is stratifiable, also knownas the filiform Lie algebra of dimension 5. The Lie brackets can be pictured with thediagram: X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 17 • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ä dx . Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x . N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (4) by Del Barco in [dB15], and as G , byMagnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y ).Since d( L x ) = − x x − x − x x − x + x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x + x ä dx + Ä x − x x ä dx . Finally, we haved( R x ) = x − x x − x x x + x − x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V = span { X , X } ,V = span { X , X } ,V = span { X } . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 19
We claim that in this Lie algebra every two complementary subspaces to the derivedsubalgebra, as in (1.8), differ by an automorphism. Indeed, when u ji varies, the matrix − u u u u u u u u is a Lie algebra automorphism and sends the complementary subspace span { X , X , X } to span { X − u X + u X + u X , X + u X + u X , X + u X + u X } , which isan arbitrary subspace as in (1.8). In particular, every ∆ as in (1.8) gives maximaldimension. The tangent cone of each of such polarizations has Lie algebra isomorphicto N , , , see page 21.The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to N , × R , where N , is the filiform algebra of step 3. N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , byde Graaf in [dG07], as (3) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 3 that is stratifiable, also known asthe free-nilpotent Lie algebra of step 3 and 2 generators. Such algebra will also bestudied in the section of free-nilpotent Lie algebras, see page 145. The Lie bracketscan be pictured with the diagram: X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ). Since d( L x ) = − x x − x x − x x x − x − x + x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − Ä x − x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx . Finally, we haved( R x ) = x − x x − x x x − x − x x + x x − x . N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , byde Graaf in [dG07], as (8) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = [ X , X ] = X . This is a nilpotent Lie algebra of rank 4 and step 2 that is stratifiable, also known asthe second Heisenberg algebra. The Lie brackets can be pictured with the diagram: X X X X X . The composition law (1.4) of N , , is given by: CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 21 • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y ).Since d( L x ) = − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx − x dx + x dx . Finally, we have d( R x ) = x − x x − x . N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , byde Graaf in [dG07], as (6) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx . Finally, we have d( R x ) = x − x x − x . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 23
6D indecomposable nilpotent Lie algebras
Among all the indecomposable nilpotent Lie algebras in dimension 6, there are 13Carnot algebras and 11 more nilpotent Lie algebras, which are gradable. N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (4) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 5 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y + x y − x y ) − x y ( x y − x y ) − ( x y + x y )( x y − x y ) + ( y − x )( x y − x y )+ ( x y − x y )( x y − x y ) . Sinced( L x ) = − x x − x x − x x x − x x + x − x x x − x x + x x x x − x x + x x − x − x x − x − x x x x + x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x + x ä ∂ x + Ä x x − x x + x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x − Ä x x + x + x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x + x ä ∂ x + x x ∂ x ; • X = ∂ x + x ∂ x + Ä x + x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x − x ä dx + Ä x x + x − x ä dx + Ä x − x x + x + x x ä dx ; • θ = dx − x dx + Ä x − x ä dx + Ä x x − x ä dx + Ä x − x x − x x + x ä dx + Ä x − x x + x x + x x + x x − x x ä dx . Finally, we haved( R x ) = x − x x − x x x − x x − x x + x x + x x x − x − x x − x x + x x + x x x − x x − x x x x − x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V i = span { X i } , i = 1 , . . . , . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 25
We claim that in this Lie algebra every two complementary subspaces to the derivedsubalgebra as in (1.8) differ by an automorphism. Indeed, when u ji varies, the matrix u u u u u u u u − u u u u u − u u − u u is a Lie algebra automorphism and sends the complementary subspace span { X , X } to span { X + u X + u X + u X + u X , X + u X + u X + u X + u X } , whichis an arbitrary subspace as in (1.8). In particular, every ∆ as in (1.8) gives maximaldimension. The tangent cone of each of such polarizations has Lie algebra isomorphicto N , , , see page 40.The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to the filiform algebra N , , of step 5, see page 32. N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (1) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 2 and step 5 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y − x y + x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y + x y − x y ) + ( y − x )( x y − x y ) − [ x y ( x y − x y ) + ( x y − x y )( x y − x y )] − y ( x y − x y ) + ( y y − x x )( x y − x y ) + ( x y y − x y )( x y − x y )] − x ( x y − x y ) . Sinced( L x ) = − x x − x x − x x x − x x + x − x x x − x x + x x x − x x + x x x x − x x − x − x x x + x − x x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x + x ä ∂ x + Ä x − x x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x − x x − x − x x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x + x ä ∂ x + Ä x + x − x x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ) ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x − x ä dx + Ä x x + x − x ä dx + Ä x − x x + x + x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx + Ä x − x x − x − x x ä dx + Ä x + x x − x x + x x − x x + x x ä dx + Ä x − x x + x − x x ä dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 27
Finally, we haved( R x ) = x − x x − x x x − x x − x x + x x + x x x − x − x x − x x + x x x + x x − x x − x x x − x x − x x + x x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V i = span { X i } , i = 1 , . . . , ,V = 0 ,V = span { X } . When u ji varies, the matrix u u u u u u u u − u u u u u u − u u − u u − u u u − u − u u is a Lie algebra automorphism when 2 u = u u , and sends the complementarysubspace span { X , X } to span { X + u X + u X + u X + u X , X + u X + u X + u X + u X } , which is an arbitrary subspace as in (1.8). The tangent cone ofeach of such polarizations has Lie algebra isomorphic to N , , if 2 u = u u , to N , , if 2 u > u u , and to N , , a if 2 u < u u , see pages 44, 40, and 41, respectively. Inparticular, each of such polarizations gives maximal Hausdorff dimension.The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to N , , , see page 34. N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (5) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , ≤ i ≤ , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 5 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − x y ( x y − x y ) + ( y − x )( x y − x y )+ ( x y − x y )( x y − x y ) . Since d( L x ) = − x x − x x − x x x − x x − x x x x x − x x + x − x x x − x − x x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x + Ä x x − x x + x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 29 • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ) dx ; • θ = dx − x dx + x dx − Ä x + x ä dx + Ä x + x x + x ä dx + Ä x − x + x x + x x − x x ä dx . Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x x − x x + x + x x x + x x − x − x x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V = span { X i } ,V = 0 ,V i = span { X i − } , i = 3 , . . . , . We claim that in this Lie algebra every two complementary subspaces to the derivedsubalgebra as in (1.8) differ by an automorphism. Indeed, when u ji varies, the matrix u u u u u u u u u u u u − u u u is a Lie algebra automorphism and sends the complementary subspace span { X , X } to span { X + u X + u X + u X + u X , X + u X + u X + u X + u X } , whichis an arbitrary subspace as in (1.8). In particular, every ∆ as in (1.8) gives maximaldimension. The tangent cone of each of such polarizations has Lie algebra isomorphicto N , , , see page 44. The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to N , , , the first-type filiform algebra of step 5, see page 32. N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (11) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 4 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y + x y − x y + x y − x y )+ ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − x y ( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x + x − x x x − x + x x + x x x , CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 31 the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x x + x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x + x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x + x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx ; • θ = dx − x dx − x dx + Ä x − x ä dx + Ä x + x + x x − x ä dx + Ä x − x + x x + x x ä dx . Finally, we haved( R x ) = x − x x − x x x − x x − x x + x x + x + x x x − x − x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V i = span { X i } , i = 1 , , ,V = span { X , X } ,V = span { X } . Every complementary subspace ∆ to the derived subalgebra is spanned by X + u X + u X + u X , X + u X + u X + u X , and X + u X + u X + u X . Sucha polarization gives maximal Hausdorff dimension if and only if u is either − u = 0 differ by an automorphism.Likewise, every two polarizations with u = − when u ji varies, the matrix − u u u u u u u
00 0 0 0 1 0 u u u − u − u u u − u u , is a Lie algebra automorphism and sends the complementary subspace(4.1) span { X , X , X } to an arbitrary one with u = 0. Instead, when u ji varies, the matrix − u u u − u u u u −
10 0 0 0 1 0 u u u − u − u u u − u u u , is a Lie algebra automorphism and sends the complementary subspace(4.2) span { X , X , X − X } to an arbitrary one with u = − N , , and N , , respectively, see pages 59 and 43.The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to N , , × R , where N , , is the filiform algebra of step 4. N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , byde Graaf in [dG07], as (3) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , ≤ i ≤ . This is a nilpotent Lie algebra of rank 2 and step 5 that is stratifiable, also knownas the filiform Lie algebra of first type of dimension 6, the second type is N , , , seepage 34. The Lie brackets can be pictured with the diagram: CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 33 X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y )+ ( y − x )( x y − x y ) + ( x y − x y )( x y − x y ) . Since d( L x ) = − x x − x x − x x x − x x − x x x − x x − x + x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x + Ä x x − x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ) dx ; • θ = dx − x dx + x dx − x dx + x dx + Ä x − x x + x x − x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x x − x x + x x − x x − x . N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , byde Graaf in [dG07], as (2) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 2 and step 5 that is stratifiable, also knownas the filiform Lie algebra of second type of dimension 6, the first type is N , , , seepage 32. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 35 • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + Ä x − y )( x y − x y )+ ( y − x )( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y )+ x y ( x y − x y ) + ( y y − x x )( x y − x y ) − y ( x y − x y ) + ( x y y − x y )( x y − x y ) − x ( x y − x y ) .Sinced( L x ) = − x x − x x − x x x − x x − x x x x − x x + x x x x − x − x x x − x x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x + Ä x − x x + x x ) ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x − x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x − x x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.6) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ) dx ; • θ = dx − x dx + Ä x + x x ä dx − Ä x + x x + x x ä dx + Ä x + x x + x x + x x ä dx + Ä x − x x − x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x x − x x + x x x + x x − x x − x − x x x + x x − x . N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (9) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X i ] = X i +1 , i = 4 , , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 3 and step 4 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) + ( y − x )( x y − x y ) − x y ( x y − x y ).Since d( L x ) = − x x − x − x x x − x x x x x − x x − x − x x x + x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x + Ä x x − x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x − x ä ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + Ä x + x x ä ∂ x ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 37 • X = ∂ x + x ∂ x + Ä x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x + x ä dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x ä dx + Ä x x − x ä dx − Ä x + x x ä dx + Ä x + x x − x x + x x ä dx .Finally, we haved( R x ) = x − x x − x x x + x − x − x x x − x x + x − x x x x − x x + x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V = span { X , X } ,V = span { X , X } ,V = span { X } ,V = span { X } . Every complementary subspace ∆ to the derived subalgebra is spanned by X + u X + u X + u X , X + u X + u X + u X , and X + u X + u X + u X . Sucha polarization gives maximal Hausdorff dimension if and only if u + u = 0. Weclaim that every two polarizations giving maximal Hausdorff dimension differ by anautomorphism. Indeed, when u ji varies, the matrix − u u u u u u − u u u u u u u is a Lie algebra automorphism and sends the complementary subspace span { X , X , X } to span { X − u X + u X + u X + u X , X + u X + u X + u X , X + u X − u X + u X } , which is an arbitrary one of maximal dimension. The tangent cone ofeach of such polarizations has Lie algebra isomorphic to N , , , see page 54.The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to N , , × R , where N , , is the filiform algebra of step 4. N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (10) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 4 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x − x − x x x x , CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 39 the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx ; • θ = dx − x dx − x dx + x dx + Ä x − x ä dx + Ä x − x x + x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x x − x x x x − x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V = span { X , X } ,V = span { X } ,V = span { X , X } ,V = span { X } . Every complementary subspace ∆ to the derived subalgebra is spanned by X + u X + u X + u X , X + u X + u X + u X , and X + u X + u X + u X .Such a polarization gives maximal Hausdorff dimension if and only if u = 0. Weclaim that every two polarizations giving maximal Hausdorff dimension differ by an automorphism. Indeed, when u ji varies, the matrix u u u u u u
00 0 0 0 1 0 u u u u u is a Lie algebra automorphism and sends the complementary subspace span { X , X , X } to span { X + u X + u X + u X , X + u X + u X + u X , X + u X + u X } ,which is an arbitrary one of maximal dimension. The tangent cone of each of suchpolarizations has Lie algebra isomorphic to N , , , see page 59.The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to N , , × R , where N , , is the filiform algebra of step 4. N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , − by de Graaf in [dG07], as (7) by Del Barco in [dB15], and as G , by Magnin [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , , [ X , X j ] = X j +2 , j = 3 , . This is a nilpotent Lie algebra of rank 2 and step 4 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − ( x y + x y )( x y − x y ). CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 41
Since d( L x ) = − x x − x x − x x x − x x x − x x − x x − x − x − x x x x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x − Ä x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx − x dx − x dx + x x dx + Ä x − x x − x x ä dx + Ä x − x x + x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x − x x + x x − x x − x x x − x x x x − x − x . N , , . The following Lie algebra is denoted as N , , a by Gong in [Gon98], as L , by de Graaf in [dG07], as (8) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 2 and step 4 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , a is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( y − x )( x y − x y )+ ( y − x )( x y − x y ) + ( x y + x y )( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x x − x x x x + x x + x x − x + x − x − x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x + Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x + x ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 43 • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx + x dx + x dx − x + x dx + Ä x x − x + x + x x ä dx − Ä x − x x + x x + x ä dx .Finally, we haved( R x ) = x − x x − x x x − x − x x x + x − x x x − x x x − x − x + x x x . N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , byde Graaf in [dG07], as (14) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ). Since d( L x ) = − x x − x x − x x + x − x x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x + x x ä dx + x dx + Ä x − x + x x ä dx .Finally, we haved( R x ) = x − x x − x x − x x + x x x x + x − x − x . N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , by de Graaf in [dG07], as (6) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 4 that is stratifiable. The Liebrackets can be pictured with the diagram:
CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 45 X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ).Since d( L x ) = − x x − x − x x x x − x − x x x x − x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x x + x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx .Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x − x x x + x − x . N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (21) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x − x x x − x x x − x x , CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 47 the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + Ä x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx ; • θ = dx − x dx − x dx + Ä x + x ä dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx .Finally, we haved( R x ) = x − x x − x x x + x − x − x x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V = span { X , X } ,V = span { X , X } ,V = span { X , X } . Every complementary subspace ∆ to the derived subalgebra is spanned by X + u X + u X + u X , X + u X + u X + u X , and X + u X + u X + u X .We claim that in this Lie algebra every two complementary subspaces to the derivedsubalgebra as in (1.8) differ by an automorphism. Indeed, when u ji varies, the matrix u u u u u u u u u u u sends the complementary subspace span { X , X , X } to span { X + u X + u X + u X , X + u X + u X + u X , X + u X + u X + u X } , which is an arbitraryone. In particular, every ∆ as in (1.8) gives maximal dimension. The tangent coneof each of such polarizations has Lie algebra isomorphic to N , , , see page 63.The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to N , , , see page 61. N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (20) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x x − x + x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 49 • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx − x dx + Ä x + x + x x ä dx − x dx .Finally, we haved( R x ) = x − x x − x x x − x − x x x + x + x − x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V = span { X , X } ,V = span { X , X } ,V = span { X , X } . Every complementary subspace ∆ to the derived subalgebra is spanned by X + u X + u X + u X , X + u X + u X + u X , and X + u X + u X + u X . Sucha polarization gives maximal Hausdorff dimension if and only if u is either − u ji varies, if we take u = 0 the matrix u u u u u u u u − u u , possibly composed with the block diagonal matrix ñ ô ⊕ ñ − −
10 1 ô ⊕ ñ − − ô , is a Lie algebra automorphism and sends the complementary subspace span { X , X , X } to span { X + u X + u X + u X , X + u X + u X + u X , X + u X + u X } ,which is an arbitrary one of maximal dimension.If instead we take u = −
1, as u ji varies, the matrix u u − u u u u − u u − u u , possibly composed with the block diagonal matrix ñ ô ⊕ ñ − −
10 1 ô ⊕ ñ − − ô , is a Lie algebra automorphism and sends the complementary subspace span { X , X , X } to span { X + u X + u X + u X , X + u X + u X + u X , X − X + u X + u X } ,which is an arbitrary one of maximal dimension. The tangent cone of each of suchpolarizations has Lie algebra isomorphic to N , , , see page 59.One should also be aware that Aut( N , , ) has two connected components (see[Gon98] on page 35).The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to N , , × R , where N , , is the free nilpotent Lie algebra of step 3 and 2generators. N , , non-stratifiable. The following Lie algebra is denoted as N , , a by Gongin [Gon98], as L , − by de Graaf in [dG07], as (18) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram:
CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 51 X X X X X X . The composition law (1.4) of N , , a is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y − x y + x y ) + ( y − x )( x y − x y ).Since d( L x ) = − x x − x x − x x − x x x x − x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x + Ä x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x − x ä ∂ x + Ä x − x x ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx ; • θ = dx − x dx − x dx + Ä x + x ä dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − Ä x + x x ä dx + Ä x + x ä dx .Finally, we have d( R x ) = x − x x − x x x + x − x − x x + x − x − x x x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , a is not stratifiable, but it is gradable as V = span { X , X } ,V = span { X , X } ,V = span { X , X } . Every complementary subspace ∆ to the derived subalgebra is spanned by X + u X + u X + u X , X + u X + u X + u X , and X + u X + u X + u X .We claim that in this Lie algebra every two complementary subspaces to the derivedsubalgebra as in (1.8) differ by an automoprhism. Indeed, when u ji varies, the matrix u u u u u u u u u u u u − u , is a Lie algebra automorphism and sends the complementary subspace span { X , X , X } to span { X + u X + u X + u X , X + u X + u X + u X , X + u X + u X + u X } ,which is an arbitrary one of maximal dimension. In particular, every ∆ as in (1.8)gives maximal dimension. The tangent cone of each of such polarizations has Liealgebra isomorphic to N , , , see page 63.The asymptotic cone of the Lie group with Lie algebra N , , a has Lie algebraisomorphic to N , , × R , where N , , is the free nilpotent Lie algebra of step 3 and2 generators. N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (19) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 53
This is a nilpotent Lie algebra of rank 3 and step 3 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x − x x x − x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + Ä x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx ; • θ = dx − x dx − x dx + Ä x + x ä dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx .Finally, we haved( R x ) = x − x x − x x x + x − x − x − x x x + x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is not stratifiable, but it is gradable as V = span { X , X } ,V = span { X , X } ,V = span { X , X } . Every complementary subspace ∆ to the derived subalgebra is spanned by X + u X + u X + u X , X + u X + u X + u X , and X + u X + u X + u X .Such a polarization gives maximal Hausdorff dimension if and only if u = 0. Weclaim that every two polarizations giving maximal Hausdorff dimension differ by anautomorphism. Indeed, when u ji varies, the matrix u u u u u u u u − u u is a Lie algebra automorphism and sends the complementary subspace span { X , X , X } to span { X + u X + u X + u X , X + u X + u X + u X , X + u X + u X } ,which is an arbitrary one of maximal dimension. The tangent cone of each of suchpolarizations has Lie algebra isomorphic to N , , , see page 61.The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebraisomorphic to N , , × R , where N , , is the free nilpotent Lie algebra of step 3 and2 generators. N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , − by de Graaf in [dG07], as (15) by Del Barco in [dB15], and as G , by Magnin in[Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , i = 2 , , [ X , X ] = [ X , X ] = X . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 55
This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x + x x ä dx + Ä x + x x ä dx − x x dx .Finally, we haved( R x ) = x − x x − x − x x x x + x x x + x − x − x . N , , . The following Lie algebra is denoted as N , , a by Gong in [Gon98], as L , by de Graaf in [dG07], as (16) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , i = 2 , , [ X , X ] = [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , a is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ). CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 57
Since d( L x ) = − x x − x x − x + x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − x + x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x + x x ä dx + Ä x + x x ä dx − x + x dx .Finally, we haved( R x ) = x − x x − x − x + x x x + x x x + x − x − x . N , , non-stratifiable. The following Lie algebra is denoted as N , , by Gong in[Gon98], as L , by de Graaf in [dG07], as (25) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 4 and step 3 that is positively gradable, yetnot stratifiable. The Lie brackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx − x dx + x dx + Ä x − x x ä dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 59
Finally, we haved( R x ) = x − x x − x x x − x x − x . Grading, polarizations of maximal dimension, and asymptotic cone.
The Lie algebra N , , is a not stratifiable, but it is gradable as V = span { X , X , X } ,V = span { X , X } ,V = span { X } . Every complementary subspace ∆ to the derived subalgebra is spanned by X + u X + u X , X + u X + u X , X + u X + u X , and X + u X + u X . We claimthat every two polarizations differ by an automorphism. Indeed, when u ji varies, thematrix u u u u − u u u u u u u is a Lie algebra automorphism and sends the complementary subspace X , X , X , X } to span { X + u X − u X + u X + u X , X + u X + u X , X + u X + u X , X + u X + u X } , which is an arbitrary one of maximal dimension. The tangent coneof each of such polarizations has Lie algebra isomorphic to N , × N , , where N , isthe first Heisenberg algebra.The asymptotic cone of the Lie group with Lie algebra N , , has Lie algebra iso-morphic to N , × R , where N , is the filiform algebra of step 3. N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , by de Graaf in [dG07], as (22) by Del Barco in [dB15], and as G , by Magnin in[Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3. The Lie brackets can be picturedwith the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx − x dx + x dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 61
Finally, we haved( R x ) = x − x − x x x + x − x x − x . N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , byde Graaf in [dG07], as (23) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx − x dx + x dx .Finally, we haved( R x ) = x − x − x x x + x − x x − x . N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , by de Graaf in [dG07], as (29) by Del Barco in [dB15], and as G , by Magnin in[Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 4 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 63 • z = x + y + ( x y − x y + x y − x y ).Since d( L x ) = − x x − x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + x dx + x dx .Finally, we have d( R x ) = x − x x x − x − x . N , , . The following Lie algebra is denoted as N , , by Gong in [Gon98], as L , byde Graaf in [dG07], as (24) by Del Barco in [dB15], and as G , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 2 that is stratifiable, also knownas the free Lie algebra of step 2 and 3 generators. The Lie brackets can be picturedwith the diagram: X X X X X X . The composition law (1.4) of N , , is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 65
Finally, we have d( R x ) = x − x x − x x − x . N , , . The following Lie algebra is denoted as N , , a by Gong in [Gon98], as L , − by de Graaf in [dG07], and as (28) by Del Barco in [dB15]. As a com-plex Lie algebra, N , , a is equivalent to the decomposable Lie algebra N , × N , ,which is why this Lie algebra is not contained in the list produced by Magnin [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra rank 4 and step 2 that is stratifiable. The Lie bracketscan be pictured with the diagram: X X X X X X . The composition law (1.4) of N , , a is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y − x y + x y ).Since d( L x ) = − x − x x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x ; • X = ∂ x − x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx − x dx + x dx + x dx ; • θ = dx − x dx + x dx − x dx + x dx .Finally, we have d( R x ) = x x − x − x x − x x − x . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 67
7D indecomposable Carnot nilpotent Lie algebras
The indecomposable Carnot Lie algebras in dimension 7 are uncountable. They canbe subdivided into 45 examples plus two families whose expressions depend on a realparameter λ . These two families are denoted as (147 E ) and (147 E ). In dimension7 there are also uncountable non-stratifiable indecomposable nilpotent Lie algebras,and not all of them are gradable. For the complete list we refer to [Gon98].(37 A ) . The following Lie algebra is denoted as (37 A ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 4 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (37 A ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx .Finally, we have d( R x ) = x − x x − x x − x . (37 B ) . The following Lie algebra is denoted as (37 B ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 4 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (37 B ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 69 • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx .Finally, we have d( R x ) = x − x x − x x − x . (37 B ) . The following Lie algebra is denoted as (37 B ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 4 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (37 B ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x x − x − x − x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x − x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx + x dx − x dx − x dx + x dx ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 71 • θ = dx − x dx − x dx + x dx + x dx ; • θ = dx − x dx + x dx .Finally, we have d( R x ) = x − x − x x x x − x − x x − x . (37 C ) . The following Lie algebra is denoted as (37 C ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 4 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (37 C ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x − x ∂ x ; • X = ∂ x − x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx .Finally, we have d( R x ) = x − x x − x x − x x − x . (37 D ) . The following Lie algebra is denoted as (37 D ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 73
This is a nilpotent Lie algebra of rank 4 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (37 D ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x − x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx .Finally, we have d( R x ) = x − x x − x x − x x − x . (37 D ) . The following Lie algebra is denoted as (37 D ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 4 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (37 D ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y + x y − x y ).Since CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 75 d( L x ) = − x x x − x − x − x x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x ; • X = ∂ x − x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx + x dx − x dx − x dx + x dx ; • θ = dx − x dx − x dx + x dx + x dx ; • θ = dx − x dx + x dx − x dx + x dx .Finally, we have d( R x ) = x − x − x x x x − x − x x − x x − x . (357 A ) . The following Lie algebra is denoted as (357 A ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (357 A ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x − x x x x − x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 77 • θ = dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx .Finally, we haved( R x ) = x − x x − x x x − x x − x x − x . (357 B ) . The following Lie algebra is denoted as (357 B ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (357 B ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ). Since d( L x ) = − x x − x − x x x x − x − x + x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx − x dx + x dx .Finally, we haved( R x ) = x − x x − x x x − x − x x + x x − x x − x . (27 A ) . The following Lie algebra is denoted as (27 A ) by Gong in [Gon98], and as G , , by Magnin in [Mag]. CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 79
The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 5 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (27 A ) is given by; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ).Since d( L x ) = − x x − x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + x dx + x dx .Finally, we haved( R x ) = x − x x x − x − x . (27 B ) . The following Lie algebra is denoted as (27 B ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 5 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (27 B ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y + x y − x y ).Since CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 81 d( L x ) = − x x − x x − x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x − x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms(1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx − x dx + x dx ; • θ = dx − x dx − x dx + x dx + x dx .Finally, we haved( R x ) = x − x x − x x x − x − x . (257 B ) . The following Lie algebra is denoted as (257 B ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 4 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (257 B ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ).Since d( L x ) = − x x − x x − x x x − x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x x + x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 83 • θ = dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx − x dx + x dx + x dx .Finally, we haved( R x ) = x − x x − x x x − x x x − x − x . (247 A ) . The following Lie algebra is denoted as (247 A ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 A ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ). Since d( L x ) = − x x − x x − x x − x x x − x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x ä ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx .Finally, we haved( R x ) = x − x x − x x − x x x − x x − x x x − x . (247 B ) . The following Lie algebra is denoted as (247 B ) by Gong in [Gon98], and as G , , by Magnin in [Mag]. CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 85
The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 B ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x − x x x − x x x − x x , the induces left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx .Finally, we haved( R x ) = x − x x − x x − x x x − x − x x x + x − x . (247 C ) . The following Lie algebra is denoted as (247 C ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 C ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ). CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 87
Since d( L x ) = − x x − x x − x x + x − x x x x − x x x − x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x + x ä ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x + x x ä dx + x dx + Ä x − x x + x ä dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx .Finally, we haved( R x ) = x − x x − x x − x x + x x x x + x − x − x x − x x x − x . (247 D ) . The following Lie algebra is denoted as (247 D ) by Gong in [Gon98], and as G , , by Magnin in [Mag]. The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 D ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x − x x x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x ä ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 89 • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx .Finally, we haved( R x ) = x − x x − x x − x x x − x − x x x x + x x x + x − x − x . (247 E ) . The following Lie algebra is denoted as (247 E ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 E ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Sinced( L x ) = − x x − x x − x x + x x − x + x x x x x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x x + x + x ä ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + x dx + x dx + Ä x + x − x x + x x ä dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx .Finally, we haved( R x ) = x − x x − x x + x − x x + x x x x − x − x − x x x x + x x x + x − x − x . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 91 (247 E ) . The following Lie algebra is denoted as (247 E ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 E ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x − x x x − x x + x x x + x x − x − x x + x , the left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x ä ∂ x − x x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x + x x − x ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x + x dx + x dx + Ä x x + x x + x ä dx − x x + x dx .Finally, we haved( R x ) = x − x x − x x − x x x − x − x x + x x x + x x + x x − x + x . (247 F ) . The following Lie algebra is denoted as (247 F ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 F ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 93 • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x + x x x − x x x − x x x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − x + x ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x + x dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx . Finally, we haved( R x ) = x − x x − x − x + x x x + x x x + x − x − x − x x x x + x x x + x − x − x . (247 F ) . The following Lie algebra is denoted as (247 F ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 F ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y − x y + x y ) + ( x − y )( x y − x y )+ ( y − x )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ). CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 95
Since d( L x ) = − x x − x x x − x x x − x x − x x x − x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x + x − x ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x − x x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x − x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx + x dx − x dx − Ä x x + x ä dx + Ä x x + x ä dx + x − x dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx .Finally, we haved( R x ) = x − x x − x x − x x x + x − x x − x − x x − x x x x + x x x + x − x − x . (247 G ) . The following Lie algebra is denoted as (247 G ) by Gong in [Gon98], and as G , , by Magnin in [Mag]. The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 G ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y + x y − x y + x y − x y )+ ( x − y + x − y )( x y − x y ) + ( x − y + x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Sinced( L x ) = − x x − x x − x x + x x + x + x − x + x x + x x − x x + x x − x x + x x + x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x x + x + x + x + x ä ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + Ä x + x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x + x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x + x ∂ x + x ∂ x ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 97 • X = ∂ x + x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x + x dx − x + x dx + Ä x + x x + x ä dx + Ä x + x x + x ä dx + Ä x + x − x x + x x + x + x ä dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx .Finally, we haved( R x ) = x − x x − x x + x − x x + x x + x + x x + x x + x x + x x + x − x + x − x + x − x x x x + x x x + x − x − x . (247 H ) . The following Lie algebra is denoted as (247 H ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 H ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y + x y − x y )+ ( x − y )( x y − x y )+ ( x − y + x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Sinced( L x ) = − x x − x x − x x + x + x − x x + x x − x x x − x x + x x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x + x + x ä ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + Ä x + x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x + x dx + Ä x x + x ä dx + Ä x + x x + x ä dx + Ä x − x x + x + x ä dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 99
Finally, we haved( R x ) = x − x x − x x − x x + x + x x + x x + x x x + x − x + x − x − x x x x + x x x + x − x − x . (247 H ) . The following Lie algebra is denoted as (247 H ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 H ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y − x y + x y )+ ( y − x )( x y − x y )+ ( x − y + x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).
00 E. LE DONNE AND F. TRIPALDI
Sinced( L x ) = − x x − x x x − x x − x − x x + x x − x x − x x x + x − x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x + Ä x − x x − x − x ä ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + Ä x + x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x − x x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x + x ∂ x + x ∂ x ; • X = ∂ x − x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx + x dx − x + x dx − Ä x x + x ä dx + Ä x + x x + x ä dx + Ä x + x − x x − x ä dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx .Finally, we haved( R x ) = x − x x − x x − x x − x + x x + x x + x − x − x x − x + x x − x x x x + x x x + x − x − x . (247 I ) . The following Lie algebra is denoted as (247 I ) by Gong in [Gon98], and as G , , by Magnin in [Mag]. CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 101
The non-trivial brackets are the following:[ X , X i ] = X i +2 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 I ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x x x − x x x − x x x − x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − x x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are:
02 E. LE DONNE AND F. TRIPALDI • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx ; • θ = dx − x dx + Ä x x + x ä dx − x dx .Finally, we haved( R x ) = x − x x − x − x x x x + x x x + x − x − x − x x x + x − x . (247 J ) . The following Lie algebra is denoted as (247 J ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The non-trivialLie brackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 J ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 103 • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Sinced( L x ) = − x x − x x − x x + x − x x + x x − x x + x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x + x ä ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x + x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x + x dx + Ä x + x x + x ä dx + Ä x − x x + x ä dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx .Finally, we haved( R x ) = x − x x − x x − x x + x x + x x + x − x + x − x x x x + x x x + x − x − x .
04 E. LE DONNE AND F. TRIPALDI (247 K ) . The following Lie algebra is denoted as (247 K ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , ≤ i ≤ , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 K ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x + x − x x x x − x x x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x + x ä ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 105 • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + x dx + Ä x − x x + x ä dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx .Finally, we haved( R x ) = x − x x − x x − x x + x x x x + x − x − x − x x x x + x x x + x − x − x . (247 N ) . The following Lie algebra is denoted as (247 N ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +2 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 N ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y );
06 E. LE DONNE AND F. TRIPALDI • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y )]; • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x − x + x x − x x x − x x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x x + x + x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + x dx + Ä x + x x ä dx + Ä x − x x + x ä dx ; • θ = dx − x dx + x dx .Finally, we haved( R x ) = x − x x − x x − x + x x x x + x x − x − x x − x . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 107 (247 P ) . The following Lie algebra is denoted as (247 P ) by Gong in [Gon98], and as G , , i λ ) with λ = 0 , X , X i ] = X i +2 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 P ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x − x x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,
08 E. LE DONNE AND F. TRIPALDI and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx .Finally, we haved( R x ) = x − x x − x x − x − x x x x + x x x + x − x − x . (247 P ) . The following Lie algebra is denoted as (247 P ) by Gong in [Gon98], and as G , , i λ ) with λ = 0 , X , X i ] = X i +2 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (247 P ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 109 • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x − x + x x x − x x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − x + x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x + x dx .Finally, we haved( R x ) = x − x x − x x − x − x + x x x + x x x + x − x − x .
10 E. LE DONNE AND F. TRIPALDI (2457 A ) . The following Lie algebra is denoted as (2457 A ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , [ X , X i ] = X i +2 , i = 4 , . This is a nilpotent Lie algebra of rank 3 and step 4 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (2457 A ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x − x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x x + x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 111 • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx ; • θ = dx − x dx + x dx − x dx + Ä x x − x x + x ä dx ; • θ = dx − x dx + x dx .Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x x − x . (2457 B ) . The following Lie algebra is denoted as (2457 B ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 3 and step 4 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (2457 B ) is given by:
12 E. LE DONNE AND F. TRIPALDI • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x − x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ä dx .Finally, we have CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 113 d( R x ) = x − x x − x x x − x x − x x − x x x − x . (2457 L ) . The following Lie algebra is denoted as (2457 L ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 4 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (2457 L ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − ( x y + x y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − ( x y + x y )( x y − x y ).
14 E. LE DONNE AND F. TRIPALDI
Since d( L x ) = − x x − x x − x x x − x x x − x x − x x − x − x x − x x + x x x − x x − x − x x − x x x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x − Ä x x + x ä ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x − Ä x x + x ä ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x + x ∂ x + x x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x x + x ä dx − x dx ; • θ = dx − x dx − x dx + x + x dx + Ä x − x x − x + x x ä dx + Ä x − x x + x x + x ä dx ; • θ = dx − x dx − x dx + x x dx + Ä x − x x − x x ä dx + Ä x − x x + x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x − x x x + x − x x − x x x − x x x + x − x − x x − x x x − x x x x − x − x . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 115 (2457 L ) . The following Lie algebra is denoted as (2457 L ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 2 and step 4 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (2457 L ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y − x y + x y ) + ( x − y )( x y − x y ) ( y − x )( x y − x y ) + ( x y − x y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y )] − ( x y + x y )( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x x − x x − x x − x x x + x x − x x − x − x x − x − x x − x x x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x − Ä x x + x ä ∂ x − Ä x + x x ä ∂ x ;
16 E. LE DONNE AND F. TRIPALDI • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x + x ä ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x − x ∂ x + x x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x − x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx + x dx − x dx + x − x dx + Ä x x − x + x x − x ä dx + Ä x − x x + x x − x ä dx ; • θ = dx − x dx − x dx + x x dx + Ä x − x x − x x ä dx + Ä x − x x + x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x − x x x + x − x x − x x x x − x x − x − x x x − x x x − x x x x − x − x . (2457 M ) . The following Lie algebra is denoted as (2457 M ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 4 that is stratifiable. The Liebrackets can be pictured with the diagram:
CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 117 X X X X X X X . The composition law (1.4) of (2457 M ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − ( x y + x y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x x − x x − x x − x − x x − x x x x x − x x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x − Ä x x + x ä ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ;
18 E. LE DONNE AND F. TRIPALDI • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx − x dx − x dx + x x dx + Ä x − x x − x x ä dx + Ä x − x x + x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x − x x x + x − x x − x x x − x x x x − x − x x − x x x − x . (23457 A ) . The following Lie algebra is denoted as (23457 A ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following[ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 5 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (23457 A ) is given by: CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 119 • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y )+ ( x y − x y )( x y − x y ) + ( y − x )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ).Since d( L x ) = − x x − x x − x x x − x x − x x x x x − x x − x − x x x − x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x x + x ä ∂ x + Ä x x − x x − x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ä dx ; • θ = dx − x dx + x dx − x dx + x dx + Ä x − x x + x x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx .
20 E. LE DONNE AND F. TRIPALDI
Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x x x − x x + x − x x − x − x x x + x − x . (23457 B ) . The following Lie algebra is denoted as (23457 B ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following[ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 2 and step 5 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (23457 B ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y − x y + x y ) + ( x − y )( x y − x y )+ ( y − x )( x y − x y ) + ( x − y )( x y − x y ) − x ( x y − x y ) CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 121 − [ x y ( x y − x y ) − x y ( x y − x y )] + ( y y − x x )( x y − x y )+ x y y ( x y − x y ) − x y ( x y − x y ) − y ( x y − x y ) ; • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ).Sinced( L x ) = − x x − x x − x x x − x x − x x x x x + x − x x x x − x x − x x − x x x x − x x − x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x x + x ä ∂ x + Ä x x + x − x x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x − x x ä ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ä dx ; • θ = dx − x dx + Ä x x + x ä dx − Ä x + x x + x x ä dx + Ä x + x x + x x + x x ä dx + Ä x − x x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx .
22 E. LE DONNE AND F. TRIPALDI
Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x x x + x − x x x x − x x + x − x − x x x x + x − x − x x x + x − x . (23457 C ) . The following Lie algebra is denoted as (23457 C ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following[ X , X i ] = X i +1 , i = 2 , , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 2 and step 5 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (23457 C ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y )+ ( x y − x y )( x y − x y ) + ( y − x )( x y − x y ); CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 123 • z = x + y + ( x y − x y − x y + x y ) + ( x − y )( x y − x y ) ( y − x )( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ) − x y ( x y − x y ) + ( x y y − x y )( x y − x y )+ ( y y − x x )( x y − x y ) − y ( x y − x y ) − x ( x y − x y ) .Sinced( L x ) = − x x − x x − x x x − x x − x x x x x − x x − x − x x x x x + x − x x x x − x x − x x − x x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x x + x ä ∂ x + Ä x x − x x − x ä ∂ x + Ä x x + x − x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x + Ä x x − x − x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x − x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ä dx ; • θ = dx − x dx + x dx − x dx + x dx + Ä x − x x + x x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − Ä x + x x + x x ä dx + Ä x + x x + x x + x x ä dx + Ä x − x x − x x ä dx .
24 E. LE DONNE AND F. TRIPALDI
Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x x x − x x + x − x x − x x x + x − x x x x − x x + x − x − x x x x + x − x . (17) . The following Lie algebra is denoted as (17) by Gong in [Gon98], and as G , , by Magnin in [Mag]. This is the 7-dimensional Heisenberg Lie algebra.The non-trivial brackets are the following:[ X , X ] = [ X , X ] = [ X , X ] = X . This is a nilpotent Lie algebra of rank 6 and step 2 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (17) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y + x y − x y ).Since d( L x ) = − x x − x x − x x , CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 125 the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x , and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx − x dx + x dx − x dx + x dx .Finally, we haved( R x ) = x − x x − x x − x . (147 D ) . The following Lie algebra is denoted as (147 D ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = [ X , X ] = [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram:
26 E. LE DONNE AND F. TRIPALDI X X X X X X X . − The composition law (1.4) of (147 D ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y + x y − x y − x y + 2 x y )+ [ x ( x y − x y ) − x ( x y − x y ) − x ( x y − x y ) − x ( x y − x y )] − [ y ( x y − x y ) − y ( x y − x y ) − y ( x y − x y ) − y ( x y − x y )].Sinced( L x ) = − x x − x x x − x x x +3 x x − x + x − x − x x x − x − x x x + x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x + x ∂ x + Ä x x +3 x x − x + x ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x − Ä x + x x ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x + Ä x − x ä ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 127 • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx + x dx − x dx ; • θ = dx − x + x dx − x dx + x dx − Ä x + x ä dx + Ä x − x x ä dx + Ä x + x + x x +3 x x ä dx .Finally, we haved( R x ) = x − x x − x − x x x x +3 x x + x + x x − x x − x − x x − x − x + x . (147 E ) . The following one-parameter family of Lie algebras is denoted as (147 E ) byGong in [Gon98], and as G , , i λ ) with λ ∈ (0 ,
1) by Magnin in [Mag].For λ ∈ (0 , X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = λX , [ X , X ] = (1 − λ ) X . Let us stress that when λ = 0 ,
1, the Lie algebra we obtain is isomorphic to (247 P ).This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . − λ − λ The composition law (1.4) of (147 E ) is given by:
28 E. LE DONNE AND F. TRIPALDI • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + [ λ ( x y − x y ) + (1 − λ )( x y − x y ) − x y + x y ]+ [(1 − λ ) x ( x y − x y ) − λx ( x y − x y ) − x ( x y − x y )] − [(1 − λ ) y ( x y − x y ) − λy ( x y − x y ) − y ( x y − x y )].Sinced( L x ) = − x x − x x x − x x + (2 λ − x x
12 (2 − λ ) x x − λx − (1 − λ ) x − (1+ λ ) x x
12 (1 − λ ) x − x λx , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x + x ∂ x + Ä x + (2 λ − x x ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x + Ä (2 − λ ) x x − λx ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x − Ä (1 − λ ) x + (1+ λ ) x x ä ∂ x ; • X = ∂ x + (1 − λ ) x ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x + λx ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx + x dx − x dx ; • θ = dx − λx dx + x dx − (1 − λ ) x dx + Ä (1 − λ ) x − (1+ λ ) x x ä dx + Ä λx + (2 − λ ) x x ä dx + Ä (2 λ − x x − x ä dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 129
Finally, we haved( R x ) = x − x x − x − x x (2 λ − x x − x − λ ) x x + λx − λ ) x − (1+ λ ) x x − (1 − λ ) x x − λx . (147 E ) . The following one-parameter family of Lie algebras is denoted as (147 E )by Gong in [Gon98], and as G , , i λ ) with λ > λ >
1, the non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = 2 X , [ X , X ] = − λX , [ X , X ] = λX , [ X , X ] = − X . Let us stress that when λ = 1, the Lie algebra we obtain is isomorphic to (247 P ).This is a nilpotent Lie algebra of rank 3 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . − λ − λ − The composition law (1.4) of (147 E ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + x y − x y − x y + x y + [ λ ( x y − x y ) − λ ( x y − x y )]+ ( λx − λy )( x y − x y ) + ( λx − λy )( x y − x y )+ ( y − x )( x y − x y ) + ( y − x )( x y − x y ).
30 E. LE DONNE AND F. TRIPALDI
Sinced( L x ) = − x x − x x x − x a a x + λx − x x + λx − x λx x − λx , where a = λx x x − λx x
12 ; a = − λx + 2 x − x x + λx x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x + x ∂ x + Ä λx + x x − λx x ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x − Ä λx +2 x + x x + λx x ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x + Ä x + λx − x x + λx ä ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x + λx ∂ x ; • X = ∂ x + x − λx ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx + x dx − x dx ; • θ = dx + λx − x dx − λx dx + x dx + Ä λx − x x + λx − x ä dx + Ä λx +2 x − λx x +2 x x ä dx − Ä λx + λx x − x x ä dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 131
Finally, we haved( R x ) = x − x x − x − x x a a λx − x x + λx − x x − λx λx − x , where a = 4 x x − λx x − λx a = λx + 2 x − x x + λx x , (137 A ) . The following Lie algebra is denoted as (137 A ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 4 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (137 A ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ).
32 E. LE DONNE AND F. TRIPALDI
Since d( L x ) = − x x − x x − x x − x x − x − x x x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + x dx + Ä x − x x ä dx + x dx + Ä x − x x ä dx .Finally, we haved( R x ) = x − x x − x x − x x x x − x x x − x − x . (137 A ) . The following Lie algebra is denoted as (137 A ) by Gong in [Gon98], andas G , , by Magnin in [Mag]. CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 133
The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = X . This is a nilpotent Lie algebra of rank 4 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (137 A ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y − x y + x y ); • z = x + y + ( x y − x y + x y − x y )+ ( x − y )( x y − x y + x y − x y )+ ( x − y )( x y − x y − x y + x y )].Sinced( L x ) = − x − x x x − x x − x x − x x + x x − x x x − x x − x x − x x x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x + x x + x x ä ∂ x ; • X = ∂ x − x ∂ x + x ∂ x + Ä x x − x x − x ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x + x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ;
34 E. LE DONNE AND F. TRIPALDI • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx − x dx + x dx + x dx ; • θ = dx − x dx + x dx − x dx + x dx ; • θ = dx − x dx − x dx + x x dx + x − x dx + Ä x + x x − x x ä dx + Ä x − x x + x x ä dx .Finally, we haved( R x ) = x x − x − x x − x x − x x − x x + x x x x − x x + x x − x x x − x − x . (137 C ) . The following Lie algebra is denoted as (137 C ) by Gong in [Gon98], and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 4 and step 3 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (137 C ) is given by: • z = x + y ; • z = x + y ; • z = x + y ; • z = x + y ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 135 • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ); • z = x + y + ( x y − x y − x y + x y )+ ( x − y )( x y − x y + x y − x y )+ ( y − x )( x y − x y )].Since d( L x ) = − x x − x − x x x x x − x x − x − x x x x + x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x + Ä x x − x x − x ä ∂ x ; • X = ∂ x + x ∂ x − x ∂ x − x x ∂ x ; • X = ∂ x + x ∂ x + Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x − x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx − x dx + x dx + x dx ; • θ = dx − x dx + x dx + x dx + Ä x x − x ä dx − x x dx + Ä x + x x − x x ä dx .Finally, we haved( R x ) = x − x x x − x − x x x − x x + x − x x x x − x x x − x .
36 E. LE DONNE AND F. TRIPALDI (12457 H ) . The following Lie algebra is denoted as (12457 H ) by Gong in [Gon98],and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , , , [ X , X j ] = X j +2 , j = 3 , , [ X , X ] = X . This is a nilpotent Lie algebra of rank 2 and step 5 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (12457 H ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − ( x y + x y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y )+ ( x − y )( x y − x y + x y − x y )+ ( x − y )( x y − x y ) + ( y − x )( x y − x y ) − [ x y ( x y − x y )+ x y ( x y − x y ) + x y ( x y − x y )] + ( y + 3 x )( x y − x y ) + ( x y + x y y − x x y )( x y − x y )+ ( y y − x x )( x y − x y ). CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 137
Sinced( L x ) = − x x − x x − x x x − x x x − x x − x x − x − x x − x x x x x x x + x x − x − x x − x − x x − x x x x − x x x + x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x − Ä x x + x ä ∂ x + Ä x x + x x − x − x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x − Ä x + x x ä ∂ x − Ä x x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x x ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x x + x ä dx − x dx ; • θ = dx − x dx − x dx + x x dx + Ä x − x x − x x ä dx + Ä x − x x + x x ä dx ; • θ = dx − x dx + x dx + Ä x x − x ä dx + Ä x + x x − x x ä dx + Ä x x − x x ä dx + Ä x + x x − x − x x + x x x − x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x − x x x + x − x x − x x x − x x x x − x − x x x + x x − x − x x + x − x x − x x x x + x x x − x x − x .
38 E. LE DONNE AND F. TRIPALDI (12457 L ) . The following Lie algebra is denoted as (12457 L ) by Gong in [Gon98], andas G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , , , [ X , X j ] = X j +2 , j = 3 , , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 2 and step 5 that is stratifiable. The Liebrackets can be pictured with the diagram: X X X X X X X . The composition law (1.4) of (12457 L ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − ( x y + x y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y + x y − x y − x y + x y )+ ( x − y )( x y − x y + x y − x y )+ ( x − y )( x y − x y + x y − x y )+ ( x − y )( x y − x y − x y + x y ) + ( x − y − x + y )( x y − x y )+ ( x y − x y )( x y − x y ) − ( x y + x y )( x y − x y ) − ( x y + x y )( x y − x y ) + ( x y + x y + x y y )( x y − x y ) CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 139 + ( x y y − x x y − x x y )( x y − x y ) + ( y − y )( x y − x y ) + ( y y + y y − x x − x x )( x y − x y ) + ( x − y )( x y − x y ) .Sinced( L x ) = − x x − x x − x x x − x x x − x x − x x − x − x x − x x x x x a a a x x + x + x x + x x − x x + x , where a = x x + x x
360 + x x − x − x x − x x − x ,a = x x − x x − x x + x − x x + x x − x ,a = x x − x x
12 + x − x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x − Ä x x + x ä ∂ x + Ä x x + x x + x x − x − x x − x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x − Ä x + x x ä ∂ x + Ä x x − x x − x x + x − x x + x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x x ∂ x + Ä x x − x x + x − x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x x + x + x ä ∂ x ; • X = ∂ x + x ∂ x + Ä x + x x − x ä ∂ x ; • X = ∂ x + x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x x + x ä dx − x dx ; • θ = dx − x dx − x dx + x x dx + Ä x − x x − x x ä dx + Ä x − x x + x x ä dx ;
40 E. LE DONNE AND F. TRIPALDI • θ = dx − x + x dx + Ä x + x + x x ä dx + Ä x x + x − x ä dx + Ä x − x + x x − x x − x x + x x ä dx + Ä x x + x x + x x x + x − x x − x x + x x + x ä dx + Ä x + x x − x − x x − x x + x x x − x x + x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x − x x x + x − x x − x x x − x x x x − x − x a a a x x + x − x x + x x + x − x + x , where a = x x + x x
360 + x x − x − x x − x x
12 + x ,a = x x − x x − x x + x − x x + x x
360 + x ,a = x x − x x
12 + x − x . (12457 L ) . The following Lie algebra is denoted as (12457 L ) by Gong in [Gon98],and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , i = 2 , , [ X , X ] = − X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = − X , [ X , X ] = − X . This is a nilpotent Lie algebra of rank 2 and step 5 that is stratifiable. The Liebrackets can be pictured with the diagram:
CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 141 X X X X X X X . The composition law (1.4) of (12457 L ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( y − x )( x y − x y )+ ( y − x )( x y − x y )] + ( x y + x y )( x y − x y ); • z = x + y + ( x y − x y − x y + x y ) + ( x − y )( x y − x y )+ ( y − x )( x y − x y + x y − x y ) + ( y − x )( x y − x y )]+ [ x y ( x y − x y ) + x y ( x y − x y ) + x y ( x y − x y )]+ ( x y + x y − x y − x y y )( x y − x y ) − y ( x y − x y ) + ( x + x x − y − y y )( x y − x y ) − x ( x y − x y ) .Sinced( L x ) = − x x − x x − x x x − x x x − x x x x + x x x + x − x + x − x − x a a x − x x − x − x x − x x , where a = x x − x x − x − x x + x x ,a = x + 2 x x
12 + x + x x ,
42 E. LE DONNE AND F. TRIPALDI the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x + Ä x x + x ä ∂ x + Ä x x − x x − x − x x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x + x x ä ∂ x + Ä x +2 x x + x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x + x ∂ x + Ä x − x x ä ∂ x ; • X = ∂ x − x ∂ x − x ∂ x ; • X = ∂ x − x ∂ x − Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x x + x ä dx − x dx ; • θ = dx + x dx + x dx − x + x dx + Ä x + x x + x x − x ä dx + Ä x x − x x + x − x ä dx ; • θ = dx − x dx + Ä x − x x ä dx − x dx + Ä x + x x − x − x x ä dx + Ä x +2 x x − x + x x ä dx + Ä x + x x − x x − x x + x x + x x + x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x − x x x + x − x x x − x x x − x − x + x x x a a − x − x x − x x − x x − x , where a = x x − x x
12 + x − x x + x x ,a = x + 2 x x
12 + x + x x . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 143 (123457 A ) . The following Lie algebra is denoted as (123457 A ) by Gong in [Gon98],and as G , , by Magnin in [Mag].The non-trivial brackets are the following:[ X , X i ] = X i +1 , ≤ i ≤ . This is a nilpotent Lie algebra of rank 2 and step 6 that is stratifiable, also knownas the filiform Lie algebra of dimension 7. The Lie brackets can be pictured with thediagram: X X X X X X X . The composition law (1.4) of (123457 A ) is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y )+ ( y − x )( x y − x y ) + ( x y − x y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y )+ ( x y − x y )( x y − x y ) + ( y − x )( x y − x y )+ x y ( x y − x y ) + ( x y + x y )( x y − x y ).
44 E. LE DONNE AND F. TRIPALDI
Since d( L x ) = − x x − x x − x x x − x x − x x x x x − x x − x − x x x x x − x x − x − x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x + Ä x x − x x − x ä ∂ x + Ä x x − x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x x − x x + x ä dx ; • θ = dx − x dx + x dx − x dx + x dx + Ä x − x x + x x − x x ä dx ; • θ = dx − x dx + x dx − x dx + x dx − x dx + Ä x − x x + x x − x x + x x ä dx .Finally, we haved( R x ) = x − x x − x x x − x x − x x x − x x x − x x + x − x x − x x x − x x + x − x x − x . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 145 Some free-nilpotent groups in low dimension
In this final section we analyze free-nilpotent groups of low dimension. We shalldenote with F rs the simply connected Lie group whose Lie algebra has rank r andis free up to nilpotency step s . In the specific, we shall study F , F , F , F ,respectively. F . The following is the free-nilpotent Lie algebra with 2 generators and nilpotencystep 3. It has dimension 5. This Lie algebra is also denoted as N , , , see page 19.The non-trivial brackets coming from the Hall basis are:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X . A representation of the left-invariant vector fields X si defined on R with respectto exponential coordinates of the second kind is the following: • X s = ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x ; • X s = ∂ x ; • X s = ∂ x .One can relate the exponential coordinates of first kind to the exponential coor-dinates of second kind. If we denote by α , . . . , α the coordinates of second kindand by a , . . . , a the coordinates of first kind, the change of coordinates are given asfollows: a = α a = α a = α − α α a = α − α α α α a = α − α α − α α , α = a α = a α = a + a a α = a + a a a a α = a + a a a a . More explicitly, we haveexp( α X ) exp( α X ) · · · exp( α X ) == exp Ç α X + α X + ( α − α α X + ( α − α α α α
12 ) X + ( α − α α − α α
12 ) X å , and viceversaexp ( a X + a X + a X + a X + a X ) == exp(( a + a a a a X ) exp(( a + a a a a X ) · exp(( a + 12 a a ) X ) · exp( a X ) · exp( a X ) .
46 E. LE DONNE AND F. TRIPALDI F . The following is the free-nilpotent Lie algebra with 2 generators and nilpotencystep 4. It has dimension 8.The non-trivial brackets coming from the Hall basis are:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = [ X , X ] = X , [ X , X ] = X . The composition law (1.4) of F is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y )] − ( x y + x y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y )Since d( L x ) = − x x − x − x x x x − x x x − x x − x − x x x x − x − x x − x − x x x x x x − x − x x x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 147 • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ä dx ; • θ = dx − x dx − x dx + x x dx + Ä x − x x − x x ä dx + Ä x − x x + x x ä dx ; • θ = dx − x dx + x dx + Ä x − x x − x x ä dx + x dx .Finally, we haved( R x ) = x − x x − x x x − x − x x x + x − x x − x x x − x x − x x x − x x x x − x − x x − x x x − x . One can relate the exponential coordinates of first kind to the exponential coor-dinates of second kind. If we denote by α , . . . , α the coordinates of second kindand by a , . . . , a the coordinates of first kind, the change of coordinates are given asfollows: a = α a = α a = α − α α a = α − α α + α α a = α − α α − α α a = α − α α + α α a = α − α α + α α + α α α + α α a = α − α α + α α and
48 E. LE DONNE AND F. TRIPALDI α = a α = a α = a + a a α = a + a a + a a α = a + a a + a a α = a + a a + a a + a a α = a + a a + a a + a a a + a a α = a + a a + a a + a a . . More explicitly, we haveexp( α X ) exp( α X ) · · · exp( α X ) == exp Ç α X + α X + ( α − α α X + ( α − α α α α
12 ) X ++( α − α α − α α
12 ) X + ( α − α α α α
12 ) X ++( α − α α + α α α α α α α
24 ) X + ( α − α α α α
12 ) X å and viceversaexp ( a X + a X + · · · + a X + a X ) == exp Ç ( a + a a a a a a X å · exp Ç ( a + a a + a a a a a a a X å ·· exp Ç ( a + a a a a a a
24 ) X å · exp Ç ( a + a a a a X å ·· exp Ç ( a + a a a a X å · exp Ç ( a + 12 a a ) X å · exp( a X ) · exp( a X ) . F . The following is the free-nilpotent Lie algebra with 3 generators and nilpotencystep 3. It has dimension 14.The non-trivial brackets coming from the Hall basis are:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X + X . The composition law (1.4) of F is given by: • z = x + y ; • z = x + y ; • z = x + y ; CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 149 • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ).Sinced( L x ) = − x x − x x − x x − x − x x x x − x − x x x x − x − x x − x − x x x x x x − x x x − x x − x x − x x x − x x x x − x x x − x x x − x x x − x x − x x x − x x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x − x ∂ x − x x ∂ x − x ∂ x ; • X = ∂ x + x ∂ x − x ∂ x + x ∂ x − Ä x x + x ä ∂ x + Ä x x − x ä ∂ x − Ä x + x x ä ∂ x + Ä x x − x ä ∂ x − x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x − x ä ∂ x + Ä x x − x ä ∂ x ;
50 E. LE DONNE AND F. TRIPALDI • X = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx − x dx + x x dx + Ä x − x x ä dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx − x dx + Ä x x + x ä dx + Ä x x + x ä dx − x x dx ; • θ = dx − x dx + Ä x x + x ä dx − x dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 151
Finally, we haved( R x ) = x − x x − x x − x x − x x x − x x − x x x − x x − x x x − x x x x − x − x − x x x + x − x x − x x x − x − x x x x + x x x + x − x − x − x x x + x − x − x x x + x − x . A representation of the left-invariant vector fields X si defined on R with respectto exponential coordinates of the second kind is the following: • X s = ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x +( x x − x ) ∂ x + x x ∂ x + x x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x .One can relate the exponential coordinates of first kind to the exponential coor-dinates of second kind. If we denote by α , . . . , α the coordinates of second kindand by a , . . . , a the coordinates of first kind, the change of coordinates are givenas follows:
52 E. LE DONNE AND F. TRIPALDI a = α a = α a = α a = α − α α a = α − α α a = α − α α a = α − α α + α α a = α − α α + α α a = α − α α + α α + α α α a = α − α α − α α a = α − α α + α α a = α + α α − α α − α α α a = α − α α − α α a = α − α α − α α and α = a α = a α = a α = a + a a α = a + a a α = a + a a α = a + a a + a a α = a + a a + a a α = a + a a + a a + a a a α = a + a a + a a α = a + a a + a a α = a + a a − a a + a a a α = a + a a + a a α = a + a a + a a . . More explicitly, we haveexp( α X ) exp( α X ) · · · exp( α X ) == exp Å α X + α X + α X + ( α − α α X + ( α − α α X ++( α − α α X + ( α − α α α α
12 ) X + ( α − α α α α
12 ) X + CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 153 +( α − α α + α α α α α X + ( α − α α − α α
12 ) X ++( α − α α α α
12 ) X + ( α + α α − α α − α α α X ++( α − α α − α α
12 ) X + ( α − α α − α α
12 ) X å and viceversaexp ( a X + a X + · · · + a X + a X ) == exp Ç ( a + a a a a X å · exp Ç ( a + a a a a X å ·· exp Ç ( a + a a − a a a a a X å · exp Ç ( a + a a a a X å ·· exp Ç ( a + a a a a X å · exp Ç ( a + a a + a a a a a X å ·· exp Ç ( a + a a a a X å · exp Ç ( a + a a a a X å · exp Ç ( a + a a X å ·· exp Ç ( a + a a X å · exp Ç ( a + a a X å · exp( a X ) · exp( a X ) · exp( a X ) . F . The following is the free-nilpotent Lie algebra with 2 generators and nilpotencystep 5. It has dimension 14.The non-trivial brackets coming from the Hall basis are:[ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = [ X , X ] = X , [ X , X ] = X + X , [ X , X ] = X , [ X , X ] = X + X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X , [ X , X ] = X . The composition law (1.4) of F is given by: • z = x + y ; • z = x + y ; • z = x + y + ( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) − ( x y + x y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y );
54 E. LE DONNE AND F. TRIPALDI • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y )+ ( y − x )( x y − x y ) + ( x y − x y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y + x y − x y ) − ( x y + x y )( x y − x y ) − x y ( x y − x y ) + ( x y y − x x y )( x y − x y )+ ( y y − x x )( x y − x y ) + ( x y − x y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y + x y − x y ) − ( x y + x y )( x y − x y ) − x y ( x y − x y ) + ( y y − x x )( x y − x y )+ ( x y y − x x y )( x y − x y ) + ( x y − x y )( x y − x y ); • z = x + y + ( x y − x y ) + ( x − y )( x y − x y ) − x y ( x y − x y )+ ( y − x )( x y − x y ) + ( x y − x y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y + x y − x y ) + ( y − x )( x y − x y )] − [ x y ( x y − x y ) + x y ( x y − x y ) + x y ( x y − x y )]+ (3 x + y )( x y − x y ) + ( y y − x x )( x y − x y ) − x x y ( x y − x y ) + ( x y + x y y )( x y − x y ); • z = x + y + ( x y − x y + x y − x y ) + ( x − y )( x y − x y )+ ( x − y )( x y − x y ) + ( y − x )( x y − x y ) − x y ( x y − x y ) − x y ( x y − x y ) + (3 x + y )( x y − x y ) + ( y y − x x )( x y − x y ) + ( x y y − x y )( x y − x y ).Sinced( L x ) = − x x − x − x x x x − x x x − x x − x − x x x x − x − x x − x − x x x x x x − x − x x x x a − x x x a a x x x x x a a x x x x x x a x x a a a a x x a a a a x , CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 155 where a = x x − x x − x a = x x − x x + x x − x a = − x x − x x − x a = x x − x x − x a = − x x − x x + x x − x a = − x x − x x − x a = x x − x + x x − x x − x a = − x x − x x a = x x − x a = x x
12 + x a = x x
720 + x x − x a = − x x − x + 2 x x
12 ; a = x x − x a = x x
12 + x , the induced left-invariant vector fields (1.6) are: • X = ∂ x − x ∂ x − Ä x + x x ä ∂ x − x ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x + Ä x x − x x − x ä ∂ x + Ä x x − x x + x x − x ä ∂ x + Ä x x − x x − x ä ∂ x + x ∂ x + Ä x x − x − x x + x x − x ä ∂ x + Ä x x + x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + Ä x x − x ä ∂ x − Ä x + x x ä ∂ x − Ä x + x x ä ∂ x − x ∂ x − Ä x x + x x + x ä ∂ x − Ä x x + x x + x x + x ä ∂ x − Ä x x + x x + x ä ∂ x − Ä x x + x x ä ∂ x − Ä x x + x +2 x x ä ∂ x ;
56 E. LE DONNE AND F. TRIPALDI • X = ∂ x + x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x + Ä x x − x ä ∂ x + Ä x x − x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x + Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x + x ∂ x + Ä x x + x ä ∂ x ; • X = ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ; • X = ∂ x ,and the respective left-invariant 1-forms (1.7) are: • θ = dx ; • θ = dx ; • θ = dx − x dx + x dx ; • θ = dx − x dx + x dx + Ä x − x x ä dx ; • θ = dx − x dx + Ä x + x x ä dx − x dx ; • θ = dx − x dx + x dx − x dx + Ä x − x x + x x ä dx ; • θ = dx − x dx − x dx + x x dx + Ä x − x x − x x ä dx + Ä x − x x + x x ä dx ; • θ = dx − x dx + x dx + Ä x − x x − x x ä dx + x dx ; • θ = dx − x dx + x dx − x dx + x dx + Ä x − x x + x x − x x ä dx ; • θ = dx − x dx + x dx − x dx + x x dx − x x dx + Ä x x + x x − x x + x ä dx + Ä x x x − x x − x x + x x − x ä dx ; • θ = dx − x dx − x dx + x x dx + x dx − x x dx + Ä x x + x x x − x x + x x − x ä dx + Ä x x − x x − x x − x ä dx ; • θ = dx − x dx + x dx − x dx + Ä x x + x x − x x − x ä dx + x dx ; • θ = dx − x dx + x dx + Ä x x − x ä dx + Ä x + x x − x x ä dx + Ä x x − x x ä dx − Ä x x − x x x + x − x x + x x + x ä dx ; • θ = dx − x dx + Ä x x − x ä dx + Ä x + x x − x x ä dx + Ä x x − x +2 x x ä dx − Ä x x − x x − x x + x ä dx . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 157
Finally, we haved( R x ) = x − x x − x x x − x − x x x + x x x − x x x − x x − x x x − x x x x − x − x x − x x x − x a − x x − x a a x x x − x − x a a x x x − x − x x a x − x a a a a x − x a a a a − x , where a = x x − x x
12 + x a = x x − x x + x x
12 + x a = − x x − x x
12 + x a = x x − x x
12 + x a = − x x − x x + x x
12 + x a = − x x − x x
12 + x a = x x − x + x x − x x
12 + x a = − x x − x x a = x x
12 + x a = x x − x a = x x
720 + x x
12 + x
58 E. LE DONNE AND F. TRIPALDI a = − x x − x + 2 x x
12 ; a = x x
12 + x a = x x − x , A representation of the left-invariant vector fields X si defined on R with respectto exponential coordinates of the second kind is the following: • X s = ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x + x x ∂ x + x x ∂ x + x ∂ x + x x ∂ x + x x ∂ x + x x ∂ x + x x ∂ x + x x x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x x ∂ x + x ∂ x + x x ∂ x + x x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x + x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x ∂ x + x x ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x ; • X s = ∂ x + x ∂ x + x ∂ x + x ∂x ; • X s = ∂ x + x ∂ x + x ∂ x + x ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x ; • X s = ∂ x .One can relate the exponential coordinates of first kind to the exponential coor-dinates of second kind. If we denote by α , . . . , α the coordinates of second kindand by a , . . . , a the coordinates of first kind, the change of coordinates are givenas follows: CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 159 a = α a = α a = α − α α a = α − α α + α α a = α − α α − α α a = α − α α + α α a = α − α α + α α + α α α + α α a = α − α α + α α a = α − α α + α α − α α a = α − α α + α α + α α + α α α − α α a = α − α α + α α + α α α + α α + α α a = α − α α + α α + α α a = α − α α + α α + α α − α α + α α α − α α a = α − α α + α α + α α α − α α + α α α − α α and α = a α = a α = a + a a α = a + a a + a a α = a + a a + a a α = a + a a + a a + a a α = a + a a + a a + a a a + a a α = a + a a + a a + a a α = a + a a + a a + a a + a a α = a + a a + a a + a a + a a a + a a a + a a α = a + a a + a a + a a a + a a + a a a + a a α = a + a a + a a + a a + a a α = a + a a + a a + a a + a a a + a a + a a a + a a α = a + a a + a a + a a a + a a + a a a + a a . More explicitly, we haveexp( α X ) exp( α X ) · · · exp( α X ) =
60 E. LE DONNE AND F. TRIPALDI = exp Ç α X + α X + ( α − α α X + ( α − α α α α
12 ) X ++( α − α α − α α
12 ) X + ( α − α α α α
12 ) X ++( α − α α + α α α α α α α
24 ) X + ( α − α α α α
12 ) X ++( α − α α α α − α α
720 ) X +( α − α α + α α α α + 2 α α α − α α
180 ) X ++( α − α α + α α α α α + α α
12 + α α
180 ) X +( α − α α α α
12 + α α
720 ) X ++( α − α α + α α α α − α α
12 + α α α − α α
120 ) X ++( α − α α + α α α α α − α α + α α α − α α
360 ) X å and viceversaexp ( a X + a X + · · · + a X + a X ) == exp Ç ( a + a a + a a a a a a a a a a a a
10 ) X å ·· exp Ç ( a + a a + a a a a + a a a a a a a a a a
20 ) X å ·· exp Ç ( a + a a a a a a
24 + a a
30 ) X å ·· exp Ç ( a + a a + a a a a a a a a a a a a
20 ) X å ·· exp Ç ( a + a a + a a a a a a a a a a a a
30 ) X å ·· exp Ç ( a + a a a a a a
24 + a a
120 ) X å ·· exp Ç ( a + a a a a a a X å · exp Ç ( a + a a + a a a a a a a X å ·· exp Ç ( a + a a a a a a
24 ) X å · exp Ç ( a + a a a a X å ·· exp Ç ( a + a a a a X å · exp Ç ( a + 12 a a ) X å · exp( a X ) · exp( a X ) . CORNUCOPIA OF CARNOT GROUPS IN LOW DIMENSIONS 161
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Francesca Tripaldi: University of Bologna, Department of Mathematics, Piazzadi Porta S. Donato 5, 40126 Bologna, Italy
E-mail address : [email protected] Enrico Le Donne: Dipartimento di Matematica, Universit`a di Pisa, Largo B. Pon-tecorvo 5, 56127 Pisa, Italy, &, University of Jyv¨askyl¨a, Department of Mathematicsand Statistics, P.O. Box (MaD), FI-40014, Finland
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