Automorphisms of Non-Singular Nilpotent Lie Algebras

Journal of Líe Theory Volume 23 (2013) 1085-1100 © 2013 Heldermann Verlag

Automorphisms of Non-Singular Nilpotent Lie Algebras

AroIdo Kaplan and Alejandro Tiraboschi

Communícated by D. Müller

Abstract. For a real, non-singular, 2-step nilpotent Líe algebra n, the group Aut(n)/ Auto(n), where Auto(n) is the group of automorphisms which act trivially on the center, is the direct product of a compact group with the l-dimensional group of dilations. Maximality of some automorphísms groups of n follows and is related to how close is n to being of Heisenberg type. For example, at least when the dimension ofthe center is two, dimAut(n) is maximal if and only if n is of Heisenberg type. The connection with fat distributions is discussed. Matbematics Subject Classification 2010: 17B30, 16W25. Key Words and Pbrases; Líe groups, Líe algebras, Heisenberg type groups.

1. Introduction

A 2-step nilpotent real Líe algebra n with center J is caUed non-sing1Llar [E], or said to satisfy hypothesis (H) [M], or be the Lie algebra of a Métivier group [MS], if ad x : n -+ J is onto for any x rt J. Equivalently, the bracket defines a vector-valued antisymmetric form

[, l: U x u-+ J,

U = n/J, such that the 2-forms >.([u, v]) on U are non-degenerate for all >. E J*, >. =f O. Here we shall call such algebras fat for short, since they are the symbols of fat distributions (as opposite to "flat", or integrable, ones [Mo]), which motivate the questions.

Let m = dim(J). While for m = 1 there is only one fat algebra up to isomorphisms, for m 2: 2 there is an uncountable number of isomorphism classes and for m 2: 3 they form a wild seto

In this paper we study the size of groups of alltomorphisms of n. Aut(n) itself is the semidirect product of the group G(n) of graded automorphisms of n = U EB J with the abelían group Hom(tl,J), times the group of dilations (t, t2 ).

Hence, we concentrate on G = G(n). We prove that there is an exact sequence

1 -+ Go -+ G -+ O(m)

ISSN 0949--5932 / $2.50 © Heldermann Verlag

1086 KAPLAN AND TIRABOSCHI

where Go is the subgroup of G of elements that act trivially on the center. In other words, there are positive-definite inner products on 3 which are invariant under all of Aut(n).

If a metric g is also given on tl, as in the case of the nilpotentization of a subriemannian structure, we also consider the subgroups Ka, K, of graded automorphisms that leave g invariant, which define a compatible exact sequence

1 --t Ko --t K --t O(m).

::\'ext, we compute the terms in this sequence and the images G/Go and K/Ko, proving that the exactness of

1 --t Lie(Ko) --t Lie(K) --t so(m) --t 1

is equivalent to n being of Heisenberg type, while the exactness of

1 --t Lie(Go) --t Lie(G) so(m) --t 1

is strictly more general. As to Go(n) , we describe it in detail for the case m = 2, leading a proof that, at least in that case, dimAut(n) is maximal if and only if n is of Heisenberg type.

In the last section we explain the connection with the Equivalence Problem for fat subriemannian distributions.

AIgebras of Heisenberg type are defined as follows [K]. If tl is a real unitary module over the Clífford algebra CI(3) associated to a qUadratic form on 3, the identity

< z, [u, v] z· U,v >u

with z E 3 e CI(3) , u, v E tl, defines a fat [ , ] : b x b --t 3. Alternatively, they are characterized by possessing a positive-definite metric such that the operator z· defined by the aboye equation satisfies z· (z . v) -lzI 2v.

lt follows from Adam's theorem on frames on spheres [H] that for any fat algebra there is an Heisenberg type algebra with the same dim 3 and dim b. That these were, in sorne sense, the most symmetric, was expected from the properties of their sublaplacians [BTV] [CaN] [aVI [K], but we found no explicit statements in this regard.

Related properties of the automorphism groups of nilpotent Lie groups are studied in [P] and [MS].

2. Automorphisms of fat algebras

Let n be a 2-step Líe algebra with center 3 and let b = n/3, so that

(1)

and the Lie algebra structure is encoded into the map


Let n = dimtJ and m = dimJ. Relative to a basis compatible with (1), the bracket becomes an )R.m -valued antisymmetric form on )R.n and an automorphism ís a matrix of the form

a E GL(n), bE GL(m), e E )R.nxm

such that b([u, vD [au, av].

Aut(n) always contains the normal subgroup :D(n) of dilations and translations

tER*, cE)R.nxm.

Let

G = G(n) = {(~ ~), a E SL(n), bE GL(m), b([u, vD = [au, av]}.

Then Aut(n) is the semídírect product of G(n) with :D(n). Let

Go Go(n) = {(~ ¡:), a E SL(n), [au, av] [u, v]},

the subgroup of automorphisms that act trivially on the center. These are Líe groups, Go ís normal in G, and the quotient group

G/Go

can be identified with the group of b E GL(J) such that b([u, v]) = [au, av] for some a E SL(b). Obviously,

dimAut(n) nm + 1 + dím(G/Go) + dim(Go). (2)

Theorem 2.1. Let n be a fat algebra with center J. Then there is a positive definite metric on J invariant under G(n).

Proof. Fix arbitrary positive inner products on b and J. For z E J, u, v E tJ

defines a linear map z 1-7 T z from J to End( tJ). Clearly,

n fat {::} Tz E GL(b) I¡/z =1- o.

Hence the hypothesis insures that the Pfaffian

pez) det(Tz )

is non-zero on b\ {O}. Thís ís a homogeneous polynomial of degree n, so it satisfies

(3)


where k, K are the mínimum and maximum values of IPI on the unít spbere, which are positive.

1et now o9a,b:= (~ ~) E Aut(n). Tben

because (ntzu, v)n = Wz, [u, vD, = (z, b([u, v]))J (z, [au, av]). (Tzau, av)o = (atTzau, v)u. Consequently

In particular, if 09 E G tben PWz) P(z). Thís implies

klWzlln:-:; IPWz)1 IP(z)l:-:; Kllzlln

for all z, therefore I¡bll :-:; :viKJk. The group of b E GL(.;) such that o9a,b E Aut(n) for sorne a E 81(\.1), is therefore bounded in End(Rm). Its dosure is a compact Lie subgroup of GL(.;) , necessarily contained in 0(.;) for sorne positive definite metric. •

From now on .; will be assumed endowed with sucb invariant metric. If a metric 09 on \.1 is also fixed, as in the case of tbe nilpotentization of a subriemannian structure, define the groups

K K(n,09) = {(~ ~), a E 80(\.1), bE 0(.;), [au, av] = b[u, v]}

Ko Ko(n, 09) = {(~ ~), a E 80(\.1), [au,av] = [u,v]}.

Let {J, {Jo, t, to be the Lie algebras of G, Go, K, Ko respectively. Tben there is the commutative diagram with exact rows

o -+ {Jo -+ {J +so(m)

t t t O -+ to -+ t +so(m)

where the vertical arrows are tbe indusions. Below we prove tbat the bottom sequence extends to

0-+ to -+ t -+ so(m) -+ O

if and only if n is of Heisenberg type. This is not the ca.'le for the top one: the condition that

O-+ {Jo -+ {J -+ so(m) -+ O

is exact defines a dass of fat algebras strictly larger than Heisenberg type algebras. We describe it in the next section for m 2.

Proposition 2.2. Let n = 1:1 +.; be an algebra of Heisenberg type. There is a metric on.; such that {JJgo ~so(m).


Proof. There is an inner product in ti such that the Ji = T;, 's satisfy the Canonical Anticommutation Relations

For IIzll = 1 let rz E 0(3) be the reflection through the hyperplane orthogonal to z and Jz E SL(ti) be as above. Then

g(J,,--Tx ) = (¿ -~J E Aut(n).

Indeed,

(w, [Jzu, Jzv]) = (JwJzu, Jzv) (-JzJwu 2(z, w)v" Jzv) -(JzJwu, Jzv) - 2(z, w)(u, Jzv) (Jwu, JzJzv) + 2(z, w)(Jzu, v)

= -(Jwu, v) + 2(z, w)(Jzu, v) = (J-w+2(Z,w)ZU, v) = (-w + 2(z, w)z, [u, vj) = (-rz(w), [u, v])

(w, -rzC[u, v])),

so that -rzC[u, vD = [JzU, JzvJ.

The Lie group generated by the -rz has finite index in 0(3). • Corollary 2.3. Let n be a fat algebra with center of dimension m. Then

dim(K/ Ko} S dím(G/Go) S m(m - 1)/2

with equality achieved for any Heísenberg type algebra of the same dimension with center of the same dimensiono

Since Aut(n)/ Auta(n) (G/Go)x (dilations), one obtaíns

Corollary 2.4. Let n be a fat algebra with center of dímension m. Then

dim(Aut(n)/ Auto(n)) S 1 + m(m - 1)/2,

with equality achíeved for any Heísenberg type algebra of the same dímension and with center of the same dimensiono

A converse for Corollary 2.3 is

Theorem 2.5. 1f n is fat with center of dimension m and

dim(K/ Ka) = m(m -1)/2

for some metric on ti, then n is of Heísenberg type.


Proof. The hypothesis implies that ~féo = {I/{lo ~ .5o(m), so that K/ Ko acts

transitively among the 1z 1 = L For (~ ~) in this group, - nz = aTz a-1,

hence Tb~ = aT;a- l . Since Tz ls invertible, we can choose the metric such thatT;o -1 for any given Zo. Therefore T; = -1 for all Izl 1, which implies the assertion. •

Maximal dimension means there are isomorphisms

Lie(K/Ko) = Lie(G/Go) ~ so(m).

....-...Therefore the simply connected covers are isomorphic: Spin(m) ~ (G/Go)e' The induced homomorphism

Spin(m) ---+ (G/GO)e

may or may not extend to a homomorphism

Pin(m) ---+ G/Go.

If it does extend, it may or may not be injective, in which case it is an isomorphism. Therefore, among the algebras for which dim(G/Go) is maximal, those for which Pin(m) f::5 G/Go can be regarded as the most symmetric.

Theorem 2.6. Suppose n is a 2-step graded algebra such that Aut(n) contains a copy 01 Pin(m) inducing the standard action on ;;. Then n is 01 Heisenberg type.

Proof. The assumption implies that there is a linear map ;; ---+ End(b) , denoted by z H J z such that J; = -lz121 for all z and

for u, v E b, z E;;, Izl 1, where rz is the reflection in ;; with respect of the Hne spanned by z. Pin(m) is the group generated by the J/s with Ilzll 1, which acts linearly on b and is compacto Fix a metric on b invariant under it.

We get, as in the proof of Theorem 2.1, that if (~ ~) E Aut(n), then

In particular:

Trx(z) JxTzJx'

If x = z, we get Tz = -JzTzJz, thus TzJz -J;ITz JzTz. If x 1. z, we get Tz JxTzJ:c, thus TzJ:c J;ITz -J:cTz' It follows that T; commutes with Jz and with Jw , w 1. z.

--


Now,let z E 3 and w..L z. Let Rw(t) the 2t-rotation from z towards w. Then Rw(t) = rzrw(t) , with w(t) cos(t)z + sin(t)w. It fol!ows that

( O)JzJW(tl

O Rw(t)

is an orthogonal automorphism and, therefore, satisfies

TRu,(tlz = (JzJw(t))tTz(JzJw(t)).

Since (JzJw(t))t = (JzJw(t))-l,

T'(Ct)z = (JzJw(t))tT;(JzJw(t)) = Jw(t) JzT} JzJw(t).

Since T; commutes with Jz and Jw ,

T'(t)z = T; Jw(t) JzJzJw(t) = -T}Jw(t)Jw(t). (4)

But J~(t) - 1, so that (4) implies that

T'(Ct)z = T;.

For al! z' E 3 we can choose w E 3, tE lR such that Rw(t)z z', so we get

T;, T;, for all z' E 3, Iz'l = 1.

The antisymmetry of the bracket implies that Tz is skew-symmetric. Rescaling the scalar product on \) we obtain that T; = -1, so T; - 1 for all z' E 3, Iz'l l. Therefore n is of Heisenberg type. •

3. The case of center of dimension 2

In this section we compute the groups G, Go, GIGo in the case m 2. The various types are parametrized by pairs

(c,r) E (1[JP¡SL(2,1R)) x Z~

where 1[J is the upper-half plane and 2é = 2 ¿; rj dim n - 2. As a corollary we conclude that Aut(n) is maximal if and only if n is of Heisenberg type. These are complex Heisenberg algebras of various dimensions regarded as real Lie algebras.

Fírst we recall the normal form for fat algebras with m = 2 deduced from [LT]. Given e = a +bi e, let

Z(c) = (~b !) . If r E Z+, set

ce)12 Z(c) A(c, r) zJ12

~ - -- -- -------------~-----~-------~ ----~- ---


A(c, r)

[cp]

which is a 28 x 28 matrix, 8 = rl + ... + rf'

Let now cp, 'l/Jec,r) be the 2-forms on ]R4s whose matrices in the standard basis are

O A(c, r)) (5)[1¡'J(c,rJl = ( -At(c,r) O .

Then

[u, v]Cc,r) (cp(u, v), 'I/J(c,r) (u, v)) =< u, [cp]v > el+ < u, ['I/J(c.r)]v > e2

is an IR2- valued antisymmetric 2-form on ]R4s. Let

n(c,r) = ]R4s E&]R2

be the corresponding Líe algebra. Define M(c.r) E End(b) by

cp(M(c,r)u, v) = 'I/J(c,r) (u, v),

whose matrix is ~A" ]_ (-A(c.rl O)[ lVl(c.r) - O -A .

(c,r)

then we have

[u, v]Cc.r) cp(u, v)el + cp(M(c,r)u, v)e2, for u, v E IR4s. (6)

One can deduce

Proposition 3.1. (a) Every fat algebra with center of dírnension 2 is ísornorphíc to sorne n(c,r) with e E Uf.

(b) Two of these are isornorphic if and only íf the r 's coincide up to perrnutations and the c's díffer by sorne Mobius transforrnation acting cornponentwise.

(e) n(c,r) is of Heisenbery type if and only if e (e, ... ,e) and r = (1, ... ,1)

Let now n = n(c,r}

be fat and let G G(n), etc. We denote n the algebra obtained replacing the matrices A(e, r) by their semi simple parts and setting all eJ The resulting

A(e, r) consists of blocks (~1 ¿) along the diagonal and n is isomorphic to the

Heisenberg type algebra n((í, ....i}.(I .... ,I)). The correspondence

n 1--+ n is functorial and seems extendable inductively to fat algebras of any dimension, although here we will maíntain the assumption m = 2.

1093KAPLAN AND TIRABOSCHI

Lemma 3.2. Go(n) e Go(tÍ) and dimAut(n)::; dimAut(tÍ).

Proof. Let c/J, 1jJ, M(c,r) E End(o) be as aboye, so that

c/J(M(c,r)U,v) = 1jJ(c,r)(u,v).

By formula (6), 9 E Go(n(c,r)) if and only if

c/J( u, v) c/J(gu, gv),

i.e., if and only if 9 E Sp(c/J) and commutes with M(c,r)' In particular it commutes with the semisimple part M(c,r). This is conjugate to a matrix having blocks

Z(c) = (_3i~(~) ~~~D for various e E e along the diagonal, and zeros elsewhere.

Every matrix commuting with such a matrix will surely commute with that having all e = 1. It follows that 9 also preserves c/J(M(c,r)U, v) and, therefore, it is an automorphism of tÍ as well. Thus,

Go(n) e Go(tÍ).

Prom Corollary 2.3, dim(G(n)/Go(n)) ::; dim(G(tÍ)/Go(tÍ)), and therefore dim G(n)= dim(G(n)/Go(n))+ dim Go(n)::; dim( G(tÍ)/Go(tÍ))+ dim Go(tÍ)= dim G(tÍ).

Formula (2) implies dimAut(n) ::; dimAut(tÍ), as claimed. •

Next we will describe go(n(c,r)) for e E ([J and r E N+, i.e., the case when the matrices A consist of a single block. Since e is SL(2, C)-conjugate to i, it is enough to take e í. Define the 2 x 2-matrices

(O-1) (0 1) (-1 0)1 1 1G~), ° ' x= ° ) y= O 1 '

and let Mr(lR(1, i)) and Mr(lR(x, y)) denote the real vector spaces of r xr matrices with coefficients in the span of 1, i and x, y respectively. Then the vector space

R(r) { (~ ~): A, DE 2',f,.(lR(1, i)), B, e E lv1r(lR(x, y))},

is a actually a matrix algebra. Note that

·t •1 = -1,

Letting At denote the transpose or an lR-matrix and At, A* the transpose and conjugate transpose of lR[i, x, y]-matrices, one obtains

for A E Mr (lR{1, i)) while

for A E A,'1r(lR(x, y)).


With the notation

J2 ['!jI((i, ...,í),(l,.. ,l))],

9o(n) {X E 1R4rx4r : JlX + XtJl = 0, J2 X + X t J2 = O}.

From [S] we know that

Changing basis,

where

J2JI (_OIr ~), = (i~r i~). This gives an alternative description of this algebra:

t9o(n) = {(~ _~*): A E Mr(JR(l,i)), B,e E M r (lR(x,y)), B t = B, e e}

We now restrict OUT attention to matrices (~ _~*) in 9o(n) where

A, B, e have the respective forms

br~.l br) (~ ° CI). ° . ~

: ° :° Cl C2 Cr° with coefficients in 1R2x2

. Let Ak (~_~.) having ak = 1 and zero otherwise

and A~ the matrix of the same form but with ak = i and zeros elsewhere. Similarly,

let Bk (resp. C k ) the matrix (~ ~) (resp., (~ ~)) with bk (resp. Ck) equal

to x and zeros elsewhere, and B~ (resp. C~) with bk (resp. Ck) equal to y and zeros elsewhere.

Theorem 3.3. Let n n(c,r) , (e,1') E 1[J x N, and rega1'd 9o(n) as a subalgebra of 9l(0). Then,

1. 90(n) is the IR-span of Ai,A~,Bi,B~,Cí,q fo1' 1:::; i:::; 1'.

2. The semisimple pan of 9o(n) is the span of A¡, A / I , B I , B /

I , C l , Ci.

3. The solvable radical is the span of A;, A~, B i , B;, C í , q with 1 < i :::; 1'.

In partícnla1', the IR-dimension the 9o(n) is equal to 61' and the semisimple pan of 9o(n) is isomorphie to 51'(1, q.


Proof. It is enough to consider the case n = n(í,,.). Let T2 [\b(i,,.)] and write T2 = J 2 + N2 where

From Lemma 3.2, 90(n) = {X E 90(ñ): T2X + X t T2 = O}. As 90(n) e 90(ñ) one obtains

90(n) {X E 90(ñ) : N2X + X t N 2 O}.

The conditions on (~ _~.) E 90(n) are, explicitly,

0= NC ctNt = NC (NC)t (7)

O NtA-ANt (8)

O=NtB BtN=NtB (lVtB)t. (9)

For the first equation, note that NC symmetric if and only ir ei,j+l = Cj,i+l

and el,j = O for i, j < n. Since C is symmetric, Cí,j+! = ej,i+! eHl,j and Cl,j O for i,j < n. We conclude:

If i + j = k ::; r, Cí,j = Ci,k-i ei-l,k-i+l = Cí-2,k-i+2' •• = el,k-l = O If i + j = k > r, Cí,) = Ci,k-í = ("i+l,k-i-l Ci+2,k-i-2' . . Cr,k-i+í-r =

c,.,k-r

Thus, the strict upper antidiagonals are zero and each lower antidiagonal have all its elements equal.

For the second equation, note that N t and A commute. This is equivalent to Cí,) et,8 when j i = s - t and Cí,l O for i > l. The first condition implies that each diagonal have all its elements equal, while the second implies that the strict lower diagonals are zero.

Equation (9) is analogous to equation (7): the condition N t B symmetric is equivalent to each antidiagonal have all its elements equal and that the strict lower antidiagonals are O.

From aH this we conclude that the span of Ai , A~, B i , B~, C il q with 1 ::; i ::; r is go(n) and (1) follows. .

(2) and (3) follow from (1) and the explicit presentation of the matrices A¡, A~, B i , B~, C il q.

• Corollary 3.4. (of the proof) Let n be fato Then dim(go(n)) is maxímal if and only if n is of Heisenberg type.

Proof. Let (c,r) ((Cl,.",C¡),(Tl,,,.,T¡)) besuchthat n=n(c,r)' Weknow that 90(n) e 90(ñ). If (''Í #- ej for sorne i,j, then there is not intertwining operator between the blocks corresponding to these invariants, so 90(n) #- .IJo(ñ).


When el e2 el we can consider ej = i for all j. Let T = I:Ti'

In this case if (~ _~*) E !Jo(n) must be satisfy the equations (7), (8), (9) but

with N such that coefficients nj+l,j are O or 1. Suppose now that !Jo(n) is not of Heisenberg type, then sorne nj+l,j Ís equal to 1. We assume that n21 = 1 and let A E M r (lll(l, i}) such that a¡2 1 and O otherwise, then

X=(~ _~*) belongs to 90(ñ) but lS not in 9o(n).

• It can be shown in general that the semisimple part of 9o(n) is isomorphic

to EBisp(mi, C), where mi is the multiplicity of the paír (ei, Ti) in (e, r).

In the case m 2, 9/90 is either O or isomorphic to so(2).

Theorem 3.5. 9(n)/90(n) ~ so(2) if el ... = el, and O otherwise.

Proof. 9/90 is a compact subalgebra of 9[(2), hence of the form gsO(2)g-1 for sorne 9 E SL(2, 1Il) and it i8 nonzero if and only if there exists X E s[(v) such that, in the notation of the proof of Theorem 3.3,

(~ 9i~-I) i8 a derivation of n. For 9 1, if T¡, T2 corre8pond to the standard basis of 3, the equations for X become

In normal form, and for a single block A(i,r) ,

O ( -Ir Ir)

O '

We decompose

with

and regard JI, J2 , , N2 as matrices with coefficients in 1Il2X2 • Note that J1> J 2

correspond to ñ, of Heísenberg type. Let

O O O O O O s O O 2i O O O s O O 1 4i O O O s . O O O 21 6i O O s O

Yo O O O 31 8i s O

O

O O O O (n - 2)1 2(n - l)i

1097 KAPLAN AND TlRABOSCHI

A straightforward calculation shows that

-y;t O ) Xo = ( Oo -yot + ilr + N

is a solution of (a), (b). We conclude that

is a derivation of n(i,r), which líes in 9(n(i,r)) but not in 90(n(i,r))'

Por any e E 1!J, n(c,r) ~ n(i,r) , hence they have the same 9/90 up to isomorphisms. In fact, for any g E SL(2, lll.), the algebra n(g.i,r) has a derivation of the form

(~ gi~-l)' Por a fixed g, these X are unique modulo 90 and come in normal formo Clearly, e determines the 2 x 2 matrix gig- l and the complex number g' i.

In the case of an arbitrary fat n(c,r), each block (ek, Tk) determines a corresponding X k such that

(~k gki~;l) is a derivation of n(Ck,rk)' If n(c,r) has a derivation in 9 that is not in 90, then its must have one which is combination of such, acting on tl as Xl + X 2 + .... This forces an the gkig;l to be the same and all the ei to be the same. The reciprocal is clear. •

In particular, all algebras n(c,r) with el ... = el and Ti > 1 maximize the dimension of g/go, but they are not Heisenberg type.

Lauret had pointed out to us that there were non Heisenberg type algebras such that g(n)/go(n) =f. O. Independently, Oscari proved that this holds whenever the ei 's aH agree.

4. Fat distributions

Let D be a smooth vector distribution on a smooth manifold M, Le., a subbundle of the tangent bundle T(M). Its nilpotentization, or symbol, is the bundle on M with fiber

ND(M)p = E9 D~j) / D~-l) j

where D~l) Dp and D~+l) = D~)+ [r(D), r(Dj)]p' The Lie bracket in r(T(M)) induces a graded nilpotent Lie algebra structure on each fiber of ND(M). If D(j) T(M) for sorne j, D is called completely non-integrable. If D(2) = T(M), the nilpotentization is 2-step, which in the notation of the previous section, is

- D Dp + [r(D), r(D)]p _ND (M)p - p ffi D - tlp + 3p,

p


It is also easy to see that D is fat in the sense of Weinstein [Mo] if and only if np = tl + 3 is non-singular, i.e., fat in the sense defined in the section lo

A subriemannian metric 9 defined on D determines a metric on tl. On 3 we put a metric a invariant under G. Let {<PI, ... , <Pm; 'lj;1' ... , 'lj;n} be a coframe on M such that

D = nker<pi,

with {<p1> ... , <Pm} and {'lj;I' ... , 'lj;n} orthonormal with respect to 9 + a. Define Tz E End(D) as before, by

a(z, [u, v]) = g(Tzu,v).

Then D is fat if and only if Tz is invertible for all non-zero z E 3. The structure equations for the coframe can be written

with the Tk 's having the property that any non-zero linear combination of them is invertible. This is deduced from the fact that if u, v E tl, then d<p[u, v] = -<p([u, v]), since u(<p(v)) = u(O) = o. The d'lj;'s are essentially arbitrary.

Let now M be a the simply connected Lie group with a fat Lie algebra n, D the left-invariant distribution on M such that De = tl. For a left-invariant coframe, the structure equations take the form

d'lj;i = O

where JI, ... , Jm are anticommuting complex structures on D. The results from the previous sections lead to consider fat distributions

satisfying

(4.1)

where the Jk are sections of End(T(M)*) satisfying the Canonical Commutation Relations

J;Jj + JjJi = -2i5ij .

The Equivalence Problem for these systems has been discussed for distributions with growth vector (2n, 2n + 1), (4n,4n + 3) and (8,15). In these cases n is parabolic, i.e., isomorphic to the Iwasawa subalgebra of a real semi simple Lie algebra 9 of real rank one. The Tanaka [T] subriemannian prolongation of such algebra is g, while in the non-parabolic case is just

n + e(n) + a(n)

where a(n) the 1-dimensional Lie algebra of dilations [Su]. In this case,Tanaka's theorem implies that, in the notation of [Z], the first pseudo G-structure pO already carries a canonical frame.


As this paper was being wrítten, E. van Erp poínted out to us hís article [Er], where fat dístributions are called polycontact and those satisfying (4.1) arise by ímposing a compatible conformal structure.

Acknowledgments. We wish to thank Professor J. Vargas for helpful discussions and M. Subils for pointing out a mistake in a previous versíon of this paper.

References

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[P] Pansu, P., Métriques de Carnot-Camthodory et quasiisométries des espaces symétriques de mng un, Ann. of Math. 129 (1989), 1-60.

[S] Saal, L., The automorphísm group of a Lie algebm of Heísenberg type, Rend. Sem. Mat. Univ. PoI. Torino 54(1996), 101-113.

[Su] Subils, M., Personal communication.

[T] Tanaka, N., On dífferential systems, gmded Líe algebms and pseudogroups J. Math. Kyoto. Univ. 10 (1970), 1-82.

[Z] Zelenko, l., On Tanaka's prolongation procedure for filtered structures of constant type SIGMA 5 (2009), Paper 094, 21pp.

Aroldo Kaplan Alejandro Tiraboschi Facultad de Matemática, Facultad de Matemática, Astronomía y Física Astronomía y Física Universidad Nacional de Córdoba Universidad Nacional de Córdoba CIEM - CONICET. (5000) CIEM CONICET. (5000)Ciudad Universitaria Ciudad Universitaria Córdoba, Argentina Córdoba, Argentina [email protected],edu.ar [email protected]

Reeeived November 22, 2011 and in final form Mareh 15, 2013

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Automorphisms of Non-Singular Nilpotent Lie Algebras

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