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In particular, we get

[’Zi,1 , Yµ,1 ] = 0
(±1 + 1)X±+1i ’11 ,k if ±1 = 0, i = 1
[’Zi,1 , X±,k ] = i
(±k + 1 + δk )Y±+1i ,1 if ±1 = 0, i = 1
±
 µj Yµ’1j +1i ,k if i = k

[’Zi,j , Yµ,k ] = Xµ’1j ,j if i = k, µj = 0

’Yµ,j if i = k, µj = 0.


Hence starting with an arbitrary linear combination of the base elements, an
iterated action of suitable vector ¬elds leads to one of the base elements Yµ,k .
Then any other base element can be reached by further actions. Therefore also
r
the modules C2 are irreducible.
r
If m = 1, then all C1 = 0 by the de¬nition and for all 0 ¤ r ¤ k ’ 1 we have
‚ ‚ ‚
C2 = gr = R with the action of g0 given by [ax ‚x , bxr+1 ‚x ] = ’rabxr+1 ‚x .
r
r r
The submodules C1 and C2 cannot be equivalent for dimension reasons. The
adjoint action Ad of GL(m) on gk is given by Ad(a)(j0 X) = j0 (a —¦ X —¦ a’1 ).
k k
m
So each irreducible component of gr has homogeneous degree ’r. Therefore the
r
modules Ci with di¬erent r are inequivalent.


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
134 Chapter IV. Jets and natural bundles


13.7. Corollary. The normal subgroup B1 ‚ Gk is generated by two closed
m
Lie subgroups D1 , D2 invariant under the canonical action of G1 . The group
m
D1 is formed by the jets of volume preserving di¬eomorphisms and D2 consists
of the jets of di¬eomorphisms keeping all the one-dimensional linear subspaces
in Rm . The corresponding Lie subalgebras are the subalgebras with grading
k’1 k’1
1 1
C1 = C1 • · · · • C1 and C2 = C2 • · · · • C2 where all the homogeneous
components are irreducible GL(m)-modules with respect to the adjoint action
and b1 = C1 • C2 .
Let us point out that an element j0 f ∈ Gk belongs to D1 or D2 if and
k
m
only if its polynomial representative is of the form f = idRm + f2 + · · · + fk
i’1 i’1
with fi ∈ C1 © Li (Rm , Rm ) = C1 or fi ∈ C2 © Li (Rm , Rm ) = C2 ,
sym sym
respectively.
13.8. Proposition. If m ≥ 2 and l > 1, or m = 1 and l > 2, then there is no
splitting in the exact sequence e ’ Bl ’ Gk ’ Gl ’ e. In dimension m = 1,
m m
there is the exceptional projective splitting G1 ’ Gk de¬ned by
2
1

bk’1
b
ax + bx2 ’ a x + x2 + · · · + k’1 xk .
(1)
a a

Proof. Let us assume there is a splitting j in the exact sequence of Lie algebra
homomorphisms 0 ’ bl ’ gk ’ gl ’ 0, l > 1. So j : g0 • · · · • gl’1 ’
m m
p
g0 • · · · • gk’1 and the restrictions jt,q of the components jq : gl ’ gq to
m
p
the g0 -submodules Ct in the homogeneous component gp are morphisms of g0 -
p p
modules. Hence jt,q = 0 whenever p = q. Since j is a splitting the maps jt,p are
the identities.
Assume now m > 1. Since [gl’1 , g1 ] equals gl in gk but at the same time this
m
bracket equals zero in gl , we have got a contradiction.
m
If m = 1 and l > 2 the same argument applies, but the inclusion j : g0 • g1 ’
g0 • g1 • · · · • gk’1 is a Lie algebra homomorphism, for in gk the bracket [g1 , g1 ]
1
equals zero. Let us ¬nd the splitting on the Lie group level. The germs of
x
transformations f±,β (x) = ±x+β , β = 0, are determined by their second jets,
so we can view them as elements in G2 . Since the composition of two such
1
transformations is a transformation of the same type, they give rise to Lie group
homomorphisms G2 ’ Gr for all r ∈ N. One computes easily the derivatives
1 1
(n) n’1 ’n
n’1
β . Hence the 2-jet ax+bx2 corresponds to f±,β with
f±,β (0) = (’1) n!±
± = ’ba’2 , β = a’1 . Consequently, the homomorphism G2 ’ Gr has the form
1 1
(1) and its tangent at the unit is the inclusion j.
We remark that a geometric de¬nition of the exceptional splitting (1) is based
on the fact that the construction of the second order jets determines a bijection
between G2 and the germs at zero of the origine preserving projective transfor-
1
mations of R.
13.9. Proposition. The Lie group Gk is solvable. Its Lie algebra gk can be
1 1
characterized as a Lie algebra generated by three elements
X1 = x2 dx ∈ g1 , X2 = x3 dx ∈ g2
d d d
X0 = x dx ∈ g0 ,

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
13. Jet groups 135


with relations

[X0 , X1 ] = ’X1
(1)
[X0 , X2 ] = ’2X2
(2)
(ad(X1 ))i X2 = 0 for i ≥ k ’ 2.
(3)


Proof. The ¬ltration gk = b0 ⊃ · · · ⊃ bk’1 ⊃ 0 from 13.2 is a descending chain
1
of ideals with dim(bi /bi+1 ) = 1. Hence gk is solvable.
1
d
Let us write Xi = xi+1 dx ∈ gi . Since [X1 , Xi ] = (1 ’ i)Xi+1 , we have

(’1)i’2
(ad(X1 ))i’2 X2 for k ’ 1 ≥ i ≥ 3
(4) Xi =
(i ’ 2)!
[Xi , Xj ] = (i ’ j)Xi+j .
(5)

¯ ¯ ¯
Now, let g be a Lie algebra generated by X0 , X1 , X2 which satisfy relations
¯
(1)“(3) and let us de¬ne Xi , i > 2 by (4). Consider the linear map ± : gk ’ g,
1
¯ ¯¯ ¯
Xi ’ Xi , 0 ¤ i ¤ k ’ 1. Then [X1 , Xi ] = (1 ’ i)Xi+1 and using Jacobi identity,
¯¯ ¯
the induction on i yields [X0 , Xi ] = ’iXi . A further application of Jacobi
¯¯ ¯
identity and induction on i lead to [Xi , Xj ] = (i ’ j)Xi+j . Hence the map ± is
an isomorphism.
13.10. The group Gk with m ≥ 2 has a more complicated structure. In par-
m
ticular Gk cannot be solvable, for [gk , gk ] contains the whole homogeneous
m mm
component g0 , so that this cannot be nilpotent. But we have
Proposition. The Lie algebra gk , m ≥ 2, k ≥ 2, is generated by g0 and any
m

element a ∈ g1 with a ∈ C1 ∪ C2 . In particular, we can take a = x2 ‚x1 .
1 1
/ 1

1 1
Proof. Let g be the Lie algebra generated by g0 and a. Since g1 = C1 • C2 is
a decomposition into irreducible g0 -modules, g1 ‚ g. But then 13.2.(4) implies
g = gk .
m

13.11. Normal subgroup structure. Let us ¬rst describe several normal
subgroups of Gk . For every r ∈ N, 1 ¤ r ¤ k ’ 1, we de¬ne Br,1 ‚ Br ,
m
Br,1 = {j0 f ; f = idRm + fr+1 + · · · + fk , fr+1 ∈ C1 , fi ∈ Li (Rm , Rm )}.
r r
sym
The corresponding Lie subalgebra in gk is the ideal C1 • gr+1 • · · · • gk’1
r
m
r
so that Br,1 is a normal subgroup. Analogously, we set Br,2 = {j0 f ; f =
idRm + fr+1 + · · · + fk , fr+1 ∈ C2 , fi ∈ Li (Rm , Rm )} with the corresponding
r
sym
r
Lie subalgebra C2 •gr+1 •· · ·•gk’1 . We can characterize the normal subgroups
k
Br,j as the subgroups in Br with the projections πr+1 (Br,j ) belonging to the
subgroups Dj ‚ Gr+1 , j = 1, 2, cf. 13.7.
m

Proposition. Every connected normal subgroup H of Gk , m ≥ 2, is one of the
m
following:
(1) {e}, the identity subgroup,
(2) Br , 1 ¤ r < k, the kernel of the projection πr : Gk ’ Gr ,
k
m m


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
136 Chapter IV. Jets and natural bundles


(3) Br,1 , 1 ¤ r < k, the subgroup in Br of jets of di¬eomorphisms keeping
the standard volume form up to the order r + 1 at the origin,
(4) Br,2 , 1 ¤ r < k, the subgroup in Br of jets of di¬eomorphisms keeping
the linear one-dimensional subspaces in Rm up to the order r + 1 at the origin,
(5) N B1 , where N is a normal subgroup of GL(m) = G1 . m

Proof. Since we deal with connected subgroups H ‚ Gk , we can prove the
m
proposition on the Lie algebra level.
Let us ¬rst assume that H ‚ B1 . Then it su¬ces to prove that the ideal in
gm generated by Cj , j = 1, 2, is the whole Cj • br+1 . But the whole algebra gk
k r r
m
2‚
is generated by g0 and X1 = x1 ‚x1 , and [g1 , gi ] = gi+1 for all 2 ¤ i < k. That
is why we have only to prove that gr+1 is contained in the subalgebra generated
r+1
r
by g0 , X1 and Cj for both j = 1 and j = 2. Since Cj are irreducible g0 -
/ r+1
r
submodules, it su¬ces to ¬nd an element Y ∈ Cj such that [X1 , Y ] ∈ C1 and
/ r+1
at the same time [X1 , Y ] ∈ C2 .
Let us take ¬rst j = 2, i.e. Y = f Y0 for certain polynomial f . Since
[f Y0 , X1 ] = (X1 f )Y0 + f [Y0 , X1 ] = (X1 f )Y0 ’ f X1 , the choice f (x) = ’xr gives
2
r+1 r+1

[Y, X1 ] = xr x2 ‚x1 which does not belong to C1 ∪ C2 , for its divergence
21
equals to 2x1 xr = 0, cf. 13.5.
2
Further, consider Y = xr+1 ‚x1 ∈ C1 and let us evaluate [xr+1 ‚x1 , x2 ‚x1 ] =
‚ ‚ ‚
r
1
2 2
’2x1 xr+1 ‚x1 . Since the divergence of the latter ¬eld does not vanish, [Y, X2 ] ∈

/
2
r+1 r+1
C1 ∪ C2 as required. Hence we have proved that all connected normal
subgroups H ‚ Gk contained in B1 are of the form (1)“(4).
m
Consider now an arbitrary ideal h in gk and let us denote n = h © g0 ‚ g0 . By
m
virtue of 13.2.(4), if h contains a vector which generates g1 as a g0 -module, then
1
b1 ‚ h. We shall prove that for every X ∈ g0 any of the equalities [X, C1 ] = 0
1
and [X, C2 ] = 0 implies X = 0. Therefore either h ⊃ b1 or n = 0 which concludes
the proof of the proposition.
‚ ‚ 1
i,j bij xj ‚xi ∈ g0 and Y = xk j xj ‚xj ∈ C2 . Then [X, Y ] =
Let X =

’( j bkj xj )Y0 . Hence [X, C2 ] = 0 implies X = 0. Similarly, for Y = x2 ‚xk ∈
1
l
1
C1 and X ∈ g0 , the equalities [X, Y ] = 0 for all k = l yield X = 0. The simple
computation is left to the reader.
13.12. Gk -modules. In the next sections we shall see that the actions of
m
the jet groups on manifolds correspond to bundles of geometric objects. In
particular, the vector bundle functors on m-dimensional manifolds correspond
to linear representations of Gk , i.e. to Gk -modules. Since there is a well known
m m
representation theory of GL(m) which is a subgroup in Gk , we should try to
m
describe possible extensions of a given representation of GL(m) on a vector
space V to a representation of Gk . A step towards such description was done
m
in [Terng, 78], we shall present only an observation showing that the study
of geometric operations on irreducible vector bundles restricts in fact to the
case of irreducible GL(m)-modules (with trivial action of the normal subgroup
B1 ). According to 5.4, there is a bijective correspondence between Lie group
homomorphisms from B1 to GL(V ) and Lie algebra homomorphisms from b1 to
gl(V ), for B1 is connected and simply connected. Further, there is the semidirect

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
13. Jet groups 137


product structure gk = gl(m) b1 with the adjoint action of gl(m) on b1 which
m
is tangent to the adjoint action of GL(m) and every representation of GL(m) on
V induces a GL(m)-module structure on gl(V ) via the adjoint action of GL(V )
on gl(V ). This implies immediately
Proposition. For every representation ρ : GL(m) ’ GL(V ) there is a bijection
between the representations ρ : Gk ’ GL(V ) with ρ|GL(m) = ρ and the set
¯m ¯
of mappings T : b1 ’ gl(V ) which are both Lie algebra homomorphisms and
homomorphisms of GL(m)-modules.
13.13. A G-module is called primary if it is equivalent to a direct sum of copies
of a single irreducible G-module.
Proposition. If V is a Gk -module such that the induced GL(m)-module is
m
primary, then the action of the normal subgroup B1 ‚ Gk is trivial.
m

Proof. Assume that the GL(m)-module V equals sW , where W is an irre-
ducible GL(m)-module. Then each irreducible component of the GL(m)-module
gl(V ) = V — V — has homogeneous degree zero. But all the irreducible compo-
nents of b1 have negative homogeneous degrees. So there are no non-zero ho-
momorphisms between the GL(m)-modules b1 and gl(V ) and 13.12 implies the
proposition.
13.14. Proposition. Let ρ : Gk ’ GL(V ) be a linear representation such
m
that the corresponding GL(m)-module is completely reducible and let V =
r
i=1 ni Vi , where Vi are inequivalent irreducible GL(m)-modules ordered by
their homogeneous degrees, i.e. the homogeneous degree of Vi is less than or equal
l’1
to the homogeneous degree of Vj whenever i ¤ j. Then W = ( i=1 ni Vi ) • nVl
is a Gk -submodule of V for all 1 ¤ l ¤ r and n ¤ nl .
m
l’1
Proof. By de¬nition, ( i=1 ni Vi ) • nVl is a GL(m)-submodule. Since every ir-
reducible component of the GL(m)-module b1 has negative homogeneous degree
and for all 1 ¤ i ¤ l the homogeneous degree of L(Vi , Vl ) is non-negative, we get

l’1 l’1
ni Vi ) • nVl ) ‚
Te ρ(X)(( ni V i
i=1 i=1


for all n ¤ nl and for every X ∈ b1 . Now the proposition follows from 13.12 and
13.5.
13.15. Corollary. Every irreducible Gk -module which is completely reducible
m
as a GL(m)-module is an irreducible GL(m)-module with a trivial action of the
normal nilpotent subgroup B1 ‚ Gk .
m

Proof. Let V be an irreducible Gk -module. Then V is irreducible when viewed
m
as a GL(m)-module, cf. proposition 13.14. But then B1 acts trivially on V by
virtue of proposition 13.13.


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
138 Chapter IV. Jets and natural bundles


13.16. Remark. In the sequel we shall often work with various subgroups in
the group of all di¬eomorphisms Rm ’ Rm which determine Lie subgroups in
the jet groups Gk . Proposition 13.2 describes the bracket and the exponential
m
map in the corresponding Lie algebras and also their gradings g = g0 • · · · •
gk’1 . Let us mention at least volume preserving di¬eomorphisms, symplectic
di¬eomorphisms, isometries and ¬bered isomorphisms on the ¬brations Rm+n ’
Rm . We shall essentially need the latter case in the next chapter, see 18.8. The
r-th jet group of the category FMm,n is Gr ‚ Gr m+n and the corresponding
m,n

Lie subalgebra gm,n ‚ gm+n consists of all polynomial vector ¬elds i,µ ai xµ ‚xi
k k
µ
with ai = 0 whenever i ¤ m and µj = 0 for some j > m. The arguments from
µ
the end of the proof of proposition 13.2 imply that even 13.2.(4) remains valid
in the following formulation.
The decomposition gk = g0 • · · · • gk’1 is a grading and for every indices
m,n
0 ¤ i, j < k it holds

(1) [gi , gj ] = gi+j if m > 1, n > 1, or if i = j.



14. Natural bundles and operators
In the preface and in the introduction to this chapter, we mentioned that
geometric objects are in fact functors de¬ned on a category of manifolds with
values in category FM of ¬bered manifolds. Therefore we shall use the name
bundle functors, in general. But the best known among them are de¬ned on
category Mfm of m-dimensional manifolds and local di¬eomorphisms and in
this case many authors keep the traditional name natural bundles. Throughout
this section, we shall use the original de¬nition of natural bundles including
the regularity assumption, see [Nijenhuis, 72], [Terng, 78], [Palais, Terng, 77],
but we shall prove in chapter V that every bundle functor on Mfm is of ¬nite
order and that the regularity condition 14.1.(iii) follows from the other axioms.
Since the presentation of these results needs rather long and technical analytical
considerations, we prefer to derive ¬rst geometric properties of bundle functors
in the best known situations under stronger assumptions. In fact the material of
this section presents a model for the more general situation treated in the next
chapter.
14.1. De¬nition. A bundle functor on Mfm or a natural bundle over m-
manifolds, is a covariant functor F : Mfm ’ FM satisfying the following con-
ditions
(i) (Prolongation) B —¦ F = IdMfm , where B : FM ’ Mf is the base functor.
Hence the induced projections form a natural transformation p : F ’ IdMfm .
(ii) (Locality) If i : U ’ M is an inclusion of an open submanifold, then
F U = p’1 (U ) and F i is the inclusion of p’1 (U ) into F M .
M M
(iii) (Regularity) If f : P —M ’ N is a smooth map such that for all p ∈ P the
˜
maps fp = f (p, ) : M ’ N are local di¬eomorphisms, then F f : P —F M ’ F N ,

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