Steps in Differential Geometry, Proceedings of the Colloquium
on Differential Geometry, 25–30 July, 2000, Debrecen, Hungary
SOME CLASSIFICATION PROBLEM ON WEIL BUNDLES
ASSOCIATED TO MONOMIAL WEIL ALGEBRAS
JIŘÍ TOMÁŠ
Abstract. A natural T -function on a natural bundle F is a natural operator
transforming vector fields on a manifold M into functions on F M . For a
monomial Weil algebra A satisfying dim M ≥ width(A) + 1 we determine all
natural T -functions on T ∗ T A M , the cotangent bundle to a Weil bundle T A M .
1. The aim of this paper is the classification of all natural T -functions defined
on the cotangent bundle to a Weil bundle T ∗ T A corresponding to a monomial Weil
algebra A. Roughly speaking, the concept of a monomial Weil algebra denotes
an algebra of jets factorized by an ideal generated only by monomial elements.
Weil algebras of this kind form a significant class of themselves, since they cover
algebras of holonomic and non-holonomic velocities as well as quasivelocities, [11].
The starting point is a general result by Kolář,[4], [5], determining all natural
operators T → T T A transforming vector fields on manifolds to vector fields on a
Weil bundle T A . Further, partial results of our general problem are solved in [3]
and [9]. We follow the basic terminology from [5].
We start from the concept of a natural T -function. For a natural bundle F ,
a natural T -function f is a natural operator fM transforming vector fields on a
manifold M to functions on F M . The naturality condition reads as follows. For a
local diffeomorphism ϕ : M → N between manifolds M , N and for vector fields X
on M and Y on N satisfying T ϕ ◦ X = Y ◦ ϕ it holds fN (Y ) ◦ F ϕ = fM (X). An
absolute natural operator of this kind, i.e. independent of the vector field is called
a natural function on F .
There is a related problem of the classification of all natural operators lifting
vector fields on m-dimensional manifolds to T ∗ T A . The solution of the second
problem is given by the solution of the first one as follows ([10]). Natural operators
AM : T M → T T ∗ T A M are in the canonical bijection with natural T -functions
gM : T ∗ T ∗ T A M → R linear on fibers of T ∗ (T ∗ T A M ) → T ∗ T A M . Using natural
equivalences s : T T ∗ → T ∗ T by Modugno-Stefani, [7] and t : T T ∗ → T ∗ T ∗ by
Kolář-Radziszewski, [6], we obtain the identification of gM with natural T -functions
fM : T ∗ T T A M → R given by fM = gM ◦tT A M ◦s−1
. Thus we investigate natural
T AM
1991 Mathematics Subject Classification. 58A05, 58A20.
Key words and phrases. natural bundle, natural operator, Weil bundle.
The author was supported by the grants No. 201/96/0079 and No. 201/99/D007 of the GA
ČR and partially supported by grants MSM 261100007 and J22/98 261100009.
363
364
JIŘÍ TOMÁŠ
T -functions defined on T ∗ T D⊗A M to determine all natural operators T → T T ∗ T A ,
where D denotes the algebra of dual numbers.
We remind the general result by Kolář, [4], [5]. For a Weil algebra A, the Lie
group AutA of all algebra automorphisms of A has a Lie algebra AutA identified
with Der A, the algebra of derivations of A. Thus every D ∈ Der A determines a one
parameter subgroup d(t) and a vector field DM on T A M tangent to (d(t))M . Hence
we have an absolute natural operator λD : T M → T T A M defined by λD X = DM
for any vector field X on M . For a natural bundle F , let F denote the corresponding flow operator, [5]. Further, let LM : A × T T A M → T T A M denote the natural
affinor by Koszul, [4], [5]. Then the result by Kolář reads
All natural operators T → T T A are of the form L(c)T A + λD for some c ∈ A
and D ∈ Der A.
Let ξ : M → T M be a vector field. Kolář in [3] defined an operation e transforming
e
a vector field on a manifold M onto a function on T ∗ M by ξ(ω)
= < ξ(p(ω)), ω >,
∗
where p is the cotangent bundle projection and ω ∈ T M . One can immediately
verify, that for a natural bundle F and a natural operator AM : T M → T F M we
eM : T ∗ F M → R defined by A
eM (X) = A
^
have a natural T -function A
M X for any
vector field X : M → T M .
2. In this section, we find all natural T -functions fM : T ∗ T A M → R for any
manifold M for m = dim M ≥ width(A) + 1. For some cases of A, [11], all natural
T -functions in question are of the form
^A , λf
h(L(c)T
D ))
c ∈ C, D ∈ D
where C is a basis of A, D is a basis of Der A and h is any smooth function
Rdim A+dim Der A → R. Let Drk denote the algebra of jets J0r (Rk , R). It can be also
considered as the algebra of polynomials of variables τ1 , . . . , τk . By [6], any Weil
algebra A is obtained as the factor of Drk by an ideal I of itself, i.e. A = Drk /I.
The contravariant approach to the definition of a Weil bundle by Morimoto sets
MA = Hom(C ∞ (M, R), A) and was studied by many authors as Muriel, Munoz,
Rodriguez, Alonso,([1] [8]). The covariant approach (Kolář, [3], [5]) defines T A M as
the space of A-velocities. Let ϕ, ψ : Rk → M , ϕ(0) = ψ(0). Then ϕ and ψ are said
to be I-equivalent iff for any germx f , f : M → R it holds germ(f ◦ ϕ − f ◦ ψ) ∈ I.
Classes of such an equivalence j A ϕ are said to be A-velocities. For a smooth map
g : M → N define T A g(j A ϕ) = j A (g ◦ ϕ). Since T A preserves products, we have
T A R = A, T A Rm = Am . The identification F : MA → T A M between those two
approaches to the definition of Weil bundle is given by
(1)
F (j A ϕ)(f ) = j A (f ◦ ϕ)
for any f ∈ C ∞ (M, R)
WEIL BUNDLES ASSOCIATED TO MONOMIAL WEIL ALGEBRAS
365
We are going to construct natural T -functions defined on T ∗ T A from natural operators T → T Tkr , since there are some additional ones on T ∗ T A , which cannot be
constructed from natural operators T → T T A .
Let p : Drk → A be the projection homomorphism of Weil algebra inducing the
natural transformation p̃M : Tkr M → T A M . There is a linear map ι : A → Drk such
that p ◦ ι = idA . By ι we construct an embedding T A M → Tkr M . Consider any
j A ϕ ∈ T A M as an element of Hom(C ∞ (M, R), A). Then domains of j A ϕ ∈ TxA0 M
can be replaced by Jxr0 (M, R). Indeed, for any f ∈ C ∞ (M, R) it holds j A ϕ(f ) =
j A (f ◦ ϕ) = [germx0 f ◦ germ0 ϕ]I , where x0 = ϕ(1), 0 ∈ Rk . Since any ideal I in
the algebra E(k) of finite codimension contains the r-th power of the maximal ideal
of E(k), the last expression can be replaced by [j0r (f ◦ ϕ)]J = j A ϕ(jxr0 f ), where J
is an ideal of Drk corresponding to I.
Further, any element jxr0 f ∈ Jxr0 (M, R) can be decomposed onto f (x0 )+jxr0 (t−1
f (x0 ) ◦
r ˜
f ) = f (x0 ) + jx0 f , where ty : R → R denotes in general a translation mapping 0
onto y. The second expression is an element of the bundle of covelocities of type
(1, r), namely an element of (T r∗ )x0 M = (T1r∗ )x0 M , the bundle of covelocities of
type (k, r) being defined as Tkr∗ M = J r (M, Rk )0 , [5] .
Select any minimal set of generators Bx0 of the algebra Txr∗0 M . For any jxr0 f˜ ∈
Bx0 define ι̃x0 : TxA0 M → (Tkr )x0 M by (ι̃x0 (j A ϕ))(jxr0 f˜) = ι̃((j A ϕ)(jxr0 f˜)). In the
second step, ι̃ can be extended onto the homomorphism Jxr0 (M, R) → Drk .
We extend the map ι̃x0 to ι̃ : T A M → Tkr M . For a general Weil algebra B
we show that any element j B ϕ ∈ Tx̄B M corresponds bijectively to some element
B
r
j B ϕ0 ∈ TxB0 M . Indeed, j B ϕ(jx̄r f ) = j B (f ◦ ϕ) = j B (f ◦ t−1
x̄ ◦ tx̄ ◦ ϕ0 ) = j ϕ0 (jx0 f0 ).
A
r
This general property extends ι̃x0 onto ι̃ : T M → Tk M . We proved the following
assertion
Proposition 1. Let A = Drk /I be a Weil algebra, p : Drk → A the projection
homomorphism with its associated natural transformation p̃ : Tkr → T A and ι :
A → Drk a linear map satisfying p ◦ ι = idA . For a manifold M and x0 ∈ M
let Bx0 be a minimal set of generators of the algebra Jxr0 (M, R)0 = Txr∗0 M . Then
there is an embedding ι̃ : T A M → Tkr M satisfying p̃M ◦ ι̃ = idT A M such that
(ι̃(j A ϕ))(jxr0 f˜) = ι((j A ϕ)(jxr0 f˜)) for any j A ϕ ∈ TxA0 M and jxr0 f˜ ∈ Bx0 .
In the following investigations, we limit ourselves to monomial Weil algebras. A
Weil algebra A = Drk /I is said to be monomial if I is generated only by monomials. We shall need the coordinate expression of some operators used later for the
construction of natural T -functions in question. Thus we introduce coordinates
on T A M and T ∗ T A M . Consider the polynomial approach to the definition of Drk .
1
Then its elements are of the form α!
xα τ α , where τ1 , . . . , τk are variables and α are
multiindices satisfying 0 ≤ |α| ≤ r. Define a linear map ι : A → Drk as follows.
For τ α , put ι(p(τ α )) = 0 if τ α ∈ I and ι(p(τ α )) = τ α otherwise. As a matter of
fact, ι : A → Drk is a zero section. Similarly as p : Drk → A, the map ι can be
extended to ι̃ : Am → (Drk )m by components. Then it coincides with the map ι̃
366
JIŘÍ TOMÁŠ
from Proposition 1, if we put M = Rm , choose x0 ∈ Rm and substitute jxr0 xi for
the elements of Bx0 , where xi are canonical coordinates on Rm . Further, define the
i
additional coordinates on T ∗ T A M by pα
i dxα .
A
Let us define operators T → T T by means of ι̃ and natural operators T → T Tkr
as follows. Every natural operator λ : T → T Tkr defines an operator
Λ : T → TTA
(2)
by
Λ = T p̃ ◦ λ ◦ ι̃
e : T ∗T A →
which does not to have be natural and neither does the functions Λ
r
R Consider a basis of natural operators T → T Tk . The non-absolute natural
operators λ together with some of the absolute ones in this basis induce natural
operators Λ : T → T T A , while the others will be used for the construction of the
additional natural functions defined on T ∗ T A .
By general theory, [5], searching for natural T -functions defined on T ∗ T A , we are
r+1
going to investigate Gr+2
T )0 Rm ×(T ∗ T A )0 Rm .
m -invariant functions defined on (J
and some of its
Therefore we state some assertions, concerning the action of Gr+2
m
subgroups on this space. It will be necessary to consider the coordinate expression
of this action as well as that of base operators Λ : T → T T A and their associated
e : T ∗ T A → R.
functions Λ
Denote by λβj a natural operator λDβ associated to a derivation of Drk defined
j
by τi → δij τ β for j ∈ {1, . . . , k} and 1 ≤ |β| ≤ r. Then we have coordinate forms
eβ , of the same form as those of Λβ and Λ
e β . We have
of λβj and λ
j
j
j
(3)
λβj =
∂
(α + β)! i
xα i
,
α!
∂xα+β−{j}
eβ = (α + β)! xi pα+β−{j}
λ
α i
j
α!
Let k be the width of a monomial Weil algebra A. For m ≥ k, define an
immersion element i ∈ T0A Rm by xiα = 0 whenever |α| ≥ 2 and xij = δji for
j ∈ {1, . . . , k}. For general r, k, remind the jet group Grk = inv J0r (Rk , Rk )0 , where
inv indicates the invertibility of maps in question. The multiplication in Grk is
defined by the jet composition. We give the coordinate form of the action of this
group on T ∗ T A . Let ail1 ,...,lq denote the canonical coordinates on Gsm and ãil1 ,...,lq
indicate the inverse. Then the transformation law of the action of Gsm on T0A Rm
is of the form
(4)
x̄iα = ail1 ...lq xlα11 . . . xlαqq
for all admissible multiindices α and their decompositions α1 , . . . , αq .
The jet group Grk is identified with Aut Drk , the group of automorphisms of the
algebra Drk , as follows. For j0r g ∈ Grk and j0r ϕ ∈ Drk define
(5)
j0r g(j0r ϕ) = j0r ϕ ◦ (j0r g)−1
Let A be a monomial Weil algebra of width k and height r and p : Drk → A be the
projection homomorphism.
In what follows, we shall consider A as Dsm /(I ∪ {τk+1 , . . . , τm }) for s ≥ r,
m ≥ k with the properly modified projection p : Dsm → A. Consider a group
WEIL BUNDLES ASSOCIATED TO MONOMIAL WEIL ALGEBRAS
367
GA = {j0s g ∈ Gsm ; p ◦ j0s g = p}, [1]. The following lemma characterizes GA as the
stability subgroup of the immersion element i.
Lemma 2. Let A = Dsm /I be a monomial Weil algebra of width k, height r and
St(i) ⊆ Gsm be the stability subgroup of the immersion element i ∈ T0A Rm under
the canonical left action of Gsm on T0A Rm . Then it holds GA = St(i).
Proof. The formula (4) implies that every element of Gsm stabilizes i if and only
if aij = δji for j ∈ {1, . . . , k} and aiα = 0 whenever |α| ≥ 2, τ α 6∈ I and τ α ∈<
τ1 , . . . , τk >.
On the other hand, GA = {j0s g ∈ Gsm ; p ◦ j0s ϕ ◦ (j0s g)−1 = p ◦ j0s ϕ ∀j0s ϕ ∈ Dsm }.
In coordinates, we have
(6)
x̄α = xl1 ,...lq ãlα11 . . . ãlαqq
where x̄α indicates the transformed value of j0s ϕ (in coordinates xα ) under an
automorphism j0s g (with coordinates aiα . Substituting an i-th projection pri for
ϕ, we obtain x̄α = ãiα and consequently ãij = aij = δji for j ∈ {1, . . . , k} and
ãiα = aiα = 0 for |α| ≥ 2, τ α 6∈ I and τ α ∈< τ1 . . . , τk >. Thus we have GA ⊆ St(i).
The converse inclusion is immediately obtained from (6), taking into account the
coordinate form of i. It proves our claim.
We remind the concept of a regular A-point of a Weil bundle MA . An element
ϕ ∈ MA is said to be regular (a regular A-point) if and only if its image coincides
with A, [1]. Taking into account the identification (1), such a concept can be
extended to an A-velocity j A ϕ ∈ T A M . Clearly, it is regular if and only if ϕ is an
immersion in 0 ∈ Rk , where k is the width of A. Further, it must hold dim M ≥ k.
In the case m = k the concept of regularity coincides with that of invertibility.
The map ι̃ from Proposition 1 preserves regularity and thus ι̃ : Ak → Rk can be
restricted to reg(N k ) → Grk , where N denotes the nilpotent ideal of A.
Alonso in [1] proved that there is a structure of a fiber bundle on reg T A M
with the standard fiber Grk /GA over a k-dimensional manifold M and therefore
reg T0A Rk is identified with Grk /GA . The elements of reg(T A )0 Rk are left classes
j0r gGA . We extend this assertion of his to m-dimensional manifolds for m ≥ k.
For ι̃ : Am → (Drk )m corresponding to a Weil algebra of width k we define a map
ι̃∗ : Am → (Drk )m by
(7)
ι̃∗ (xiα τ α ) = xiα τ α + δip τp
p≥k+1
Then we have a lemma, giving the decomposition of any j0r g ∈ Grm onto its projection from ι̃∗ ◦ p̃(Grm ) and the component in GA .
Lemma 3. Let A = Drk /I be a Weil algebra of width k and j0r g ∈ Grm , m ≥ k.
There is an element j0r h ∈ GA such that
(8)
j0r g = ι̃∗ ◦ p̃(j0r g) ◦ j0r h
368
JIŘÍ TOMÁŠ
Proof. The proof of the assertion is done in coordinates and it is based on the
iterated application of (4). We do it for only for k, since for m ≥ k it is almost the
same. Let ciγ denote the coordinates of j0r g, aiγ the coordinates of ι̃ ◦ p̃(j0r g) and
biγ the coordinates of j0r h to be found. Clearly, aiγ = ciγ whenever τ γ 6∈ I. In the
first step suppose that α is a minimal multiindex such that τ α ∈ I. It follows from
(4), that ciα = ail blα , if we consider the conditions for j0r h. The unique solution is
given by the invertibility of j0r g. Suppose the assertion being proved for |α| ≤ p.
We prove it for |α| = p + 1. By (4) we have ciα = ail1 ...ls blα11 . . . blαss + ail blα , s ≥ 2.
From the regularity of j0r g we obtain again the unique solution blα , which proves
our claim.
In the proof of the assertion giving the main result, we need to describe the
∂
). The transformation laws for the action of Gr+2
stability group of j0r+1 ( ∂xm+1
m+1
r+1
m
on (J
T )0 R has the coordinate expression
(9)
X̄αi = ailγ1 Xγl 2 ãγα ,
∂
where Xαi , |α| ≤ r + 1 denote the canonical coordinates of j0r+1 ( ∂xm+1
). Further,
any multiindex γ including the empty one is decomposed into γ1 , γ2 and the notation ãγα denotes the system of all ãlα11 . . . ãlαss for l1 , . . . , ls forming the multiindex
γ and decompositions α1 , . . . , αs forming α. It follows, that in coordinates any
i
i
i
element of Gr+2
m+1 must satisfy aj = δm+1 and aα = 0 whenever the multiindex α
formed by all 1, . . . , m + 1 contains any m + 1 for |α| ≥ 2. To describe the stability
∂
) by terms of Lemma 2 and Lemma 3, denote Asm+1 the Weil
group of j0r+1 ( ∂xm+1
s
algebra of Dm+1 /I for I =< τm+1 τ α >, |α| ≥ 1. Thus we have proved the following
lemma
r+2 m+1
∂
Lemma 4. The stability group of j0r+1(∂xm+1
) in Gr+2
)
m+1 is of the form ι̃((Am+1)
r+2
r+1
∂
∩Gm+1 . Moreover, the stability group of j0 ( ∂xm+1 ) and the immersion element
m+1
i ∈ T0A Rm+1 is of the form GA;m+1 = GA ∩ ι̃((Ar+2
).
m+1 )
e defined on T ∗ T A (not natural in
Let us consider the base Be of all T -functions Λ
general ), constructed from the non-absolute natural operators L(τ α )T A and from
the absolute operators Λβj with the coordinate expression given by (3). Let Be1
denote the subbasis of Be formed by natural operators T → T T A . It follows from
Lemma 3, that any element j A g ∈ reg T A M is identified with ι̃∗ (j A g) ∈ Gr+1
m+1 , the
only representative of the left class j A gGA in the sense of Lemma 3. Therefore we
have
(10)
i = l((ι̃∗ (j A g))−1 , j A g)
A m+1
where l is the symbol for the left action of Gr+1
to be used also for
m+1 on T0 R
r+1
m+1
∗ A
the action of this group on (J
T )0 R
× (T T )0 Rm+1 . Let us define a map
Imm : T ∗ (reg T A )0 Rm+1 → (Ti∗ T A )0 Rm+1 as follows
(11)
Imm(w) = l((ι̃∗ (q(w)))−1 , w),
WEIL BUNDLES ASSOCIATED TO MONOMIAL WEIL ALGEBRAS
369
w ∈ T ∗ (reg T A )0 Rm+1 .
Proposition 5. Let A be a monomial Weil algebra and (T ∗ (reg T A ))0 Rm+1 →
(reg T A )0 Rm+1 be the restriction of the natural bundle T ∗ T A Rm+1 → T A Rm+1
to the opened submanifold (reg T A )0 Rm+1. Then all operators from Be − Be0 are
Gr+2
m+1 -invariant in respect to the map Imm.
Proof. We prove the assertion from the transformation laws of the action of Gr+2
m+1
on (J r+1 T )0 Rm+1 × (T ∗ T A )0 Rm+1 . We complete them for pα
.
Denote
γ
=
α
−
{j}
j
the multiindex from (3). Then we have
p̄βj =
(12)
(β + γ)! l
ã
x̄l1 . . . x̄lγss pβγ
j
β!γ1 ! . . . γs ! jl1 ...ls γ1
when the sum is made for all decompositions γ1 , . . . , γs of multiindices γ. The
formula is obtained from (3) and the standard combinatorics. To accent Imm(w)
as a transformed value for any w ∈ T ∗ (reg T A )0 Rm+1 , use p̄α
i for the additional
coordinates (obviously, the coordinates x̄iα coincide with those of i). Then we have
e β (Imm(w)) = Λ
e β (x̄iα )p̄α+β−{j} = β!p̄β = β! (β+γ)! ãi pβγ if we put γ = α − {j},
Λ
jγ i
j
j
i
j
β!γ!
which follows from (12). If we consider the coordinate expression of ι̃(Am+1 ) and
βγ
i
the formula (3), we obtain that the last expression coincides with (β+γ)!
γ! xjγ pi =
αj (α+β)! i α+β−j
e i , pα ) = Λ(w).
e
x p
= Λ(x
It proves our claim.
αj +βj
α i
α!
α
i
The following lemma specifies a certain class of functions, among which all investigated ones must be contained.
Lemma 6. Let m ≥ k. Then every natural T -function f : T ∗ T A Rm+1 → R is of
^
α )T A , Λ
e β ) for some smooth function h of the suitable type.
the form h(L(τ
j
Proof. By general theory, we are searching for all Gr+2
m+1 -invariant functions defined on (J r+1 T )0 Rm+1 × (T ∗ T A )0 Rm+1 . Let w ∈ (T ∗ T A )0 Rm+1 and xiα denote
the coordinates of q(w), q : T ∗ T A → T A being the cotangent bundle projection. By a general lemma from [5], Chapter VI, the natural T -function must
satisfy f (j0r+1 X, w) = h(Xγi pβi , xiα pβi ) for any non-zero j0r+1 X of a vector field
X on Rm+1 . The coordinates used in the recent identity coincide with those
defined before Lemma 2. The last expression can be considered in the form
^
α )T A , X i pβ , Λ
e β , xi pβ ) for |β| ≥ 0, |γ| ≥ 1 and |δ| ≥ 2. Identify q(w)
h(L(τ
γ i
j
δ i
with j A g for any w ∈ T ∗ (reg T A )0 Rm+1 , i.e. q(w) = l(ι̃(j A g), i) and put j0r+1 Y =
^
α )T A , Y i p̄β , Λ
e β , 0, p0 ) for |γ| ≥ 1
l(ι̃(j A g)−1 , j r+1 Y ). Then f (j r+1 X, w) = h(L(τ
γ
0
0
i
j
i
and i ∈ {1, . . . , k}. Here p̄βi indicate the transformed values of pβi under the map
Imm. The last identity follows from Proposition 5. Further, there is j0r+2 g ∈ GA ∩
∂
GAr+2 such that l(j0r+1 g, j0r+1 ( ∂xm+1
)) = j0r+1 Y . Then we have f (j0r+1 X, w) =
m+1
370
JIŘÍ TOMÁŠ
^
α )T A , 0, Λ
e β , 0, p0 ) for i ∈ {1, . . . , k}. The excessive coordinates p0 are annih(L(τ
i
i
j
m+1
hilated by an element of Ker πrr+1 ∩ ι̃((Ar+2
), namely by an element satism+1 )
fying in coordinates aiα = 0 except of α = (i, . . . , i) . Such an element stabilizes
| {z }
(r+1)−times
∂
j0r+1 ( ∂xm+1
) as well as i, which completes the proof.
Searching for all natural T -functions T ∗ T A Rm+1 → R among those from Lemma
e
6, we state the basis B of functions defined on Ti∗ T A Rm+1 and identify it with B.
By general theory, [5], every natural T -function in question is determined by its
∂
value over j0r+1 ( ∂xm+1
) on (T ∗ T A )0 Rm+1 . Further, it follows from Lemma 4 and
∂
the formula (11) that the map Imm stabilizes j0r+1 ( ∂xm+1
) in the following sense.
∗
A
m+1
For any w ∈ T (reg T )0 R
the action of ι̃(q(w)) on (J r+1 T )0 Rm+1 stabilizes
∂
j0r+1 ( ∂xm+1
).
Set B the basis of functions defined on Ti∗ T A Rm+1 obtained by the restriction
∂
of Be over j0r+1 ( ∂xm+1
) onto Ti∗ T A Rm+1 . Conversely, B determines Be by
(13)
e r+1 (
B(j
0
∂
∂xm+1
), w) = B ◦ Imm(w)
f1 . Moreover, for any w ∈ T ∗ (reg T A )0 Rm+1 ,
Analogously, we construct B1 from B
i
the values formed by B(w) coincide with the coordinates pβj of w defined before
(2) for j = 1, . . . , k except that of p0j for the absolute functions and pβm+1 for the
non-absolute ones. Thus any base T -function of B defined on Ti∗ (reg T A )0 Rm+1 corresponds to some projection prβj : Ti∗ (reg T A )0 Rm+1 → R. It follows from Lemma 4
^
α )T A are natural that all natural T -functions (T ∗ T A )Rm+1 →
and the fact that L(τ
R from Lemma 6 are in the canonical bijection with GA -invariant functions defined
^
α )T A )(Λ
e β ) for Λ
e β : T ∗ T A Rm+1 → R.
on Ti∗ T A Rm+1 which are of the form h(L(τ
i
j
j
β
Using coordinates, we find all GA -invariants of pj , j ∈ {1, . . . , k}, |β| ≥ 1. Then
^
^
α )T A )(pβ ) with h(L(τ
α )T A )(Λ
e β ) and by (12), we
we identify the functions h(L(τ
j
j
obtain all natural T -functions on T ∗ T A Rm+1 .
This way we have deduced that our problem can be reduced to the problem
of searching for all GA -invariant functions defined on Ti∗ T A Rm+1 which can be
identified with a smooth function h : RN → R for a suitable integer N . The
coordinate expression of the action of GA on Ti∗ T A Rm+1 is induced by (12) and it
is of the form
(14)
p̄βj = pβj − C(β + γ, β)aljγ pβγ
l
for τj τ γ ∈ I
and τ β τ γ 6∈ I
where C indicates the multicombinatorial number. Clearly, Ti∗ T A Rm+1 is identified with the space RN endowed with the action (14) of GA . We are going to
r+1
investigate GA ∩ Grm+1 -orbits on RN , since only p0j depend on Bm+1
and they
can be annihilated by this subgroup. For those orbits, we construct all functions
WEIL BUNDLES ASSOCIATED TO MONOMIAL WEIL ALGEBRAS
371
distinguishing them and then we express the corresponding invariants by terms of
e
elements from B.
The following assertion describes an important property of (GA ∩ Ker πsr )-orbits
to bee necessary in the proof of the main result. Denote by Bs ⊆ B the set of all
(GA ∩ Ker πsr )-invariants selected from B and denote by Ns the number of elements
in Bs . Clearly, B1 ⊆ B2 ⊆ · · · ⊆ Br−1 ⊆ Br . Further, denote Bts = Bs − Bt and
Nts = Ns − Nt . Then we have
Proposition 7. Let w ∈ RN and Orbs (w) be its (GA ∩ Ker πsr )-orbit. Then
s+1
Bss+1 (Orbs (w)) has the structure of an affine subspace of RNs , the modelling
s+1
vector space of which being determined by the formula (14) restricted to Bm+1
∩GA .
Proof. is done directly applying the formula (14). Let w1 and w2 be elements
of Bss+1 (Orb(w)). Then w1 can be achieved from w by the action of an element
s+1
of Bm+1
∩ GA . The coordinate expression of such a transformation is given by
β
β
β
¯β
p̄j = pj − C(β + γ, β)aljγ pβγ
l . Analogously for w1 and w2 , we have p̄j = p̄j − C(β +
βγ
β
β
βγ
−→
−→
−→
γ, β)bljγ p̄l . Then p̄¯j = pj − (aljγ + bljγ )pl , which follows ww
2 = ww1 + w1 w2 . It
proves our claim.
e of natural functions from B.
e The conIn what follows, we construct a basis D
struction is given by a procedure, generating step by step a base of GA -invariants
determining the base of natural functions. We start the procedure selecting ele1 = Be1 . For any w ∈ T ∗ T A Rm+1 , consider its orbit
ements of B1 and put D
i
Orb(w) = Orb1 (w).
In the second step, consider B12 (Orb1 (w)), which is by Proposition 7 a k2 2
dimensional affine subspace of the affine space RN1 for some k2 ≤ N12 . For almost every GA -orbit in the sense of density, such an affine subspace contains a
unique point I C2 satisfying prj (I C2 ) = 0 for j ∈ C2 . The remaining components
C2
identified with natural functions
of I C2 determine GA -invariants I1C2 , . . . , IN
2
1 −k2
C
C
Ie 2 , . . . , Ie 22
.
1
N1 −k2
In order to express them in formulas, we notice the following property of
Bss+1 (Orbs (w)) for any s = 1, . . . , r −1. Proposition 7 and its proof imply that if an
s+1
element of Bss+1 (Orbs (w)) is stabilized by j0s+1 g ∈ Bm+1
under the canonical left
s+1
action then the whole Bss+1 (Orbs (w)) is stabilized. Denote Sts+1
s;m+1 ⊆ GA ∩ Bm+1
the stability group of Bss+1 (Orbs (w)). One can easily deduce that Sts+1
s;m+1 satisfies
the stability property of this kind for almost every w ∈ RN . Clearly, Sts+1
s;m+1 is a
s+1
s+1
s+1
closed and normal subgroup of GA ∩Bm+1 and thus Hs;m+1 = GA ∩Bm+1 / Sts+1
s;m+1
s+1
s+1
is a Lie group. It follows the existence of a section σs+1;m+1 : Hs;m+1
→ GA ∩Bm+1
.
N
2
2
2
Hence for any w ∈ R we have a unique j0 h ∈ σ2;m+1 (H1;m+1 ) ' H1;m+1 such
2
that B12 (l(j02 h, w)) = I C2 (w). Thus we have a map αC2 : RN → H1;m+1
. Therefore,
372
JIŘÍ TOMÁŠ
any w ∈ Ti∗ T A Rm+1 is transformed onto
(15)
l(αC2 (w), w) = lαC2 (w) = (IjCs2 (w)),
s = 1, . . . N12 − k2 , js 6∈ Cs
C2
e2 = D
e1 ∪
Applying the identification (13), we obtain Ie1C2 , . . . , IeN
and put D
2
1 −k2
C
C
{Ie1 2 , . . . , IeN22 −k2 }.
1
es of natuIn the (s + 1)-th step of the procedure we come out from the basis D
ral functions and an element ws = lαCs ◦ · · · ◦ lαC2 (w) ∈ Orb1 (w) instead w from
the second step. By Proposition 7, Bss+1 (Orbs (ws )) is an affine subspace of dis+1
mension ks+1 of RNs for some ks+1 . Select Cs+1 ⊆ {1, . . . , Nss+1 }. For almost
∗ A m+1
every ws ∈ Ti T R
there is a unique point I Cs+1 (ws ) = I Cs+1 (Bss+1 (Orbs (ws ))
Cs+1
such that prj ◦I
= 0 for j ∈ Cs+1 . The remaining components of I Cs+1 determine analogously to the second step of the procedure GA -invariants and by (13)
Cs+1 ,...,C2
natural functions Iels+1
for ls+1 6∈ Cs+1 . Analogously to the second step,
s+1
for any ws under discussion there is a unique element j0s+1 h ∈ σs+1;m+1 (Hs;m+1
)
s+1
s+1
Cs+1
N
such that l(j0 h, Bs (ws )) = I
(ws ). Hence we have a map αCs : R →
s+1
σs+1;m+1 (Hs;m+1
) such that l(αCs+1 (ws ), ws ) = I Cs+1 (ws ) = lαCs+1 ◦· · ·◦lαC2 (w) =
IeCs+1 ,...,C2 (w) taking into account the identification (13). Hence we obtained the
es+1 = D
es ∪ {IeCs+1 ,...,C2 ; ls+1 6∈ Cs+1 }. We proved the main result, given by
basis D
ls+1
the following Proposition
Proposition 8. Let A = Drk /I be a monomial Weil algebra of width k, dim M =
m ≥ k + 1. Let ι̃ : T A M → Tkr M be an embedding described in Proposition 1.
Consider a basis C of A and a basis B0 of Der(Drk ). Further, let Be be a basis
of functions defined on T ∗ T A M constructed from operators T p̃ ◦ λD ◦ ι̃ by the
operation e defined in the very end of Section 1, D ∈ B0 . Then all natural T functions fM : T ∗ T A M → R are of the form
A , Ie , IeC2 , . . . , IeCs ,...,C2 )
h(LM^
(c)TM
l1 l2
ls
where h is any smooth function of a suitable type, Iel1 are natural functions selected
directly from Be and IelCs s ,...,C2 (ls 6∈ Cs ) are obtained by the procedure.
I would like to express my gratitude to professor Kolář for his precious advise
and comments.
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Department of Mathematics, Faculty of Civil Engineering, Technical University
Brno, Žižkova 17 602 00 Brno, Czech Republic
E-mail address: Tomas.Jemail.fce.vutbr.cz