DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 247–254

ESTIMATES OF THE ACTION ORDER OF SOME JET GROUPS

KAROLINA LUBAŚ AND ANDRZEJ ZAJTZ

Abstract. As we know, if the general k-jet group G(n, k) in R^{n} acts continuously on a manifold of finite dimension m > 0 and the highest k-th order derivatives act nontrivially, then k is bounded by a known function of m and n [4]. In geometric interpretation the estimate function translates into the order of the natural bundle in which the group G(n, k) acts on the standard fibre (meant as the space of geometric objects). On manifolds with extra structure the same question arises for corresponding subgroups of G(n, k), preserving the structure. In this note the sharp upper estimates are found for actions of unimodular, homothetic and symplectic subgroups.

1. Introduction

Given a volume form ω on a smooth n-dimensional manifold, the k-jet group of local diffeomorphisms of R^{n} preserving the form ω_{o} = dx^{1} ∧ ⋯ ∧ dx^{n} is the unimodular differential group Sl(n, k). It may be seen as group of polynomial diffeomorphisms of R^{n} of degree k, without constant term, with determinant equal to 1. Its Lie algebra sl(n, k) consists of polynomial, of degree ≤ k, vector fields of divergence zero.

A complementary subgroup to Sl(n, k) in G(n, k) is the group of nonlinear homotheties of variable scale ρ(x) ≠ 0. Denote it by H(n, k). Its elements are polynomial diffeomorphisms of the form f(x) = ρ(x)x.

The Lie algebra h(n, k) is made up of all vector fields X = aX_{o} with X_{o} = ∑_{i} x_{i} \frac{∂}{∂x_{i}} and a(x) = ∑_{1≤|α|≤k} a_{α}x^{α}.

The Lie algebras of Sl(n, k) and H(n, k) are direct summands of the Lie algebra of G(n, k),

sl(n, k) ⊕ h(n, k) = g(n, k) .

The symplectic k-jet group Sp(n, k) is formed by all local polynomial (of degree ≤ k) diffeomorphisms in R^{2n} preserving the origin and the symplectic form ω =

1991 Mathematics Subject Classification. 58G99. Key words and phrases. Unimodular, homothetic, symplectic jet groups and Lie algebras,

bounds to codimension of subalgebras, action order.

This paper is in final form and no version of it will be submitted for publication elsewhere.

∑_{1≤i≤n} dx_{i} ∧ dx_{n+i}. Its Lie algebra is formed by Hamiltonian vector fields

X_{f} = ∑_{i=1}^{n} \frac{∂f}{∂x_{n+i}} \frac{∂}{∂x_{i}} − \frac{∂f}{∂x_{i}} \frac{∂}{∂x_{n+i}} , (1.1)

where f is an arbitrary polynomial of the form ∑_{2≤|α|≤k+1} a_{α}x^{α}.

A k-jet group acts k-essentially means that the action does not factorize to an action of its truncation to order k − 1.

Theorem 1 (Main Result). Let G ⊂ G(n, k) be one of the following k-jet groups

1. Sl(n, k) of unimodular diffeomorphisms of R^{n}, (n ≥ 2),

2. H(n, k) of nonlinear homotheties (n ≥ 1),

3. Sp(2n, k) of symplectic diffeomorphisms.

Suppose G acts continuously and k-essentially on a manifold of dimension m ≥ 1. Then, respectively,

(1) k ≤ \frac{m}{n−1},  (2) k ≤ 2m + 1,  (3) k ≤ \frac{m}{n},

and the estimates are sharp.

The main idea of the procedure to get the estimates is as for the action of the general group G(n, k) (then k ≤ max\{\frac{m}{n−1}, \frac{m}{n} +1\}). For details see the monograph [2] or the article [4]. It consists in the estimation of the lowest possible codimension d of ”admissible” Lie subalgebras g of the Lie algebra g of G.

Roughly we justify it as follows. Let m be the dimension of the manifold M on which the group G acts continuously. If y ∈ M and G_{y} is the isotropy subgroup at the point y then

m ≥ dim(G/G_{y}) ≥ d . (1.2)

This is so because the homogeneous space G/G_{y} is mapped injectively and continuously into M. Solving the extreme inequality we get an estimation of k.

For technical reason, we determine the lower bound d by passing to the graded subalgebra

Gr(g) = g_{o} + g_{1} + ⋯ + g_{k−1}, dim Gr(g) = dim g

induced by the natural filtration in jet groups. The elements of g_{p} are represented by homogeneous polynomial vector fields of degree p+1, which belong to g modulo polynomials of higher degree (if any). The admissibility of g means that g_{k−1} ≠ g_{k−1}, i.e. it is not maximal. By the above equality of dimensions, it suffices to consider only graded subalgebras of Gr(g). The results will be true also for the original Lie algebras. Another essential simplification is the use of the following algebraic lemma ([2], [4]).

Lemma 1. Let E, F be finite dimensional real vector spaces and let f : E×F → R be a bilinear form . Denote by E^{o} or F^{o} the annihilators of E or F related to f, respectively. Let M ⊂ E and N ⊂ F be subspaces satisfying f|(M × N) = 0. Then

codim M + codim N ≥ codim E^{o} .

If g_{k−1} is not maximal, there is a non-zero linear form C on g_{k−1} with g_{k−1} ⊂ ker C. Given p, q, p + q = k − 1, we define a bilinear form f : g_{p} × g_{q} → R by f(X, Y) = C([X, Y]) and we study the dimensions of the annihilators. By Lemma 1 we get the estimate

codim g_{p} + codim g_{q} ≥ codim h

where h ⊂ g_{p} is the annihilator of g_{q} related to f.

For the coordinate expression, let

X = ∑_{i,λ} x_{λ}^{i} \frac{∂}{∂x^{i}} ∈ g_{p}, Y = ∑_{i,μ} y_{μ}^{i} \frac{∂}{∂x^{i}} ∈ g_{q} .

Then

[X, Y]_{γ}^{i} = ∑_{\begin{matrix}γ=λ+μ−1_{j} \\ |λ|=p+1 \\ |μ|=q+1 \end{matrix}}(λ_{j}x_{λ}^{i}y_{μ}^{j} − μ_{j}y_{μ}^{i}x_{λ}^{j}),

C([X, Y]) = ∑_{i,γ} C_{i}^{γ}[X, Y]_{γ}^{i}, |γ| = k .

Thus we shall study the equation

∑_{\begin{matrix}i,j \\ |λ|=p+1 \\ |μ|=q+1 \end{matrix}}(C^{μ+λ−1_{j}}λ_{j} − C^{μ+λ−1_{i}}μ_{i})x_{λ}^{i}y_{μ}^{j} = 0 . (1.3)

The required codimension of h is bounded below by the number of linearly independent coefficients at independent y_{μ}^{j}, i.e. by the rank of the linear system

∑_{\begin{matrix}1≤i≤n \\ |λ|=p+1 \end{matrix}}(C^{μ+λ−1_{j}}λ_{j} − C^{μ+λ−1_{i}}μ_{i})x_{λ}^{i} = 0. (1.4)

where the indices j, μ are fixed and stay at independent y’s.

2. The case g=sl(n, k)

In this case the polynomial vector fields X, Y have divergence zero. In coordinates it writes

∑_{i}(μ_{i} + 1)y_{μ+1_{i}}^{i} = 0 for each μ. (2.1)

Therefore the variables y_{μ}^{j} with j ∉ μ are independent. Hence the equations (1.4) hold for such couples (j, μ). We shall use the symbol (1) for a multiindex μ with μ_{1} = |μ|.

Denote by E_{s} the (LHS of) (s, (1))-equation for 2 ≤ s ≤ n and by F_{k} the (n, (1) + 1_{k})-equation for 1 ≤ k < n. They are

E_{s} = ∑_{i,λ}(C_{i}^{(1)+λ−1_{s}}λ_{s} − C_{s}^{(1)+λ−1_{i}}δ_{i}^{1}(q + 1))x_{λ}^{i} .

F_{k} = ∑_{i,λ}(C_{i}^{(1)+1_{k}+λ−1_{n}}λ_{n} − C_{n}^{(1)+1_{k}+λ−1_{i}}(δ_{i}^{1}q + δ_{i}^{k}))x_{λ}^{i} .

Now, let E_{1} be the joint coefficient at y_{(1)}^{1} after elimination

y_{(1)+1_{n}}^{n} = −(q + 1)y_{(1)}^{1} − y_{(1)+1_{2}}^{2} − ⋯ − y_{(1)+1_{n−1}}^{n−1}

(see (2.1)). We have

E_{1} = ∑_{i,λ}[(C_{i}^{(1)+λ−1_{1}}λ_{1} − C_{1}^{(1)+λ−1_{i}}(δ_{i}^{1}(q + 1))

− (q + 1)(C_{i}^{(1)+1_{n}+λ−1_{n}}λ_{n} − C_{n}^{(1)+1_{n}+λ−1_{i}}(δ_{i}^{1}q + δ_{i}^{k}))]x_{λ}^{i} .

We claim that the subsystem E_{1} = 0, E_{s} = 0, F_{k} = 0 of 2n − 1 equations is of rank ≥ 2n − 2. For this consider a linear combination

a^{1}E_{1} + ∑_{2≤s≤n} a^{s}E_{s} + ∑_{1≤k≤n−1} b^{k}F_{k} = 0 a^{i}, b^{k} ∈ R . (2.2)

We may assume that C_{n}^{(1)} ≠ 0 (any linear form C which does not annihilates the whole subalgebra sl(n, k) can satisfy this condition in appropriate coordinate system; we deduce it from properties established in [2] or [4]).

The couple (i, λ) = (n, (1)) stands at independent variable x_{(1)}^{n}; on substituting it in (2.2) we get

C_{n}^{(1)}(p + q + 1)a^{1} = 0 ,

which gives a^{1} = 0. Then, if we substitute λ = (1) + 1_{v} and i = n in (2.2) we obtain

∑_{2≤s≤n} C_{n}^{(1)+1_{v}−1_{s}}λ_{s}a^{s} = 0 .

For s ≠ v λ_{s} = 0, and for s = v λ_{v} = 1, so it gives a^{v} = 0 for 2 ≤ v ≤ n − 1. Assume now λ = (1) and 2 ≤ i ≤ n − 1. From (2.2) it remains

∑_{2≤k≤n−1} C_{n}^{(1)+1_{k}−1_{i}}δ_{i}^{k}b^{k} = 0 .

Hence b^{i} = 0 for 2 ≤ i ≤ n − 1. Consequently, (2.2) reduces to

a^{n}E_{n} + b^{1}F_{1} = 0

or explicitely

∑_{i,λ}(C_{i}^{(1)+λ−1_{n}}λ_{n} − C_{n}^{(1)+λ−1_{i}}δ_{i}^{1}(q + 1))(a^{n} + b^{1})x_{λ}^{i} = 0 .

Taking account of condition (2.1) we find the joint coefficient at x_{(1)+1_{n}}^{1} to be

−(p + q + 2)C_{n}^{(1)}(a^{n} + b^{1}). Hence a^{n} + b^{1} = 0.

All together we proved that

a^{1} = ... = a^{n−1} = 0, b^{2} = ... = b^{n−1} = 0, a^{n} + b^{1} = 0 ,

so that the rank of the system (2.2) is ≥ 2n − 2.

In view of Lemma 1 we get the estimation

codim g_{p} + codim g_{k−p−1} ≥ 2n − 2, 0 ≤ p ≤ k − 1 ,

and consequently we have proved

Proposition 1. Let g be a subalgebra of g = sl(n, k) which does not contain g_{k−1}. Then

codim g ≥ k(n − 1) .

The following example shows that the estimate is the best possible.

Example I Let g ⊂ sl(n, k) be defined by

g = \{∑_{i,ν} a_{ν}^{i}x^{ν} \frac{∂}{∂x_{i}} ∈ sl(n, k); a_{(1)}^{j} = 0, for j = 2, ..., n, 1 ≤ |(1)| ≤ k\} .

The codimension of this subalgebra is k(n − 1).

3. The case g = h(n, k)

We apply the same procedure as in the previous section. In coordinates the homogeneous elements X ∈ g_{p}, Y ∈ g_{q}, p + q = k − 1, write now

X = ∑_{i,λ} a_{λ}x^{λ+1_{i}} \frac{∂}{∂x_{i}} , Y = ∑_{i,μ} b_{μ}y^{μ+1_{i}} \frac{∂}{∂x_{i}}

and

[X, Y] = ∑_{\begin{matrix}1≤i≤n \\ |λ|=p, |μ|=q \end{matrix}} a_{λ}b_{μ}(|λ| − |μ|)x^{λ+μ+1_{i}} \frac{∂}{∂x_{i}} ∈ g_{k−1} . (3.1)

Assume, correspondingly,

C([X, Y]) = ∑_{λ+μ} C^{λ+μ}a_{λ}b_{μ}(|λ| − |μ|) = 0,

for X ∈ g_{p} and Y ∈ g_{q} with p ≠ q. Since b_{μ} are arbitrary, we get the system of linear equations in x_{λ}:

∑_{|λ|=p} C^{λ+μ}a_{λ} = 0, for all |μ| = q .

For any non-zero form C ∈ S^{k}(R^{n^{∗}}), this system has rank at least 1. Hence we deduce

codim g_{p} + codim g_{k−p−1} ≥ 1, p ≠ \frac{k − 1}{2} .

Summing up over p yields

2 codim g ≥ \begin{cases}k & if k is even, \\ k − 1 & if k is odd. \end{cases}

It gives codim g ≥ [\frac{k}{2} ].

Proposition 2. Any admissible Lie subalgebra of h(n, k) has codimension at least [\frac{k}{2} ] .

The following example shows that the estimate is sharp.

Example II

g = \{∑_{i,ν} a_{ν}x^{ν+1_{i}} \frac{∂}{∂x_{i}} ∈ h(n, k); a_{(1)} = 0, for |(1)| = 1, ..., [\frac{k}{2} ]\} .

Its codimension is [\frac{k}{2} ].

4. The case g = sp(2n, k)

The map f → X_{f} defined in (1.1) is an isomorphism of sp(2n, k) and the Lie algebra F(2n, k + 1) of all polynomials of degree k + 1 without terms of degree ≤ 1, with the Poisson bracket

\{f, g\} = ∑_{1≤i≤n} \frac{∂f}{∂x_{i}} \frac{∂g}{∂x_{n+i}} − \frac{∂f}{∂x_{n+i}} \frac{∂g}{∂x_{i}} . (4.1)

Then X_{{f,g}} = [X_{f}, X_{g}].

To introduce filtration and the grading

F(n, k + 1) = h_{o} + h_{1} + ... + h_{k−1} ,

corresponding to that of sp(n, k), we define ord(f) by that for monomials ord(x^{α}) := |α| − 2. For any polynomial function f ∈ F, ord(f) is defined to be the minimum of orders of non-zero monomials.

Equivalently, it suffices to find the lowest codimension of admissible Lie subalgebras h in F(n, k + 1). The algorithm is analogous as before.

Take homogeneous polynomials

f = ∑_{|μ|=p+2} a_{μ}x^{μ} ∈ h_{p}, g = ∑_{|ν|=q+2} b_{ν}x^{ν} ∈ h_{q} , p + q = k − 1

and compute the Poisson bracket

\{f, g\} = ∑_{\begin{matrix}1≤i≤n \\ |μ|=p+2 \\ |ν|=q+2 \end{matrix}}(μ_{i}ν_{n+i} − μ_{n+i}ν_{i})a_{μ}b_{ν}x^{μ+ν−1_{i}−1_{n+i}} ∈ h_{k−1} .

Take a non-zero form C ∈ S^{k+1}(R^{n^{∗}}) and suppose that C^{γ}{f, g}_{γ} = 0. Explicitely, it follows

∑_{i,μ,ν} C^{μ+ν−1i−1n+i}(μ_{i}ν_{n+i} − μ_{n+i}ν_{i})a_{μ}b_{ν} = 0 .

Hence, as b_{ν} are arbitrary, we get the linear system

∑_{i,μ} C^{μ+ν−1_{i}−1_{n+i}}(μ_{i}ν_{n+i} − μ_{n+i}ν_{i})a_{μ} = 0, |ν| = q + 2 . (4.2)

Notice that, up to a linear change of coordinates in R^{n}, we may assume C^{(1)} ≠ 0.

Denote by E_{1}, F_{i}, G_{i}, H_{i} the ν-equations of (4.1) for ν equal respectively to

(1), (1) + 1_{i} + 1_{n+i}, (1) + 1_{i}, (1) + 1_{n+i} .

We claim that the subsystem is of rank ≥ 2n.

For this consider linear combination

e_{1}E_{1} + f_{i}F_{i} + g_{i}G_{i} + h_{i}H_{i} = 0 (4.3)

with real coefficients. Set μ = (1) + 1_{i} + 1_{n+i}, then the combination (4.3) reduces to

C^{(1)}(−q − 2)e_{1}a_{(1)+1_{n+1}} = 0 .

As a_{μ} are arbitrary and C^{(1)} ≠ 0, it follows e_{1} = 0. Set μ = (1). From (4.2) we get C^{(1)}(p + 2)h_{1}a_{(1)} = 0, whence h_{1} = 0. For μ = (1) + 1_{n+i}, using (4.2) and the so far results, we have

∑_{2≤i≤n} C^{(1)}g_{i}a_{(1)+1_{i}} = 0 ,

hence g_{i} = 0 for 2 ≤ i ≤ n.

For μ = (1) + 1_{i} we obtain

∑_{2≤i≤n} C^{(1)}f_{i}a_{(1)+1_{i}} = 0 ,

hence f_{i} = 0 for 2 ≤ i ≤ n.

Thus, jointly, 2n coefficients in the linear combination (4.3) vanish, so that the rank of the considered subsystem is at least 2n. It gives codim h_{p}+codim h_{k−p−1} ≥ 2n for 0 ≤ p ≤ k − 1. Hence codim h ≥ kn. It translates into

Proposition 3. Any admissible Lie subalgebra of sp(2n, k) has codimension at least kn

The following example makes the estimate the best possible.

Example III

Let g ⊂ sp(2n, k) be defined by

g = {X_{f}; f = ∑_{μ} a_{μ}x^{μ}; a_{(1)+1_{i}} = 0 for 1 ≤ i ≤ n, 1 ≤ |(1)| ≤ k} .

Its codimension is kn.

In view of (1.2), the Propositions 1,2 and 3 followed by the examples imply immediately the assertions of the Theorem 1.

Some consequences

(i) If m < n − 1, the unimodular k-jet group SL(n, k) acts trivially on mdimensional manifolds. The first positive case m = n − 1 is represented by the natural action of SL(n, k) on the projective space P^{n−1}(R^{n}).

In unimodular geometry on n-manifolds every field of geometric objects valued in R^{m}, m < n − 1, is a collection of functions.

If m < n − 1, the linear group GL(n) can act nontrivially on m-manifolds only via the determinant.

It has been known, that for linear representations of Gl(n), this assertion is true already for m < n.

(ii) Similar conclusions can be formulated for the action of the symplectic k-jet group Sp(2n, k) in the case m < n.

(iii) Since for n = 1 the homothetic jet group H(1, k) coincides with the general differential group G(1, k), the Theorem 1 translates into the following known theorem, proved first by J. E. Penzov (1950) [3] and independently by D. B. A. Epstein and W. P. Thurston (1979) [1].

If a differential group G(1,k) acts continuously on a manifold of dimension m, then k ≤ 2m + 1 and the estimate is sharp.

References

[1] Epstein, D. B. A., Thurston, W. P., Transformation groups and natural bundles, Proc. London Math. Soc. 38(1979), 219–237.

[2] Kolář, I., Michor, P. W., Slovák, J., Natural operations in differential geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1993.

[3] Penzov, J. E., The classification of geometric objects of class v in X_{1} (Russ.), Math. Sbornik 26 (68) (1950), 161–182.

[4] Zajtz, A., The sharp upper bound on the order of natural bundles of given dimensions, Bull. Soc. Math. Belg. Ser B 39 (1987), 347–357.

Institute of Mathematics, Pedagogical University of Cracow Podchorazych 2, 30-084 Kraków, POLAND E-mail: smzajtz@cyf-kr.edu.pl