Singularities of bihamiltonian systems and the multidimensional rigid body, страница 3

A similar statement holds forskew-symmetric forms:Theorem 4 (Jordan-Kronecker theorem, see [38]). Let A, B be two skewsymmetric forms on a complex vector space V . Assume that B is a genericform in the pencil αA + βB, i.e.rank B ≥ rank (αA + βB) for all α, β.Then there is a basis in V such that A, B will have the following blockdiagonal form:0 −JkT1 ,λ1A=Jk1 ,λ10...0Jkm ,λm−JkTm ,λm0AK10,B=0−Ek1Ek10...0−EkmEkm0BK,where• Jk,λ is a k × k Jordan block with the eigenvalue λ.• Ek is the k × k identity matrix.• rank (AK + λBK ) does not depend on λ.Remark 1.4.1. If B is not a generic form, one should replace it with a suitablelinear combination αA + βB.Remark 1.4.2.

The theorem also gives a normal form for AK and BK , whichwe do not need.The following property of the Jordan-Kronecker form is very important:Proposition 1.4.1.rank (A + λB) < max rank (A + νB) ⇔ λ ∈ {λi , 1 ≤ i ≤ m}.νNote that we will not use the Jordan-Kronecker form in formulations oftheorems, because this form can hardly be calculated explicitly in examples.Nevertheless, it will be convenient to use this form for the illustration of somenotions we are going to introduce.Also note that we will not use the Jordan-Kronecker theorem in theproofs, preferring an invariant style.111.5Bihamiltonian systems and integrals ininvolutionDefinition 6.

Two Poisson brackets P0 , P∞ (on a smooth manifold M ) arecalled compatible if any linear combination of them is a Poisson bracket again.The set of non-zero linear combinations Π = {αP0 + βP∞ } (with complexcoefficients) is called in this case a Poisson pencil.Since it only makes sense to consider Poisson brackets up to proportionality, we will write Poisson pencils in the formΠ = {Pλ = P0 + λP∞ }λ∈C .Definition 7. Rank of a pencil Π at a point x is the numberrank Π(x) = max rank Pλ (x).λRank of a pencil Π on a manifold M is the numberrank Π = max rank Π(x) = max rank Pλ = max rank Pλ (x).xλλ,xDefinition 8.

A vector field v is called bihamiltonian with respect to a pencilΠ, if it is hamiltonian with respect to all brackets of the pencil.Let Π be a Poisson pencil on M and v be a vector field which is bihamiltonian with respect to Π. We want to construct a complete family ofcommuting integrals for v. The main idea for this is provided by the followingwell-known statement.Proposition 1.5.1 (see [36]). Let Π = {Pλ } be a Poisson pencil. Then1.

If f is a Casimir function of Pλ for some λ, then f is an integral ofany vector field bihamiltonian with respect to Π.122. If f is a Casimir function of Pλ , g is a Casimir function of Pν , andλ 6= ν, then f and g are in involution with respect to all brackets of thepencil.3. If f and g are Casimir functions of Pλ , and rank Pλ (x) = rank Π(x),then f and g are in involution with respect to all brackets of the pencilat the point x.Remark 1.5.1. The third statement can be false if rank Pλ (x) < rank Π(x).For example, let P0 be an arbitrary non-zero Poisson bracket and P∞ = 0.The set of Casimir functions of P∞ coincides with the set of all smoothfunctions. Obviously, not all of them are in involution with respect to P0 .However, the statement is true if λ is such that rank Pλ (x) = rank Π foralmost all x.

This can be proved by the continuity argument.LetBad = {x ∈ M : rank Π(x) < rank Π}.Further we will only consider x ∈/ Bad. For such x we can find α such thatrank Pα (x) = rank Π. Moreover, we can find ε > 0 and a neighbourhoodU (x) such that rank Pν (y) = rank Π for |ν − α| < ε, y ∈ U (x), and all localCasimir functions of Pν where |ν − α| < ε are defined in U (x). Consider afamily F = Fα,ε generated by all these Casimir functions. Proposition 1.5.1implies the following.Proposition 1.5.2. F is a (local) family of integrals in involution for anysystem bihamiltonian with respect to our pencil.Remark 1.5.2.

The choice of α and ε is not important which means that ourresults remain true for any choice of α, ε. Moreover, under some additionalconditions we will get the same family of integrals for all α, ε. What is13important in this construction, is the fact that F is generated by the Casimirfunctions of brackets which are regular at the point x (see Example 2.4.1).1.6CompletenessLetΛ(x) = {λ ∈ C : rank Pλ (x) < rank Π(x)}.andS = {x : Λ(x) 6= ∅}.Definition 9.

We will say that Π is micro-Kronecker (or simply Kronecker ),if the set S has measure zero. Giving this definition we follow I.Zakharevich[40] and A.Panasyuk [33].Theorem 5 (A.V.Bolsinov [5], the criterion of completeness of F on a regularsymplectic leaf). Let O(α, x) be a symplectic leaf of the bracket Pα passingthrough a point x such that Pα is regular at x. Then F |O(α,x) is complete atx if and only if x ∈/ S.Corollary 1.6.1. F is complete on O(α, x) if and only if the set S ∩ O(α, x)has measure zero.Corollary 1.6.2. If Π is Kronecker, then F is complete on almost all regularsymplectic leaves.The theorem also implies that singular points of F |O(α,x) are exactly thepoints where the rank of some bracket Pβ drops.

A question arises: Howdo we check non-degeneracy of these points and determine theirtype?It turns out that the answer can be given in terms of the so-called linearization of the pencil Π, which will be defined later.14Chapter 2Definitions and thenon-degeneracy criteria2.1Linear pencilsDefinition 10. Let g be a Lie algebra and A be a skew-symmetric bilinearform on it. Then A can be considered as a Poisson tensor on the dual space g∗ .Assume that the corresponding bracket is compatible with the Lie-Poissonbracket. The Poisson pencil Πg,A = {Pλg,A }, wherePλg,A (x)(ξ, η) = hx, [ξ, η]i + λA(ξ, η), for ξ, η ∈ g,will be called the linear pencil associated with the pair (g, A).Giving this definition, we are motivated by the fact that linear pencilsarise as a linearization of a general Poisson pencil at a singular point (seeSection 2.3).

Now we shall discuss some properties of linear pencils.Proposition 2.1.1. A form A on g is compatible with the Lie-Poissonbracket if and only if this form is a 2-cocycle in terms of the Chevalley15Eilenberg complex, i.e.dA(ξ, η, ζ) = A([ξ, η], ζ) + A([η, ζ], ξ) + A([ζ, ξ], η) = 0(2.1)for any ξ, η, ζ ∈ g.The proof is straightforward.Corollary 2.1.1.

If a form A is compatible with the Lie-Poisson bracket ong∗ , then Ker A is a subalgebra in g.Proof. Let ξ, η ∈ Ker A. Then last two terms in identity (2.1) vanish, therefore so does the first one, which means that [ξ, η] ∈ Ker A, q.e.d.Remark 2.1.1. A definition of the Chevalley-Eilenberg complex can be foundin [10].Example 2.1.1 (Exact forms or “argument shift method”). Let g be an arbitrary Lie algebra and Aa (ξ, ψ) = ha, [ξ, ψ]i, where a ∈ g∗ . It is easy to seethat Aa is compatible with the Lie-Poisson bracket.The condition A = Aa is equivalent to the fact that A = da in theChevalley-Eilenberg complex, therefore the compatibility of Aa and the LiePoisson bracket simply means that d2 = 0.The pencils with A = Aa are the ones which one should consider whenconstructing an integrable system by the argument shift method (see [30]),therefore we will call them pencils of the argument shift type.

It also seemsreasonable to call such pencils exact.We want to use linear pencils to construct commuting functions followingthe general scheme of Section 1.5. In order to be able to do so, we need thefollowing property of regularity:Definition 11. We will say that a cocycle A on g is regular, if rank Πg,A =rank A.16It is easy to see that if A = Aa , then rank Pλg,A = dim g − ind g for anyλ 6= ∞, therefore regularity of A means that rank A = dim g − ind g, i.e.

it isequivalent to regularity of the element a. To give a criteria for regularity inthe general case, we need a (standard) construction of the central extensionof g, associated with a form A. Let us consider the space gA = g + K1 , whereK1 = hzi is a one-dimensional vector space, and define a commutator [ , ]Aon gA by the following rule:[x, y]A = [x, y] + A(x, y)z, for any x, y ∈ g ⊂ gA ,[z, gA ]A = 0.It is easy to see that if A is closed, then the commutator [ , ]A turns gA intoa Lie algebra. Also note that g = gA /hzi and the lift of A to gA is an exactform. This means that every closed 2-form on a Lie algebra becomes exactafter a lift to a certain one-dimensional central extension.eProposition 2.1.2.

A 2-cocycle A on g is regular if and only if it its lift Ato gA is a differential of a regular element.e is regular. SinceProof. It suffices to show that A is regular if and only if Aee = rank A, our proposition is proved.rank ΠgA ,A = rank Πg,A and rank AProposition 2.1.3. If A is a regular cocycle on g, then Ker A is abelian.e be the lift of A to gA . Since Ae is exact, Ker Ae is abelian. ButProof. Let Ae where π is the natural projection. Therefore, Ker A isKer A = π(Ker A),abelian as well.We will see that in good examples Ker A is not just an abelian subalgebra,but a Cartan subalgebra.Suppose that A is regular.