Singularities of bihamiltonian systems and the multidimensional rigid body, страница 4

Then we can apply the construction of Section1.5 to the pencil Πg,A in the neighbourhood of the origin (because the origin17does not belong to Bad due to regularity of A) and get a family F whichis involutive with respect to all brackets of the pencil. In particular, withrespect to the constant bracket A. Consider the symplectic leaf of A passingthrough the origin. This is simply a linear subspace of g∗ . The restriction ofF to this symplectic leaf is complete if and only if Λ(x) is empty for almostall x belonging to this subspace (see Corollary 1.6.1).

This leads us to thefollowingDefinition 12. A pencil Πg,A with regular A will be called integrable if themeasure of the set S ∩ O is zero whereS = {x : there exists λ such that rank Pλg,A (x) < rank Πg,A }and O is the symplectic leaf of A passing through the origin.We see that if a pencil Πg,A is integrable then there is a well-definedintegrable system on the symplectic leaf of A passing through the origin.2.2Singularities associated with integrablelinear pencilsSuppose that a pencil Πg,A is integrable. Then the Casimir functions of theregular brackets of the pencil define an integrable system on a symplecticleaf of A passing through the origin. The origin is a zero-rank singular pointfor this system.

This means that every integrable linear pencil canonicallydefines a zero-rank singularity (i.e. a germ of an integrable system at asingular point). Denote the singularity associated with Πg,A by Sing(Πg,A ).Example 2.2.1 (Argument shift on semisimple Lie algebras). Let g be asemisimple Lie algebra with two or four-dimensional coadjoint orbits and18Figure 2.1: Singularity corresponding to so(3)Figure 2.2: Singularity corresponding to sl(2), shift by a hyperbolic elementA = Aa be an “argument shift” form, where a ∈ g∗ is a regular element.Below is the list of the corresponding singularities:1.

Two-dimensional orbits: one degree of freedom.• so(3) - elliptic singularity (center). See Figure 2.1.• sl(2) - hyperbolic singularity (saddle) if the Killing form is positive on a (Figure 2.2), elliptic if it is negative (Figure 2.3), anddegenerate if it is zero (Figure 2.4).2. Four-dimensional orbits: two degrees of freedom.19Figure 2.3: Singularity corresponding to sl(2), shift by an elliptic elementFigure 2.4: Singularity corresponding to sl(2), shift by a nilpotent element20• so(4) ' so(3) ⊕ so(3) - center-center singularity (a product of twoelliptic singularities).• so(2, 2) ' sl(2) ⊕ sl(2) - saddle-saddle (a product of two hyperbolic singularities), saddle-center (a product of an elliptic and ahyperbolic singularity), center-center (a product of two ellipticsingularities) or degenerate singularity (depending on a).• so(3, 1) ' so(3, C) ' sl(2, C) - focus-focus singularity if a issemisimple, degenerate otherwise.We will see further that no semisimple Lie algebras except for the sumsof so(3), sl(2) and so(3, 1) give rise to non-degenerate singularities.

Thecorresponding fact in the theory of integrable systems is the Eliasson theorem:all non-degenerate singularities are products of elliptic, hyperbolic and focusfocus singularities (see Theorem 3).There are also solvable Lie algebras which give rise to non-degeneratesingularities.Example 2.2.2.1. Any regular linear pencil on e(2) = so(2) i R2 which is not of theargument shift type gives rise to an elliptic singularity.

The argumentshift pencil gives rise to a degenerate singularity.2. Any regular linear pencil on e(1, 1) = so(1, 1) i R2 which is not ofthe argument shift type gives rise to a hyperbolic singularity. Theargument-shift pencil gives rise to a degenerate singularity.3. Any regular linear pencil on e(2, C) ' e(1, 1, C) which is not of the argument shift type gives rise to a focus-focus singularity. The argumentshift pencil gives rise to a degenerate singularity.21Definition 13.

An integrable linear pencil Πg,A will be called nondegenerate, if the singularity Sing(Πg,A ) is non-degenerate.Before giving a classification theorem of non-degenerate pencils, we needto define three special algebras.Definition 14.1. Denote by g♦ the Lie algebra generated by e, f, h, t with the followingrelations:[e, f ] = h,[h, g♦ ] = 0,[t, e] = f,[t, f ] = −e.This algebra is known as the “diamond Lie algebra” (see [19]).2. Denote by gh♦ the Lie algebra generated by e, f, h, t with the followingrelations:[e, f ] = h,[h, gh♦ ] = 0,[t, e] = e,[t, f ] = −f.This algebra may be called the “hyperbolic diamond Lie algebra”.3. gC♦ = C ⊗ g♦ ' C ⊗ gh♦ - the common complexification of these twoalgebras.22Remark 2.2.1. The algebras g♦ and gh♦ are (the only non-trivial) onedimensional central extensions of e(2) and e(1, 1).On the other hand, g♦ and gh♦ may be obtained in the following way:There is a natural action of sp(2) on three-dimensional Heisenberg algebrah(3).

sp(2) has subalgebras so(2) and so(1, 1). The semi-direct sums of h(3)with these subalgebras are exactly g♦ and gh♦ .Definition 15. A complex Lie algebra g will be called non-degenerate if itcan be represented asg'M M Cso(3, C) ⊕g♦ /h0 ⊕ V,where V is abelian, and h0 is a central ideal.A real Lie algebra g will be called non-degenerate if it can be representedasg'MMMso(3) ⊕sl(2) ⊕so(3, C) ⊕MMM ⊕g♦ ⊕gh♦ ⊕gC♦ /h0 ⊕ V,where V is abelian, and h0 is a central ideal1 .A type of non-degenerate Lie algebra g with respect to its Cartan subalgebra2 h is the triple (ke , kh , kf ), where• ke = is the number of so(3) terms in the decomposition of g + thenumber of g♦ terms + the number of sl(2) terms such that the Killingform on sl(2) ∩ h is negative.1In other words, a real Lie algebra is non-degenerate if its complexification is non-degenerate.2By definition, a Cartan subalgebra of an arbitrary Lie algebra is a nilpotent subalgebra which coincides with its normaliser.

In the case of non-degenerate algebras Cartansubalgebras are simply maximal diagonalizable abelian subalgebras.23• kh = is the number of gh♦ terms + the number of sl(2) terms such thatthe Killing form on sl(2) ∩ h is positive.• kf = is the number of so(3, C)terms + the number of gC♦ terms.Theorem 6 (Classification of non-degenerate linear pencils).1. A linear pencil Πg,A is non-degenerate if and only if g is non-degenerateand Ker A is a Cartan subalgebra.2. Assume that Πg,A is non-degenerate. Then the type of Sing(Πg,A ) coincides with the type of g with respect to Ker A.The proof of the first statement in the complex case can be found inSection 5.2, and in the real case - in Section 5.3. The proof of the secondstatement is given in Section 5.4.Remark 2.2.2.

It follows from the theorem that if g is non-degenerate andKer A is a Cartan subalgebra, then A is automatically regular and the pencilΠg,A is automatically integrable.2.3Linearization of a Poisson pencilLet P be a Poisson bracket on a manifold M , x ∈ M . It is well-known thatthe linear part of P defines a Lie algebra structure on the kernel of P at thepoint x.The commutator in this algebra is defined as follows: let ξ, η ∈ Ker P (x).Choose any functions f, g such that df = ξ, dg = η, and define[ξ, η] = d{f, g}.The following is well-known (see [10], for example).24Proposition 2.3.1.1. The commutator [ , ] is well-defined and indeed turns Ker P (x) into aLie algebra.2. If P is regular at the point x, then this algebra is abelian.Now consider a Poisson pencil {Pλ = P0 +λP∞ } and fix a point x. Denoteby gλ (x) the Lie algebra on the kernel of Pλ at the point x. For regular λ(i.e.

for λ ∈/ Λ(x)) the algebra gλ is abelian. For singular λ (λ ∈ Λ(x)) this isnot the case in general, therefore gλ (x) carries non-trivial information aboutthe behaviour of the pencil in the neighbourhood of x.Remark 2.3.1. For real λ the algebra gλ (x) is real, while for complex λ it iscomplex.It turns out that apart from the Lie algebra structure gλ carries one moreadditional structure.Proposition 2.3.2. For any α, β the restrictions of Pα (x), Pβ (x) on gλ (x)coincide up to a multiplicative constant.Proof. Since Pλ vanishes on gλ , all other brackets of the pencil are proportional.The restriction Pα |gλ is a 2-form on gλ . Therefore, it can be interpretedas a constant Poisson bracket on g∗λ .Theorem 7. The bracket Pα |gλ is compatible with the Lie-Poisson bracketon g∗λ (i.e.

Pα |gλ is a 2-cocycle on gλ ).Proof. Since Pα and Pλ are compatible, we have{{f, g}α , h}λ + {{g, h}α , f }λ + {{h, f }α , g}λ ++ {{f, g}λ , h}α + {{g, h}λ , f }α + {{h, f }λ , g}α = 0.25But if df, dg, dh ∈ Ker Pλ then the first three terms vanish and we can writedown{{f, g}λ , h}α + {{g, h}λ , f }α + {{h, f }λ , g}α = 0.orPα ([df, dg], dh) + Pα ([dg, dh], df ) + Pα ([dh, df ], dg) = 0,i.e. Pα is a 2-cocycle on gλ , q.e.d.Consequently, Pα |gλ defines a linear pencil on g∗λ .

Since Pα |gλ is definedup to a multiplicative constant, the pencil is well-defined. Denote this pencilby dλ Π(x).Definition 16. The pencil dλ Π(x) will be called the λ-linearization of thepencil Π at the point x.2.4The non-degeneracy criterionDefinition 17. A pencil Π will be called diagonalizable at a point x, if foreach λ ∈ Λ(x) and any α 6= λ the following is truedim Ker Pα (x) |Ker Pλ (x) = corank Π(x).Remark 2.4.1.

In terms of the Jordan-Kronecker decomposition for the pencilΠ at a point x this means that all Jordan blocks Jki ,λi have size 1 × 1, i.e.all ki are equal to 1.Theorem 8 (The non-degeneracy criterion). Let O(α, x) be a symplectic leafof a bracket Pα passing through x. Assume that Pα is regular at x (or, whichis the same, O(α, x) is a symplectic leaf of maximal dimension). Then thesingular point x of the integrable system F |O(α,x) is non-degenerate if andonly if the following two conditions hold:261. Π is diagonalizable at point x.2. For each λ ∈ Λ(x) the linear pencil dλ Π(x) is non-degenerate.The proof of this theorem is given in Section 5.5.The following example shows that Theorem 8 is wrong if we add to thesystem F a Casimir function of a bracket, singular at x (recall that F, bydefinition, is generated by Casimir functions of regular brackets).Example 2.4.1 (A.V.Bolsinov).