DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 581–592

THE BIHAMILTONIAN STRUCTURE OF

THE DRINFEL’D–SOKOLOV HIERARCHIES

RELATED TO THE AFFINE TWISTED LIE ALGEBRAS

PAOLO CASATI

Abstract. This paper presents a bihamiltonian construction of the Drinfel’d– Sokolov hierarchies corresponding to the twisted affine Lie algebras. We show, indeed, that these hierarchies can be framed in the general theory of bihamiltonian manifolds, by applying an extended bihamiltonian reduction theorem stated and proved in the paper.

1. Introduction

At least since the seminal paper of Drinfel’d and Sokolov [1] it is a well known fact, that each affine Lie algebra is associated with an integrable hierarchy of soliton equations, usually referred in the literature as hierarchy of KdV type or GD hierarchies [2] [3] [4], or finally (which seems the more appropriate name) Drinfel’d–Sokolov hierarchies [3], [4]. These hierarchies can be divided by their properties in two classes: the hierarchies corresponding to the non twisted affine Lie algebras, and those corresponding to the twisted ones. While the hierarchies of the first class share (up to some technicalities) all the properties usually owned by the integrable systems, the hierarchies corresponding to the twisted case seem to lack some theoretical features.

In particular, it is widely accepted, that the soliton hierarchies corresponding to the case of the twisted affine Lie algebras do not posses a bihamiltonian structure. This statement is, strictly speaking, obviously true, but it is certainly not the end of the story.

In this paper, it will be shown that actually, also these “exceptional” hierarchies can be seen as reduced theories of bihamiltonian theories defined on loop-algebras, by performing an appropriate bihamiltonian reduction process, which extends to this case the bihamiltonian reduction already developed in [5] [6] [7].

1991 Mathematics Subject Classification. 58F, 17B67.

Key words and phrases. Bihamiltonian manifolds, twisted affine Kac–Moody algebras.

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

This new reduction process, which contains the usual one presented in [6] and [7] as particular case, is based on a different geometrical construction. As a consequence of this fact one of the two Poisson tensors of the reduced bihamiltonian theory is simply the trivial structure, and not all the vector fields of the bihamiltonian hierarchy can be projected to the quotient space, but only those belonging to particular subclasses. Therefore on the level of the soliton phase space these theories appear to be only Hamiltonian and not bihamiltonian, although, they descend from bihamiltonian theories, from which they inherit all their remarkable properties, like the existence of an infinite number of commuting flows, or an infinite number of conserved quantities.

This paper is organized as follows. In the next section we will present our extended bihamiltonian reduction theorem. In the third one we will apply this theorem to the bihamiltonian manifold associated with the twisted affine Lie algebras. The last section, is devoted to study of the behavior of the twisted bihamiltonian hierarchies under the reduction process.

I whish to thank Prof. Magri, who suggested to study the bihamiltonian aspects of the twisted hierarchies and Marco Pedroni, Gregorio Falqui and Carlo Morosi for many useful discussions on this subject.

2. The Extended bihamiltonian reduction theorem

This section is devoted to state and prove the main result of the paper, an extended bihamiltonian reduction theorem, which allows a bihamiltonian interpretation of the twisted Drinfel’d–Sokolov hierarchies. Although we are mainly interested in these latter hierarchies, we will formulate the theorem in a more general fashion, which contains both reductions (i.e. for the untwisted and the twisted as well) as particular cases.

Let 𝔤 be a simple (finite–dimensional) Lie algebra, and let us denote by 𝔤̃ the loop–algebra C^{∞}(S^{1}, 𝔤). This algebra is an infinite dimensional flat manifold (see for example [10]), which, therefore can be identified with its tangent space at any point: 𝔤̃ ≃ T_{U} 𝔤̃, ∀U ∈ 𝔤̃. Moreover by means of the separating bilinear form:

(2.1)  ⟨V, U⟩ = ∫_{S^{1}}(V (x), U(x))_{𝔤}dx  U, V ∈ 𝔤̃ ,

induced on 𝔤̃ by the Killing form (·, ·)_{𝔤} of 𝔤, the cotangent spaces T_{U}^{∗} 𝔤̃, ∀U ∈ 𝔤̃ can be also identified with the tangent spaces T_{U} 𝔤̃. So we have the following chain of identifications:

(2.2)  𝔤̃ ≃ T_{U} 𝔤̃ ≃ T_{U}^{∗} 𝔤̃ ≃ 𝔤̃^{∗}  ∀U ∈ 𝔤̃ ,

which allows us to define on the manifold 𝔤̃ the two Poisson tensors:

(2.3)  (P_{0})_{U} V = [A, V]
(2.4)  (P_{1})_{U} V = V_{x} + [V, U]

BIHAMILTONIAN STRUCTURES ON TWISTED AFFINE … 583

where V ∈ T_{U}^{∗} 𝔤̃ ≃ T_{U} 𝔤̃ ≃ 𝔤̃,U ∈ 𝔤̃, A ∈ 𝔤 is a constant loop, and V_{x} is the derivative of the loop V along S^{1}. It is well–known that, for any choice of A, P_{0} and P_{1} are compatible (i.e. the Poisson pencil P_{0} − λP_{1} is still a Poisson tensor ∀λ ∈ ℝ), so that 𝔤̃ has several bihamiltonian structures (see, for instance, [5] [11]).

This means that the corresponding Poisson brackets {·, ·}_{0} and {·, ·}_{1} satisfy for any triple of functions (f, g, h) on M the cyclic compatibility condition [11]:

(2.5)  \begin{matrix}\{f, \{g, h\}_{0}\}_{1} & + \{h, \{f, g\}_{0}\}_{1} + \{g, \{h, f\}_{0}\}_{1}+ \\ + & \{f, \{g, h\}_{1}\}_{0} + \{h, \{f, g\}_{1}\}_{0} + \{g, \{h, f\}_{1}\}_{0} = 0 . \end{matrix}

Both theories of bihamiltonian reductions are based on the Marsden–Ratiu reduction theorem for a Poisson manifold [12]. This theorem asserts that if M is a Poisson manifold with Poisson tensor P and Poisson bracket {·, ·}; S is a submanifold of M; i_{S} : S ↪ M is the canonical immersion of S into M; and finally D is a distribution on M such that:

(1) E = D ∩ T S is an integrable distribution of S;

(2) the foliation induced by E on S is regular, so that N = S/E is a manifold and π : S → N is a submersion;

(3) the Poisson bracket {F, G}^{M} of functions F, G which are constant along D, is constant along D;

(4) P (D^{0}) ⊂ T S + D, where D^{0} is the annihilator of D; then N is a Poisson manifold with bracket {·, ·}^{N} given by

(2.6)  {f, g}^{N} ∘ π = {F, G}^{M} ∘ i_{S} ,

where F, G are extensions of f ∘ π, g ∘ π which are constant along D.

As for the usual bihamiltonian reduction theorem, the central idea of the extended one is to use the pair of compatible Poisson tensor in order to determine the geometry of the Marsden–Ratiu reduction, or more precisely, to choose the submanifold S and the distribution D. Moreover in our case we need an additional algebraic structure, which is the decomposition of the Lie algebra 𝔤:

(2.9)  𝔤 = 𝔤^{0} ⊕ 𝔤^{1}

where 𝔤^{0} is a subalgebra of 𝔤 such that 𝔤^{0} ∩ (𝔤^{0})^{⊥} = {0} (or in other words (𝔤^{0})^{⊥} = 𝔤^{1}, so that 𝔤^{0} is a reductive subalgebra of 𝔤). Observe that any simple Lie algebra 𝔤 admits at least the trivial decomposition: 𝔤^{0} = 𝔤 and 𝔤^{1} = {0}, and that from the ad–invariance of the Killing form follows immediately [𝔤^{0}, 𝔤^{1}] ⊂ 𝔤^{1}. The decomposition (2.9) can obviously extended to the whole loop–algebra 𝔤̃ by setting

(2.10)  𝔤̃_{i} = C^{∞}(S^{1}, 𝔤_{i})  i = 0, 1 .

We are now in the position to define the submanifold S and the distribution D. Let us start, by characterizing the symplectic foliation of 𝔤̃ induced by the tensor P_{0}.

In our case, P_{0} is a skew–symmetric tensor with respect to the bilinear separating form (2.2) on 𝔤̃ and the characteristic distribution of P_{0} is given by

(2.11)  𝔤̃_{A}^{⊥} = {v ∈ 𝔤̃|⟨V, U⟩ = 0 ∀U ∈ 𝔤̃_{A}}

where 𝔤̃_{A} is the kernel of the Poisson tensor P_{0}: 𝔤̃_{A} = Ker(ad A). This distribution 𝔤̃_{A}^{⊥}, (see [13]) is obviously integrable, and because 𝔤̃_{A}^{⊥} is a flat manifold its integral leaves are the affine submanifolds S′_{V} of 𝔤̃ modeled on 𝔤̃_{A}^{⊥} passing through any point of 𝔤̃. As submanifold S we take any submanifold of S′_{V}:

(2.12)  S = S′_{V} ∩ 𝔤̃^{0} = B + S′_{B} ∩ 𝔤̃^{0}

where B is any point of 𝔤̃^{0} so that S is the intersection of the symplectic leaf S′_{B} with the loop–subalgebra 𝔤̃^{0}.

It remains to define the distribution D such that condition (4) holds for both Poisson tensors. Since we have restricted the submanifold S and therefore its tangent bundle T S, our idea is to enlarge the distribution D. So let us define

(2.13)  D = P_{1}(𝔤̃_{A}) ∩ 𝔤̃_{0} ⊕ 𝔤̃^{1} .

Then the following new bihamiltonian reduction holds

Theorem 2.1. Let (M = 𝔤̃, P_{0}, P_{1}) be the bihamiltonian manifold defined above, where A belongs either to 𝔤̃^{0} or 𝔤̃^{1}, let S be any submanifold of the type (2.12), and let D be the distribution (2.13). Then

(1) E = D ∩ T S = P_{1}(𝔤̃_{A}) ∩ T S is an integrable distribution;

(2) Points (3) and (4) of the Marsden–Ratiu theorem hold for both Poisson tensors, so that, if N/E is a manifold, then N is a bihamiltonian manifold and the corresponding reduced Poisson brackets {·, ·}_{0}^{N}and λ{·, ·}_{1}^{N} are obtained from those of M by means of formula (2.6).

Proof. 1) E = D ∩ T S = P_{1}(𝔤̃_{A}) ∩ T S is an integrable distribution since it is the intersection of two integrable ones: P_{1}(𝔤̃_{A}) which is integrable because it is the characteristic distribution of the Poisson tensor P_{1} of the subalgebra 𝔤̃_{A}; and T S which is obviously integrable.

2) a) In order to prove that the functions constant on D build a Poisson algebra with respect both brackets {·, ·}_{0}, {·, ·}_{1} it is sufficient to prove that

(2.14)  L_{X}{F, G}_{i} = 0  ∀X ∈ D  i = 0, 1

for all F and G constant on D (i.e. such that dF, dG ∈ D^{0}), where L_{X} denote the Lie derivative along X. Let us decompose the vector field X as

X = X^{0} + X^{1} where X^{0} ∈ 𝔤̃^{0}, and X^{1} ∈ 𝔤̃^{1}

and let us prove (2.14) separately for X^{0} and X^{1}. First of all, 𝔤̃^{1} ⊂ D, implies that X^{1} ∈ D, and therefore, since X ∈ D also that X^{0} ∈ D. Moreover dF ∈ D^{0} ⊂ 𝔤̃^{0} implies

(2.15)  {F, G}_{i} = ⟨dF, (P_{i}dG)^{0}⟩  i = 0, 1

(where (P_{i}dG)^{0} denotes the component in 𝔤̃^{0} of (P_{i}dG)). Thus to prove that L_{X^{1}}{F, G}_{i} = 0 it is sufficient to show that

L_{X^{1}}⟨dF, Y ⟩ = 0  ∀dF ∈ D^{0}  ∀Y ∈ 𝔤̃^{0} .

But this is easily checked writing L_{X^{1}}⟨dF, Y ⟩ as (see [14])

L_{X^{1}}⟨dF, Y ⟩ = ⟨L_{X^{1}}(dF), Y ⟩ + ⟨dF, L_{X^{1}}(Y)⟩ ,

because dF ∈ D^{0} and X^{1} ∈ D imply that L_{X^{1}}(dF) = 0, and thus that L_{X^{1}}⟨dF, Y ⟩ = ⟨dF, L_{X^{1}}(Y)⟩, and finally

⟨dF, [X^{1}, Y]⟩ = 0

because [X^{1}, Y] ∈ 𝔤̃_{1}, whereas dF ∈ D^{0} ⊂ 𝔤̃^{0}.

It remains to prove that

L_{X^{0}}{F, G}_{i} = 0

but in this case X^{0} ∈ D ∩ 𝔤̃^{0} = P_{1}(Ker P_{0}) ∩ 𝔤̃^{0} and we can therefore write X^{0} as X^{0} = P_{1}dk where k is a Casimir of P_{0}, (recall that Ker P_{0} is integrable see [5]). Then

L_{P_{1}(dk)}{F, G}_{1} = {L_{P_{1}(dk)}(F), G}_{1} + {dF, L_{P_{1}(dk)}(G)}_{1} = 0

because L_{P_{1}(dk)}(F) = L_{P_{1}(dk)}(G) = 0. Similarly since L_{P_{0}(dk)} = 0,

L_{P_{1}(dk)}{F, G}_{0} = L_{P_{1}(dk)}{F, G}_{0} − {L_{P_{1}(dk)}(F), G}_{0} − {F, L_{P_{1}(dk)}(G)}_{0} =

=^{(2.5)} −L_{P_{0}(dk)}{F, G}_{1} + {L_{P_{0}(dk)}(F), G}_{1} + {F, L_{P_{0}(dk)}(G)}_{1} =

= 0 .

b) Let us finally check condition (4) of the Marsden Ratiu Theorem. Since A is either in 𝔤̃^{0} or in 𝔤̃^{1} we have that either P_{0}(X) ∈ 𝔤̃^{1} ⊂ D ∀X ∈ D^{0}, or P_{0}(X) ∈ 𝔤̃^{0} ∩ 𝔤̃_{A}^{⊥} ∀X ∈ D^{0}, and thus in both cases P_{0}(D^{0}) ⊂ D + T S. Further under our assumption on A any Y ∈ 𝔤̃_{A} can be decomposed as Y = Y^{0}+Y^{1} with Y^{0} ∈ 𝔤̃^{0} ∩ 𝔤̃_{A} and Y^{1} ∈ 𝔤̃^{1} ∩ 𝔤̃_{A}. Now, since on the points of S, P_{1}(𝔤̃_{i}) ⊂ 𝔤̃_{i}, we have that P_{1}(X) ∈ 𝔤̃^{0}, and thus

⟨P_{1}(X), Y ⟩ = ⟨P_{1}(X), Y^{0} + Y^{1}⟩ = ⟨P_{1}(X), Y^{0}⟩ = −⟨P_{1}(Y^{0}), X⟩ = 0

because P_{1}(X) ∈ P_{1}(Ker P_{0}) ∩ 𝔤̃_{A}. This equation implies finally that P_{1}(X) ∈ 𝔤̃_{A}^{⊥} ∩ 𝔤̃^{0} = T S, and therefore that condition (4) of Marsden Ratiu theorem holds in our case. □

Let us close this section by the remark that if the decomposition (2.9) of 𝔤 is the trivial one then Theorem 2.1 coincides with the usual bihamiltonian reduction theorem [5].

3. The “Twisted” Bihamiltonian Manifolds

The aim of this section is to apply our extended bihamiltonian reduction theorem to a particular case of bihamiltonian manifolds, which turns out to be strictly related to the twisted Drinfel’d Sokolov hierarchies.

These bihamiltonian manifolds, defined on loop–algebras, are constructed by making a opportune choice of the gradation 𝔤 = 𝔤^{0} ⊕ 𝔤^{1} (2.9), and of the elements A and B considered in the previous section. These choices will depend heavily on the algebraic data carried by an associated twisted affine Lie algebra. Therefore, we need some results of the theory of the twisted affine Lie algebras, which for the convenience of the reader are briefly summarized in the next subsection. A complete and well made exposition of these algebras can be found in the well known book by Kac ([8] chapter VIII).

The twisted Kac–Moody affine Lie algebras.

Let 𝔤̂ be an affine (twisted) Lie algebra, whose Dynkin diagram is denoted by X_{n}^{(k)}, let 𝔤 be the (finite) simple Lie algebra associated with the Dynkin diagram X_{n}, and let us finally denote by {E_{1}, … E_{n}, F_{1}, … F_{n}} a set of Chevalley generators of 𝔤, then the following holds

Theorem 3.1. Let us label with (0, 1, …, n) the dots of the Dynkin diagram X_{n}^{(k)} and let s = (s_{0}, s_{1}, …, s_{n}) be a corresponding sequence of non negative relatively prime integers, define m = k ∑_{i=0}^{n} a_{i}s_{i} (where a_{i} is the Kac label of X_{n}^{(k)} associated with the dot i see [8] and [9]), then

(1) the relations

(3.1)  \begin{aligned}σ_{s,k}(E_{j}) & = e^{2πs_{j}/m}E_{j} \\ σ_{s,k}(F_{j}) & = e^{−2πs_{j}/m}F_{j}  j = 1, 2, … n \end{aligned}

define a m–th automorphism σ_{s,k} of 𝔤 (i.e. σ^{m} = 1, σ^{q} ≠ 1 if q ≠ 0 mod m);

(2) up to conjugation by an automorphism of 𝔤, the automorphisms σ_{s,k} exhaust all m–the order automorphism of 𝔤;

(3) the automorphisms σ_{s,k} and σ_{s′,k′} are conjugate by an automorphism of 𝔤 if and only if k = k′ and the sequence s can be transformed into the sequence s′ by an automorphism of the diagram X_{n}^{(k)}.

The Lie twisted affine algebra 𝔤̂ can be constructed starting from the Lie algebra 𝔤, by means of this Theorem of 𝔤 as follows. An automorphism σ_{s,k} of order m of 𝔤 induces on 𝔤 a ℤ/mℤ–gradation:

(3.2)  𝔤 = ⨁_{j∈ℤ/mℤ} 𝔤_{j}

where 𝔤_{j} is the eigenspace of σ_{s,k} corresponding to the eigenvalue e^{\frac{2πi}{m}}.

One elementary properties of the gradation (3.2), which will be useful in the following, is that (𝔤_{i}, 𝔤_{j})_{𝔤} = 0 if i+j ≠ 0 mod m, and 𝔤_{i} and 𝔤_{j} are nondegenerate paired if i + j = 0 mod m.

With the automorphism σ of 𝔤, we associate the subalgebra L_{λ}(𝔤, σ_{s,k}) of the loop algebra L_{λ}(𝔤):

(3.3)  L_{Λ}(𝔤, σ_{s,k}) = ⨁_{j∈ℤ} 𝔤_{j mod m} ⊗ λ^{j} ,

which is the subalgebra of L_{Λ}(𝔤) formed by the fixed point of the automorphism σ̃_{s,k} of L_{λ}(𝔤) extending to the whole loop algebra L_{λ}(𝔤) the automorphism σ_{s,k} as follows:

(3.4)  σ̃_{s,k}(X ⊗ λ^{j}) = σ_{s,k}(X) ⊗ ϵ^{−j}λ^{j}  ∀j ∈ ℤ X ∈ 𝔤 .

Then it can be proved (see for example [8] for the details) that the Lie algebra 𝔤̂ is isomorph to the algebra

(3.5)  𝔤̂ ≃ L̂_{λ}(𝔤, σ_{s,k}) = L_{λ}(𝔤, σ_{s,k}) ⊕ ℂc ⊕ ℂd ,

where c and d are respectively the central extensions and the derivation of L_{λ}(𝔤, σ_{s,k}) defined in [8]. Therefore if X_{i} = X ⊗λ^{i} (with obviously X ∈ 𝔤_{i mod m}) is a typical element of L̂_{λ}(𝔤, σ)_{s,k}) then the algebra–multiplication of 𝔤̂ may be written as

(3.6)  \begin{aligned}[X_{i}, Y_{j}] & = [X, Y]_{i+j} + (X, Y)_{𝔤}cδ_{i+j,0} \\ [d, X_{i}] & = iX_{i} \\ [d, d] & = [c, d] = [c, c] = [c, X_{n}] = 0 . \end{aligned}

The ℤ/mℤ–gradation associated with σ_{s,k} can be obviously extended to a ℤ– gradation of the whole affine Lie L̂_{λ}(𝔤, σ_{s,k}) setting:

(3.7)  L̂_{λ}(𝔤, σ_{s,k}) = ⨁_{j∈ℤ} L̂_{λ}(𝔤, σ_{s,k})_{j}

where L̂_{λ}(𝔤, σ_{s,k})_{j} = 𝔤_{j mod m} ⊗ λ^{j} and L̂_{λ}(𝔤, σ_{s,k})_{0} = 𝔤_{0} ⊗ λ^{j} ⊗ c ⊗ d.

This gradation can be also introduced from another point of view, perhaps more useful for us, as follows.

Let {E_{0}, … E_{n}, F_{0}, … F_{n}} a set of Chevalley generators of 𝔤̂ = L̂_{λ}(𝔤, σ_{s,k}) (which can be constructed starting from a set of Chevalley generators of 𝔤 (see [8] chapt. VII and VIII) then the gradation corresponding to the automorphism σ_{s,k}, with s = (s_{0}, s_{1}, …, s_{n}) is given by the formula:

(3.8)  deg(E_{i}) = s_{i}  deg(F_{i}) = −s_{j}  i = 0, 1, … n .

Among them only the following two will be considered in the paper:

(1) the principal gradation, for which s_{0} = s_{1} = ⋯ = s_{n} = 1;

(2) the standard gradations (also called “minimal gradation” see [3]), for which, for some j, s_{j} = 1, while s_{i} = 0 if i ≠ j.

The above collected facts on the twisted affine Lie algebras allow us to define a bihamiltonian manifold, which satisfies the hypothesis of Theorem 2.1. This amounts to describe the choice of the decomposition (2.9), and of the element A and B depends on the twisted affine Lie algebra 𝔤̂.

Let 𝔤 be a simple Lie algebra, whose Dynkin diagram X_{n} can be extended to a twisted affine Dynkin diagram X_{n}^{(k)} (k = 2, 3), (thus 𝔤 is of the type A_{n}, D_{n}, E_{6}, because the only twisted affine Lie algebra are A_{n}^{(2)}, D_{n}^{(2)}, E_{6}^{(2)}, D_{4}^{(3)}).

We have seen in the previous subsection, how to realize explicitly the affine Lie algebra 𝔤̂ associated to the Dynkin diagram X_{n}^{(k)} starting from the Lie algebra 𝔤 as

(3.9)  𝔤̂ = L_{λ}(𝔤, σ_{s,k})

where L_{λ}(𝔤, σ_{s,k}) is the subloop–algebra (3.5), and σ_{s,k} is supposed here and in what follows to be a standard automorphism of 𝔤, which will be simply denoted by σ. The automorphism σ induces a ℤ/mℤ–gradation (3.2) of 𝔤 and of 𝔤̃ as well (where m is the finite order of σ, defined in the Theorem 3.1 and 𝔤̃_{i mod m} = C^{∞}(S^{1}, 𝔤_{i mod m})). Thanks to the elementary properties recalled above, this gradation allows us to define a nontrivial decomposition (2.9) of 𝔤 by setting

(3.10)  𝔤^{0} = 𝔤_{0 mod m}  𝔤^{1} = ⨁_{\begin{matrix}i∈ℤ/mℤ \\ i≠0 mod m \end{matrix}} 𝔤_{i} .

Once the decomposition (2.9) is given, it remains only to define the elements A and B. This is made as follows.

With respect to the chosen standard gradation we can find a Chevalley basis of 𝔤̂: {E_{0}, E_{1}, …, E_{n}}, {F_{0}, F_{1}, …, F_{n}} such that deg(E_{j}) = s_{j} and deg(F_{j}) = −s_{j}, where, being σ a standard automorphism, s_{j} = δ_{ij} for some i, (j = 0, 1, …, n). Beside the standard gradation of 𝔤̂, we can endowed 𝔤̂ by the principal gradation obtained by putting:

(3.11)  deg(E_{i}) = 1  deg(F_{i}) = −1  ∀i = 0, 1, …, n .

Then the element Λ defined by

(3.12)  Λ = ∑_{i=0}^{n} E_{i}

is an element of degree 1 with respect to the principal gradation, while with respect to the standard gradation Λ can be written as the sum of two homogeneous elements Λ_{0}, Λ_{1} of degree 0 and 1 respectively. The properties of the element Λ, which play a crucial role in our construction, are collected in the following Proposition 3.3. The element Λ of 𝔤̂ (where 𝔤̂ is of the type A_{n}^{(2)}, D_{n}^{(2)}, E_{6}^{(2)}, D_{4}^{(3)}) defined by (3.12) is a regular element ( i.e., (see [1], [3]), Ker(ad Λ) is a Heisenberg subalgebra of 𝔤̂, or in other words its restriction Ker (ad Λ|_{𝔤}) to 𝔤 is a Cartan subalgebra). It has degree 1 with respect to the principal gradation of 𝔤̂. Its decomposition in the standard gradation used to realized explicitly 𝔤̂ from 𝔤, is

(3.13)  Λ = Λ_{0} + Λ_{1}

with Λ_{0} in 𝔤, whereas Λ_{1} can be written in the form

(3.14)  Λ_{1} = A ⊗ λ

where A ∈ 𝔤. A and Λ_{1} are elements of 𝔤 of degree 1 and 0 respectively, in the ℤ/mℤ–gradation of 𝔤 (3.2) induced from the standard automorphism σ.

Proof. We need only to prove the last assertion, but this follows easily from the properties of the standard gradation of 𝔤̂ and the fact proved by Kac [8] that it is always possible to find a Chevalley basis with the properties described in the paragraph before the proposition. Indeed by definition Λ_{0} ∈ 𝔤̂_{0} ≃ 𝔤_{0} and Λ_{1} ∈ 𝔤̂_{1} ≃ 𝔤_{1} ⊗ λ. □

The above stated proposition allows us to make explicitly our choice of A and B in such a way that all the hypothesis of the Theorem 2.1 are satisfied. We namely choose A as defined by the formula (3.14) and put B = Λ_{0}. We can therefore apply the general bihamiltonian reduction Theorem 2.1, in our particular case obtaining a reduced “bihamiltonian” manifold N, which coincides (see [7]) with the phase space of the soliton hierarchies associated with the twisted affine Lie algebras.

Unfortunately the choice made of the element A implies P_{0}(𝔤̃_{i}) ⊂ 𝔤̃_{i+1} i = 0, 1 mod 2 and then from equation (2.8) follows immediately that the reduced Poisson tensor P_{0}^{N} is identically equal to zero. Therefore the bihamiltonian reduction theorem for the twisted case is not so powerful as in the non twisted case. On the phase space level the bihamiltonian structure virtually disappears. The soliton hierarchies of the twisted case still, however, remain the “shadows” of bihamiltonian hierarchies defined on the loop-algebra level. Moreover each remarkable properties of the twisted soliton hierarchies can be recovered by the above described reduction process from the properties usually owned by the bihamiltonian hierarchies. In this paper we limit ourself to study the reduction process of the bihamiltonian hierarchy.

4. The twisted reduced hierarchy

In this section we want study what happens to a twisted bihamiltonian hierarchy defined at the level of the loop–algebra 𝔤̃, when one tries to reduced it on the manifold N. Unfortunately, since one of the Poisson tensors, namely P_{0}, is reduced to the trivial one, we may not expect that the whole bihamiltonian hierarchies can be recovered on the reduced space, as happens in the non–twisted case.

A bihamiltonian hierarchy on a bihamiltonian manifold (M, P_{0}, P_{1}) is a sequence of functions {H_{k}}_{k≥0} verifying the Lenard’s recursion relation

(4.1) P_{0}(dH_{k+1}) = P_{1}(H_{k}) or {F, H_{k+1}}_{0} = {F, H_{k}}_{1}  ∀F ∈ C^{∞}(M, ℝ)

together with P_{0}(dH_{0}) = 0 or {F, H_{0}}_{0} = 0 ∀F ∈ C^{∞}(M, ℝ). The main property of these hierarchies is that the functions H_{k} are in involution with respect to both Poisson brackets [5]:

(4.2)  {H_{j}, H_{k}}_{1} = {H_{j}, H_{k}}_{0} = 0 .

A bihamiltonian hierarchy gives rise to a hierarchy of bihamiltonian vector fields, pairwise commuting, defined by the relations X_{k} = P_{1}dH_{k} = P_{0}dH_{k+1}. This vector fields generate (if independent) the whole hierarchy of partial differential equations of the integrable system associated with the bihamiltonian manifold (M, P_{0}, P_{1}).

These remarks are the basis of the bihamiltonian approach to integrable systems.

Unfortunately the problem to construct a bihamiltonian hierarchy for a given bihamiltonian variety is perhaps the hardest one of the whole theory, and it is not yet solved in the most general case. Therefore we do not tackle it in this paper preferring to refer to the papers [1], [3] and [16], where this problem is discussed and solved in the case that the bihamiltonian variety is a loop–algebra 𝔤̃ = C^{∞}(S^{1}, 𝔤) with 𝔤 simple, and the elements A and B defined in Sect. 2 are such that A+B is a regular element (as it happens in our case, as stated by Proposition 3.3). In this important case one has to find first a Casimir of the Poisson pencil P_{λ} = P_{1} − λP_{0} i.e., a 1–form V (λ) = ∑_{k=−j}^{∞} V_{k}λ^{−k} which solves the stationary equation:

(4.3)  P_{λ}(V (λ)) = 0 ,

and then showing that V (λ) is exact 1–form, so that a Laurent series H(λ) = ∑_{k=−j}^{∞} H_{k}λ^{−k} exists such that dH(λ) = V (λ). Then the coefficients H_{k} of the Laurent series H(λ) satisfy the Lenard recursion relations (4.1). (Here we have changed the enumeration of the bihamiltonian hierarchies in such a way that it matches with that of the powers of λ.) In the particular case studied in this work the results of those papers can be reformulated stating, that there exist on S exact Casimirs V (λ) = ∑_{k=−j}^{∞} V_{k}λ^{−k} of the pencil P_{λ} which belong to the algebra 𝔤̂ so that V_{k} ∈ 𝔤̃_{k} ∀k = −j, −j + 1, −j + 2, . . .. Since the Casimir V (λ) is exact, there exists a corresponding Laurent series of functions H(λ) = ∑_{k=−j}^{∞} H_{k}λ^{−k} with dH(λ) = V (λ). To each of these Casimirs V (λ) corresponds a infinite hierarchy of vector fields defined by X_{k} = P_{1}dV_{k} = P_{0}dV_{k+1} k = −j, −j + 1, . . .. The vector fields of these hierarchies which can be reduced on the quotient submanifold N form the integrable hierarchy of the solitonic theory. These last vector fields are selected by the algebraic data encoded in the twisted affine Lie algebra 𝔤̂. It holds, indeed Theorem 4.1. Let V = ∑_{k=−j}^{∞} V_{k}λ^{−k} be a exact Casimir of the stationary equation (4.3) belonging to 𝔤̂, and let X_{k} = P_{1}V_{k} = P_{0}V_{k+1} the corresponding hierarchy of vector fields. Then only the vector fields X_{k} with k = 0 mod m are reducible on N.

Proof. Obviously the vector fields X_{k} with k ≠ 0 mod m cannot be reduced on the quotient space N, because they are not even tangent to the leave S, belonging to ⨁_{k=1}^{m−1} 𝔤̃_{k} ≃ 𝔤̃^{1} while T S ∈ 𝔤̃_{0} ≃ 𝔤̃^{0}.

The vector fields on S: X_{k} = P_{0}(V_{k+1}) = P_{1}(V_{k}) with k = 0 mod m are reducible on N. In order to be reducible the vector fields X_{k} have to be on S tangent to the leave S, and projectable with respect to the projection π : S → N. The first condition is immediately satisfied. Indeed, each X_{k} belongs to the image of P_{0} by the relation X_{k} = P_{0}V_{k+1}, but being X_{k} bihamiltonian k = −j, −j+1, …, we can write it also as X_{k} = P_{1}(V_{k}), so that X_{k} ∈ 𝔤̃^{0} if k = 0 mod m, because V_{k} ∈ 𝔤̃_{0}, and moreover P_{1} has degree 0 on S as observed in the proof of Theorem 2.1, (part (2) b).

In order to prove the second one, we have to show [11] that if k = 0 mod m then

(4.4)  [X_{k}, Y] ∈ E  ∀Y ∈ E .

From the definition of E follows that its elements have the form Y = P_{1}(η) with η ∈ Ker(P_{0}) and therefore we get [11] [16] that

[X_{k}, Y] = [P_{1}(V_{k}), P_{1}(η)] = P_{1}{V_{k}, η}_{1} .

Thus, since [X_{k}, Y] belongs to T S being S a submanifold, it remains only to prove that {V_{k}, η}_{1} is in Ker(P_{0}). But this follows immediately from the compatibly condition [11]

{V_{k}, η}_{1} = {V_{k+1}, η}_{0} .

Indeed

P_{0}({V_{k}, η}_{1}) = P_{0}({V_{k+1}, η}_{0}) = [P_{0}(V_{k}), P_{0}(η)] = 0 ,

because η ∈ Ker(P_{0}). □

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Dipartimento di Matematica, Università di Milano Via C. Saldini 50, I-20133 Milano, ITALY E-mail: Casati@ares.mat.unimi.it