Algebraic Nijenhuis operators and Kronecker Poisson pencils

This paper is devoted to a method of constructing completely integrable systems based on the micro-local theory of bihamiltonian structures [GZ89, GZ91, Bol91, GZ93, GZ00, Pan00, Zak01]. The main tool are the so-called microKronecker bihamiltonian structures [Zak01], which will be called Kronecker in this paper for short (in [GZ00] the term Kronecker was used for the micro-Kronecker structures with some additional condition of ”flatness” which will not be essential in this paper). A Kronecker bihamiltonian structure on a manifold M is a Poisson pencil {s1θ1 + s2θ2}(s1,s2)∈K2 , i. e. a twodimensional linear space over a base field K in the set of all Poisson structures on M , satisfying an additional condition of the constancy of rank: rankC θ s = const, s := (s1, s2) ∈ C 2 \ (0, 0), θ := s1θ1 + s2θ2 (in the real case we should pass to the complexification of the pencil). The kroneckerity condition is important due to the fact that it automatically implies the existence (at least locally) on M of the complete involutive with respect to any bivector θ set of functions. This set is functionally generated by the Casimir functions of the bivectors θ (see Proposition 2.4). Geometrically this set corresponds to the intersection over s ∈ K \ {(0, 0)} of all symplectic leaves of maximal dimension of the Poisson structures θ and the completeness of this set reflects the fact that this intersection is lagrangian in any fixed symplectic leaf (see [GZ91, Pan00]). The main result of this paper (Theorem 2.5) gives a criterion of kroneckerity for the Poisson pencils related to diagonalizable algebraic Nijenhuis operators. An algebraic Nijenhuis operator N on a Lie algebra g (see [KSM90, CGM01], for example) is a linear operator N : g → g with the condition of the vanishing of the so-called Nijenhuis torsion (see Definition 1.1). Given a linear operator N : (g, [, ]) → (g, [, ]), the condition of vanishing of its Nijenhuis torsion guarantees that the infinithesimal part [, ]N of the trivial deformation (Id+λN) [(Id+λN)·, (Id+λN)·] of the Lie bracket [, ] is again a Lie bracket. This new Lie bracket [, ]N is automatically compatible with [, ], thus any Nijenhuis operator N ”produces” the pencil of Lie brackets [, ] := s1[, ] + s2[, ]N and, consequently, the corresponding pencil {θ N}s∈K2 of the Lie-Poisson structures on g . Let us look more closely at the problem of the kroneckerity of the Poisson pencil {θ N}. It can be shown (Proposition 1.2) that if N is Nijenhuis, (N − λ Id)[(N − λ Id)·, (N − λ Id)·] = [·, ·]N − λ[·, ·]. In particular, all the Lie brackets [, ] are isomorphic to [, ] except those corresponding to s = (s1, s2) with λ = −s1/s2 belonging to SpN , the spectrum of N . Thus the problem of kroneckerity of {θ N} (modulo some not very restrictive assumption on the codimension of the set of singular coadjoint orbits of g, see (2.5.1)) reduces to the problem of calculating the dimension of the coadjoint orbits of the exceptional brackets [, ]N − λi[, ], i = 1, . . . , n, where λ1, . . . , λn are the eigenvalues of N . In fact, due to the semicontinuity of the function rank θ, in order to prove the kroneckerity it is sufficient to find for any i a particular coadjoint orbit Oi of a Lie bracket [, ]N − λi[, ] such that dimOi = dimO, where O is the generic coadjoint orbit of [, ]. One possibility of finding the orbits Oi is the following. If N is a Nijenhuis operator, then N : (g, [, ]N ) → (g, [, ]) is a homomorphism of Lie algebras [KSM90]. Hence imN is a subalgebra of (g, [, ]) and we have a Poisson inclusion N : ((imN), θst) →֒ (g , θN ), where θst is the standard Lie–Poisson structure on (imN) ∗ and θN corresponds to [, ]N . In particular, one can take Oi to be a symplectic leaf in ([im(N − λi Id)] , θst) ⊂ (g , θ(N−λi Id)) (the operator N − λi Id is also Nijenhuis). Choosing Oi to be a generic coadjoint orbit and passing to codimensions we get the following sufficient condition of kroneckerity: if ind im(N − λi Id) + codim im(N − λi Id) = ind g for any i, where ind stands for the index of a Lie algebra, i. e. the codimension of a generic coadjoint orbit, then the Poisson pencil {θ N} is Kronecker (cf. Corollary 2.6). In general, however, this condition is not necessary because it may happen that the generic coadjoint orbits in (imN) are not generic in (g, θN ). For example, take g = sl(2) = n− ⊕ b+, where b+ is the upper Borel subalgebra and n− is the lower nilpotent subalgebra. Let N = Pn− be the projector to the first summand along the second one. Then coadjoint orbits of imN are points, whereas the algebra (g, [, ]N ) is nonabelian and has also coadjoint orbits of dimension 2. So our main theorem generalizes the above mentioned sufficient condition and gives necessary and sufficient conditions of the kroneckerity of the pencil {θ N} (for a diagonalizable N). The method of proof of this result consists