On a canonical basis of a pair of compatible Poisson brackets on a symplectic Lie algebra

Abstract

Every reductive complex Lie algebra g is equipped with the canonical Poisson structure {φ,ψ}(x) = (x, [dxφ, dxψ]), where φ and ψ are smooth functions on g, while dxφ and dxψ are treated as elements of g, which is identified with g∗ using the invariant inner product. Moreover, for every a ∈ g, a Poisson structure ‘with frozen argument’ is defined: {φ,ψ}a(x) = (a, [dxφ, dxψ]). An approach described in [1] makes it possible to work with Poisson structures using the language of linear algebra. The Poisson brackets { · , · }a and { · , · } are regarded as skew-symmetric bilinear forms fa and fx over the field K = C(g) on the space g⊗K of rational vector fields on g, where the element a is fixed and x is a generic element. Namely, if φ and ψ are polynomials, then dφ and dψ can be regarded as elements of g⊗K, and then {φ,ψ}(x) = fx(dφ, dψ) and {φ,ψ}a(x) = fa(dφ, dψ). The above approach can be used to solve an important problem in Hamiltonian mechanics, namely, the search for complete families of functions in bi-involution, that is, maximal families of functions commuting with respect to both Poisson brackets. The polynomials φ1, . . . , φs define a complete family of functions in bi-involution with respect to { · , · }a and { · , · } if and only if their differentials dφ1, . . . , dφs form a basis of a bi-Lagrangian subspace (that is, a maximal subspace which is isotropic with respect to both bilinear forms). Thus, to obtain a complete family of functions in bi-involution, it suffices to find a basis for a bi-Lagrangian subspace and ‘integrate with respect to x’. If a basis is found (we call it canonical) in which the matrices of both forms fa and fx are reduced simultaneously to the canonical Jordan–Kronecker form (with blocks of two types, Jordan and Kronecker, see [1]), then a basis of the bi-Lagrangian subspace is formed by the second halves of the bases in each block. The second (‘larger’) halves of the bases in Kronecker blocks span a subspace L, which is the intersection of all bi-Lagrangian subspaces for the forms fa and fx. In [4] a basis of the subspace L and the corresponding functions in bi-involution were constructed for the Lie algebras gln and sp2n. In [3] the Kronecker part of a canonical basis and the corresponding part of the complete system of functions in bi-involution were constructed for the Lie algebra gln. Now we introduce our notation and formulate the result. Let {λ1, . . . , λs} be distinct eigenvalues of a matrix A ∈ gln. Assume that Jordan cells of order nk,1 ⩾ · · · ⩾ nk,ik correspond to an eigenvalue λk. We write