Jordan–Kronecker invariants of Borel subalgebras of semisimple Lie algebras

Abstract

In the theory of bi-Hamiltonian systems, the generalized Mischenko–Fomenko conjecture is known. The conjecture states that there exists a complete set of polynomial functions in
involution with respect to a pair of naturally defined Poisson structures on a dual space of a Lie algebra. This conjecture is closely related to the argument shift method proposed by
A. S. Mishchenko and A. T. Fomenko in [10]. In research works devoted to this conjecture, a connection was found between the existence of a complete set in bi-involution and the algebraic
type of the pencil of compatible Poisson brackets, defined by a linear and constant bracket. The numbers that describe the algebraic type of the generic pencil of brackets on the dual space
to a Lie algebra are called the Jordan–Kronecker invariants of a Lie algebra. The notion of Jordan–Kronecker invariants was introduced by A. V. Bolsinov and P. Zhang in [2]. For some
classes of Lie algebras (for example, semisimple Lie algebras and Lie algebras of low dimension), the Jordan–Kronecker invariants have been computed, but in the general case the problem of
computation of the Jordan–Kronecker invariants for an arbitrary Lie algebra remains open. The problem of computation of the Jordan–Kronecker invariants is frequently mentioned among the
most interesting unsolved problems in the theory of integrable systems [4, 5, 6, 11]. In this paper, we compute the Jordan–Kronecker invariants for the series 𝐵𝑠𝑝(2𝑛) and construct complete sets of polynomials in bi-involution for each algebra of the series. Also, we calculate the Jordan–Kronecker invariants for the Borel subalgebras 𝐵𝑠𝑜(𝑛) for any 𝑛. Thus, together with the results obtained in [2] for 𝐵𝑠𝑙(𝑛), this paper presents a solution to the problem of computation of Jordan–Kronecker invariants for Borel subalgebras of classical Lie algebras.