On a family of Poisson brackets on gl(n) compatible with the Sklyanin bracket

Vladimir V. Sokolov; Dmitry V. Talalaev

Home/Archive/arXiv

arXiv · 2502.16925

On a family of Poisson brackets on gl(n) compatible with the Sklyanin bracket

Vladimir V. Sokolov·Dmitry V. Talalaev

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study a family of compatible quadratic Poisson brackets on gl(n), generalizing the Sklyanin one. For any of the brackets in the family, the argument shift determines the compatible linear bracket. The main interest for us is the use of the bi-Hamiltonian formalism for some pencils from this family, as a method for constructing involutive subalgebras for a linear bracket starting by the center of the quadratic bracket. We give some interesting examples of families of this type. We construct the centers of the corresponding quadratic brackets using the antidiagonal principal minors of the Lax matrix. Special attention should be paid to the condition of the log-canonicity of the brackets of these minors with all the generators of the Poisson algebra of the family under consideration. A similar property arises in the context of Poisson structures consistent with cluster transformations.

Explore related subjects

nlin.SI math-ph math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Vladimir V. Sokolov, Dmitry V. Talalaev. 2025-02-24. On a family of Poisson brackets on gl(n) compatible with the Sklyanin bracket. https://arxiv.org/abs/2502.16925

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the dispersionless limit of the Manakov system, its Riemann invariants, and the modulational stability of its counterpropagating plane waves

We study the dispersionless limit of the Manakov system, the integrable two-component generalization of the nonlinear Schrödinger equation. We derive the resulting four-component genus-zero Manakov-Whitham system, characterize its hydrodynamic structure, and show that it passes the Haantjes tensor test for integrability. We show that the branch points of the spectral curve associated with plane wave solutions of the Manakov system are the local Riemann invariants of the dispersionless system. We also use the characteristic speeds to classify the baseband modulational stability/instability of the plane waves, and we study a direct linearization of the Manakov system to characterize their finite-wavenumber stability and verify agreement with the Whitham prediction in the long-wave limit. Finally, we validate the predictions by comparing them with the results of direct numerical simulations.

nlin.SI↗

arXiv2026

Geometric, algebraic and analytic properties of $\mathrm{al}_{ab}$ function for hyperelliptic curves of genus $g$

In this paper, we investigate the geometric, algebraic and analytic properties of the hyperelliptic $\mathrm{al}_{ab}$ functions of a hyperelliptic curve $X$ with genus $g$ as the $\mathrm{al}_{ab}$ functions together with the $\mathrm{al}_a$ functions are a generalization of the Jacobi elliptic $\mathrm{sn}$, $\mathrm{cn}$, and $\mathrm{dn}$ functions. We then demonstrate the differential identities of the $\mathrm{al}_{ab}$ function. These identities are novel integrable partial nonlinear differential equations as an extension of the differential identities in terms of the $\mathrm{al}_a$ function known as the hyperelliptic solutions of the modified Korteweg-de Vries equation. Thus, we also show that by the identities, the $\mathrm{al}_{ab}$ function is useful for expressing hyperelliptic solutions to the nonlinear Schrödinger and complex modified Korteweg-de Vries equations in an explicit form as an extension of the elliptic $\mathrm{sn}$ function solutions.

nlin.SI↗

arXiv2026

Equations of state of hydrodynamic type and particle statistics of a Dyson gas in an analytic confining potential

We investigate the equilibrium thermodynamics of a Dyson gas in connection with a set of integrable statistical mechanical observables satisfying the Toda Lattice hierarchy. We prove that in the thermodynamic limit, the integrable observables are state functions satisfying a set algebraic equations of state in closed form, obtained from direct integration of the Toda Lattice hierarchy in the continuum limit. We then explore the connection between regularity and critical behaviour of the state functions and the Dyson gas particle statistics via Monte Carlo simulations. We show that the properties of the integrable observables, such as regularity, multivaluedness, cusp singularities, carry information on the macroscopic particle statistics and its qualitative changes but with some limitations.

nlin.SI↗