Painlevé VI, Rigid Tops and Reflection Equation

Abstract

We show that the Painlevé VI equation has an equivalent form of the non-autonomous Zhukovsky-Volterra gyrostat. This system is a generalization of the Euler top in C 3 and include the additional constant gyrostat momentum. The quantization of its autonomous version is achieved by the reflection equation. The corresponding quadratic algebra generalizes the Sklyanin algebra. As by product we define integrable XYZ spin chain on a finite lattice with new boundary conditions. June 4, 2018 PVI was discovered by B.Gambier [10] in 1906. He accomplished the Painlevé classification program of the second order differential equations whose solutions have not movable critical points. This equation and its degenerations P V − P I have a lot of applications in Theoretical and Mathematical Physics. (see, for example [41]).We prove here that PVI can be write down in a very simple form as ODE with a quadratic non-linearity. It is a non-autonomous version of the SL(2, C) Zhukovsky-Volterra gyrostat (ZVG) [42,39]. The ZVG generalizes the standard Euler top in the space C 3 by adding an external constant rotator momentum. The ZVG equation describes the evolution of the momentum vector S = (S 1 , S 2 , S 3 ) lying on a SL(2, C) coadjoint orbit. We consider Non-Autonomous Zhukovsky-Volterra gyrostat (NAZVG)where J(τ ) · S = (J 1 S 1 , J 2 S 2 , J 3 S 3 ). Three additional constants ν ′ = (ν ′ 1 , ν ′ 2 , ν ′ 3 ) form the gyrostat momentum vector, and the vector J = {J α (τ )}, (α = 1, 2, 3) is the inverse inertia vector CRDF RM1-2545. The work of A.Z. was also partially supported by the grant MK-2059MK- .2005.2. We are grateful for hospitality to the Max Planck Institute of Mathematics at Bonn where two of us (A.L. and M.O.) were staying during preparation of this paper. We would like to thank a referee for valuable remarks, that allows us to improve the paper.

Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices

“…Its applications to classical integrable systems, 1+1 models and monodromy preserving equations can be found in [16,17]. Consider the following matrix g ∈ Mat(N, C):…”

Classification of isomonodromy problems on elliptic curves

“…Механическая мо-дель описывается гамильтонианом 3. Неавтономный гиростат Жуковского-Вольтерра был определен в [64]. Там же была доказана его эквивалентность уравнению Пенлеве VI.…”

“…2) неавтономный аналог гиростата Жуковского-Вольтерра [64]: 12) где S обозначает sl * (2, C)-значную динамическую переменную, а J и ν ′ -неди-намические, но τ -зависимые. Линейные (2 × 2)-задачи для (1.11) и (1.12) бы-ли описаны в [65] и [64] соответственно.…”

“…Линейные (2 × 2)-задачи для (1.11) и (1.12) бы-ли описаны в [65] и [64] соответственно. Подробности изложены в разделе 8.…”

Quadratic algebras related to elliptic curves

“…The proof is based on direct calculation of the left-and right-hand sides of (4.27) (see [9] for the details). If all the ν α = 0 in relations (4.28)-(4.30), then the associative algebra coincides with the Sklyanin algebra.…”