Relativistic Interacting Integrable Elliptic Tops

Abstract

We propose relativistic generalization of integrable systems describing M interacting elliptic gl(N ) tops of the Euler-Arnold type. The obtained models are elliptic integrable systems, which reproduce the spin elliptic GL(M ) Ruijsenaars-Schneider model for N = 1 case, while in the M = 1 case they turn into relativistic integrable GL(N ) elliptic tops. The Lax pairs with spectral parameter on elliptic curve are constructed. M k:k =iand for its non-diagonal part (1.4) yieldṡ( 1.7) These equations can be viewed as relativistic deformation [16] (the deformation parameter is η ∈ C) of the spin elliptic Calogero-Moser model [3]:(1.8)The system (1.8) admits anisotropic gl(NM) generalization, where the spin variables S ij are replaced by matrix-valued variables S ij ∈ Mat(N, C). Equations of motion are of the form:(1.9)2 Spin Ruijsenaars-Schneider modelIn this Section we derive equations of motion. It will help to simplify the proof of the more complicated statement related to interacting tops. More precisely, we prove the following Proposition 2.1 Equations of motioṅ

Integrable system of generalized relativistic interacting tops

“…In the elliptic case the above given Lax pairs and equations of motion are reproduce 1 the results of our previous paper [17], and in the non-relativistic limit the results of [6] are reproduced as well. For N = 1 the R-matrix operators under consideration become the scalar functions from (1.1), thus reproducing the spin Ruijsenaars-Schneider model (1.21)-(1.27).…”

“…which turns into zero on-shell constraints [7], the detailed derivation of (1.26) can be also found in [17]. This derivation is convenient for consideration of a more general system, where the functions entering (1.21)-(1.25) are replaced by their R-matrix analogues.…”

“…This paper is a continuation of series of articles [6,14,17], where known integrable systems and related structures are extended through the use of quantum R-matrices (in the fundamental representation of GL(N) Lie groups) being interpreted as matrix generalizations of the Kronecker function. It is convenient to give its explicit form the very beginning in the rational, trigonometric and elliptic cases since the identities to be used are valid in all these cases: All the functions are complex valued.…”

“…The definitions and properties of elliptic functions can be found in [15] (see also Appendix from [17]).…”

Trigonometric Real Form of the Spin RS Model of Krichever and Zabrodin

“…The unbroken interest in integrable many-body systems of Calogero-Moser-Sutherland [8,38,54] and Ruijsenaars-Schneider (abbreviated RS) [50] types is due to their ubiquity in physical applications and rich web of connections to important areas of mathematics [3,12,40,49,55]. The same can be said about spin extensions of these models, which currently attract attention [5,6,10,[14][15][16]28,33,42,[45][46][47]51,57].…”