We construct the classical -matrix structure for the Lax formulation of Ruijsenaars–Schneider systems proposed in Commun. Math. Phys., 115, 127 (1988). The -matrix structure takes a quadratic form similar to the Ruijsenaars–Schneider Poisson bracket behavior, although the dynamical dependence is more complicated. Commuting Hamiltonians stemming from the Ruijsenaars–Schneider Lax matrix are shown to be linear combinations of particular Koornwinder–van Diejen “external fields” Ruijsenaars–Schneider models, for specific values of the exponential one-body couplings. Uniqueness of such commuting Hamiltonians is established once the first of them and the general analytic structure are given.
REFERENCES
1.
S. N. M.
Ruijsenaars
and H.
Schneider
, Ann. Phys. (N.Y.)
170
, 370
(1986
).2.
3.
4.
5.
6.
7.
E. K.
Sklyanin
, “On the complete integrability of the Landau–Lifchitz equation,” preprint E-3-79 LOMI, Leningrad (1979
); Funct. Anal. Appl.
16
, 263
(1982
).8.
9.
10.
11.
12.
V. G. Drinfel’d, Proc. I.C.M. (MSRI, Berkeley, 1986), p. 798.
13.
A. Yu.
Alekseev
and A. Z.
Malkin
, Commun. Math. Phys.
162
, 147
(1994
).14.
Calogero–Moser–Sutherland Models, CRM Series in Mathematical Physics, edited by J. F. Van Diejen and L. Vinet (Springer, Berlin, 2000), see in particular contributions by H. Awata, K. Hasegawa, and V. I. Inozemtsev
15.
16.
17.
18.
19.
J. F.
Van Diejen
, q-alg/9504012;20.
21.
22.
23.
24.
25.
26.
27.
H. W. Braden, in Ref. 15, pp. 77 sqq.
28.
G.
Felder
, hep-th/9407154, in Proceedings of the ICM 1994 (Birkhäuser, Berlin, 1994), pp. 1247–1255.29.
O.
Babelon
, D.
Bernard
, and E.
Billey
, Phys. Lett. B
375
, 89
(1996
).30.
A.
Antonov
, K.
Hasegawa
, and A.
Zabrodin
, Nucl. Phys. B
503
, 747
(1997
);31.
32.
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