Lecture 2:
Classical R-Matrix theory on loop algebras and groups
Classical Yang--Baxter equations, commuting invariants and Lax equations.
Split $R$-matrices and loop algebras.
Loop groups and quadratic Poisson brackets.
(refs. (1), (2), (3), (6))
Lecture 3 & 4:
Separation of variables in Spectral Darboux coordinates.
Rational coadjoint orbits in $\lgl^*(r)_R$.
Construction of abelian differentials.
Spectral Darboux coordinates.
Liouville--Arnold integration and abel map linearization.
(refs. (4), (5), (7))
Lecture 5:
Introduction to separation of variables in quantum
integrable systems.
Quantized Moser systems and separation of variables in $\lgl^*(r)_R$
(refs. (3), (5), (8), (9))
Classical:
1) Faddeev, L.D. and Takhtajan, L.A. Hamiltonian Methods in the
Theory of Solitons, Springer-Verlag, Berlin (1986).
2) Olshanetsky, M.M., Perelomov, A.M., Reyman, A.G. and
Semenov-Tian-Shansky, M.A., Integrable Systems II.
Encyclopaedia of Mathematical Sciences Vol. 16,
Springer-Verlag, Berlin, Heidelberg, New York (1994).
Classical and Quantum:
3) Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. Quantum inverse
scattering method and correlation functions. Cambridge Monographs on
Mathematical Physics. Cambridge University Press, Cambridge, 1993.
Chapts. V-VII.
II. Lecture Notes:
Classical:
4) Harnad, J., "Isospectral Flow and Liouville-Arnold Integration in
Loop Algebras", in: Geometric and Quantum Methods in Integrable
Systems, Springer Lecture Notes in Physics 424, Pgs 1-42. (ed.
G. Helminck, Springer-Verlag, N.Y., Heidelberg, (1993)).
Quantum:
5) Sklyanin, Evgueni K. Separation of variables - new trends. Quantum
field theory, integrable models and beyond (Kyoto, 1994). Progr.
Theoret. Phys. Suppl. No. 118, 35-60 (1995).
III. Research Papers:
6) Adams, M.R., Harnad, J. and Hurtubise, J.,
"Dual Moment Maps to Loop Algebras", Lett. Math. Phys. 20,
294-308 (1990).
7) Adams, M.R., Harnad, J. and Hurtubise, J., "Darboux
Coordinates and Liouville-Arnold Integration in Loop Algebras",
Commun. Math. Phys. 155, 385-413 (1993).
8) Harnad, J. and Winternitz, P., Classical and Quantum Integrable
Systems in Lgl(2)^{+*} and Separation of Variables,
Commun. Math. Phys. 192, 263-285 (1995).
9) Harnad, J. and Winternitz, P., Harmonics on Hyperspheres,
Separation of Variables and the Bethe Ansatz,
Lett. Math. Phys. 33, 61-74 (1995).