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Lax operator algebras and Hamiltonian integrable hierarchies. (English. Russian original) Zbl 1246.17029
Russ. Math. Surv. 66, No. 1, 145-171 (2011); translation from Usp. Mat. Nauk. 66, No. 1, 151-178 (2011).
Author’s summary: This paper considers the theory of Lax equations with a spectral parameter on a Riemann surface, proposed by Krichever in 2001. The approach here is based on new objects, the Lax operator algebras, taking into consideration an arbitrary complex simple or reductive classical Lie algebra. For every Lax operator, regarded as a map sending a point of the cotangent bundle on the space of extended Tyurin data to an element of the corresponding Lax operator algebra, a hierarchy of mutually commuting flows given by the Lax equations is constructed, and it is proved that they are Hamiltonian with respect to the Krichever-Phong symplectic structure. The corresponding Hamiltonians give integrable finite-dimensional Hitchin-type systems. For example, elliptic $$A_n$$, $$C_n$$, and $$D_n$$ Calogero-Moser systems are derived in the framework of our approach.
##### MSC:
 17B66 Lie algebras of vector fields and related (super) algebras 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 14H10 Families, moduli of curves (algebraic) 14H15 Families, moduli of curves (analytic) 14H55 Riemann surfaces; Weierstrass points; gap sequences 30F30 Differentials on Riemann surfaces 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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