Special Quasigraded Lie Algebras and Integrable Hamiltonian Systems

Abstract

We construct a family of special quasigraded Lie algebras g F of functions of one complex variables with values in finite-dimensional Lie algebra g, labeled by the special 2-cocycles F on g. The main property of the constructed Lie algebras g F is that they admit Kostant-Adler-Symes scheme. Using them we obtain new integrable finite-dimensional Hamiltonian systems and new hierarchies of soliton equations.

Infinite-dimensional Lie algebras, classical r-matrices, and Lax operators: Two approaches

“…where the Poison brackets among the dynamical variables -components of the Lax operator repeat the commutation relations of the basis of g − r , i.e., have the following form: (31) and (33) are written in the tensor form as follows:…”

“…In the present paper we do not consider concrete examples of classical r-matrices r (u, v) and corresponding algebras g r preferring to work in the most general setting. Interested reader may consult our papers [28][29][30][31][32][33] for concrete examples in case of non-skew-symmetric r-matrices and our papers 36,37 for concrete examples in case of skew-symmetric classical r-matrices.…”