The Argument Shift Method and Maximal Commutative Subalgebras of Poisson Algebras
نویسندگان
چکیده
Let q be a Lie algebra over an algebraically closed field k of characteristic zero. The symmetric algebra S(q) has a natural structure of Poisson algebra, and our goal is to present a sufficient condition for the maximality of Poisson-commutative subalgebras of S(q) obtained by the argument shift method. Study of Poisson-commuttive subalgebras of S(q) has attracted much attention in the last years, see [2, 6, 14, 15, 16]. This is related to commutative subalgebras of the enveloping algebra U(q), fine questions of symplectic geometry, and integrable Hamiltonian systems. Commutative subalgebras of U(q) (e.g., the famous Gelfand-Zetlin subalgebra of U(sln)) occur in the theory of quantum integrable systems and have interesting application in representation theory.
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