Abstract
We prove that all classical affine W-algebras \U0001d4b2(\U0001d524; f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G2, one in F4, and five in E8.
Highlights
In order to define a Hamiltonian ODE one needs a Poisson algebra, that is a commutative associative algebra P, endowed with a Poisson bracket P ⊗ P → P, a ⊗ b → {a, b}, and an element h ∈ P, called a Hamiltonian function
Due to the first sesquilinearity axiom (i), the RHS of equation (1.2) is well defined
In the paper [Mag78] Magri proposed a simple algorithm, called nowadays the Lenard–Magri scheme, which allows one to prove that integrals of motion of a Hamiltonian PDE (1.3) are in involution, provided that the same equation can be written using a different Poisson structure H1(∂) in place of H(∂), and a different local funtional h1 in place of h. In such a case one obtains a bi-Hamiltonian hierarchy of PDEs
Summary
In order to define a Hamiltonian ODE one needs a Poisson algebra, that is a commutative associative algebra P, endowed with a Poisson bracket P ⊗ P → P, a ⊗ b → {a, b}, and an element h ∈ P, called a Hamiltonian function. In the paper [Mag78] Magri proposed a simple algorithm, called nowadays the Lenard–Magri scheme, which allows one to prove that integrals of motion of a Hamiltonian PDE (1.3) are in involution, provided that the same equation can be written using a different Poisson structure H1(∂) in place of H(∂), and a different local funtional h1 in place of h. In such a case one obtains a bi-Hamiltonian hierarchy of PDEs. See [BDSK09] for details. Throughout the paper the base field F is an algebraically closed field of characteristic zero
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