Daniele Valeri

Classical W-Algebras and Generalized Drinfeld-Sokolov Bi-Hamiltonian Systems Within the Theory of Poisson Vertex Algebras

We describe of the generalized Drinfeld-Sokolov Hamiltonian reduction for the construction of classical

Bethe Ansatz and the Spectral Theory of Affine Lie algebra–Valued Connections II: The Non Simply–Laced Case

{\mathcal{W}}

\mathcal W

W -algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.

-algebras

We construct an analogue of Dirac’s reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac’s reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld–Sokolov hierarchy.

Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras

We assess the ODE/IM correspondence for the quantum

Double Poisson vertex algebras and non-commutative Hamiltonian equations

{\mathfrak{g}}