Reduction by stages for finite \(W\)-algebras (Q6590686)

In this paper, the authors study Hamiltonian reduction by stages for the Slodowy slices of a simple complex Lie algebra.\par \N\NLet us recall that the dual \(\mathfrak{g}^*\) of a Lie algebra \(\mathfrak{g}\) is a Poisson manifold and that one can construct a new Poisson structure by Hamiltonian reduction which is isomorphic to some subvariety of \(\mathfrak{g}^*\). In this work, the authors recall the construction of Slodowly slice as Hamiltonian reduction of the dual space of a simple Lie algebra. Then, given two nilpotent elements \(f_1\) and \(f_2\), the authors prove the reduction by stages. Then, they recall the construction of finite \(W\)-algebras and show their relations with the Slodowy slices.\N\NGeneralization of the Skryabin equivalence is then obtained and open questions for affine \(W\)-algebras are finally discussed.