Mathematics

In this seminar I will report about an ongoing research with D. Valeri. It is known that classical affine W-algebras arise from a generalized Drinfeld-Sokolov reduction of a simple Lie algebra. Using the language of Poisson Vertex Algebra we are able to perform such reduction for any simple Lie algebra and nilpotent orbit, with limited computational effort. As already observed by O. Pavlyk and Y. Dinar for the D4 subregular case, the biHamiltonian structures of classical W-algebras do not admit a dispersionless limit when the reduction has been obtained from any but the principal nilpotent orbit. A suitable Dirac reduction, however, allows to perform the limit and to obtain a biHamiltonian pencil of hydrodynamic type. I will present Pavlyk and Dinar examples, as well as some new ones obtained by A type Lie algebras.