Dispersionless limit for the BiHamiltonian structure of classical affine W-algebras: scouting for algebraic Frobenius manifolds
Matteo Casati (University of Loughborough)
Thursday 2nd March, 2017 16:00-17:00 Maths 522
It is known that classical affine W-algebras arise from a generalized Drinfeld-Sokolov reduction of simple affine Lie algebras. Using the language of Poisson vertex algebras, we are able to perform such reduction starting from any simple Lie algebra and nilpotent orbit, with limited computational effort.
As already observed by O. Pavlyk and Y. Dinar for the D_4 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. In some cases, the reduced pencil generates the principal hierarchy of a Frobenius manifold, with algebraic prepotential.
This is ongoing joint work with D. Valeri; for the computational aspects see arXiv/1603.05028