Low dimensional bihamiltonian structures of topological type

TL;DR

This paper constructs specific low-dimensional bihamiltonian structures from classical W-algebras linked to nilpotent elements in Lie algebras, revealing their geometric and algebraic properties and connecting them to integrable systems.

Contribution

It introduces new local bihamiltonian structures of topological type derived from classical W-algebras for certain Lie algebra elements, expanding the understanding of their geometric structures.

Findings

Formed exact Poisson pencils with dispersionless limits

Defined logarithmic or trivial Dubrovin-Frobenius manifolds

Calculated constant central invariants

Abstract

We construct local bihamiltonian structures from classical $W$-algebras associated to non-regular nilpotent elements of regular semisimple type in Lie algebras of type $A_2$ and $A_3$. They form exact Poisson pencil, admit a dispersionless limit and their leading terms define logarithmic or trivial Dubrovin-Frobenius manifolds. We calculate the corresponding central invariants which are expected to be constants. In particular, we get Dubrovin- Frobenius manifolds associated to the focused Schr\"{o}dinger equation and Hurwitz space $M_{0;1,0}$ and the corresponding bihamiltonian structures of topological type.