On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket

Details On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket

2011-04-14

arxiv.org/abs/1104.2722v1

Physics

Mathematical Physics

21 pages

Scientific paper

In our recent paper we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of PDE's associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the first Poisson structure of their hierarchy. Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental-Y.-P. Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we reprove it using a deformation formula for weak quasi-Miura transformation that relates our hierarchy of PDE's with its dispersionless limit.

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