A twisted generalization of linear Poisson brackets of hydrodynamic type

Abstract

We study a twisted generalization of linear Poisson brackets of hydrodynamic type, in which the Lie bracket is replaced by a Hom–Lie bracket. With some natural additional conditions, such structures correspond to the Hom–Novikov algebras introduced by Yau, which is the twisted version of Balinsky–Novikov's approach of constructing a Lie algebra from a Novikov algebra. Certain central extensions of this twisted generalization of linear Poisson brackets of hydrodynamic type are obtained from the bilinear forms on their corresponding Hom–Novikov algebras satisfying some invariance conditions. Finally, we give some examples of the infinite-dimensional Hom–Lie algebras constructed from the Hom–Novikov algebras. In particular, there is an interesting twisted generalization of the Virasoro algebra.