Balinskii, A. A.; Novikov, S. P. Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras. (English. Russian original) Zbl 0606.58018 Sov. Math., Dokl. 32, 228-231 (1985); translation from Dokl. Akad. Nauk SSSR 283, 1036-1039 (1985). Poisson brackets on function spaces specified by formulae of the type \[ \{u^ i(x),u^ j(x)\}=g^{ij}(u(x)\delta '(x-y)+u^ k_ x b_ k^{ij}(u(x))\delta (x-y) \] are studied. In particular the case when \(g^{ij}\) is linear in \(u\) and \(b_ k^{ij}\) are constant is considered. The properties of such a bracket are related to the properties of the finite dimensional algebra with structure constants \(b_ k^{ij}\). Reviewer: D.J.Simms Cited in 4 ReviewsCited in 97 Documents MSC: 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures 17A30 Nonassociative algebras satisfying other identities 17B63 Poisson algebras 76A02 Foundations of fluid mechanics 58D25 Equations in function spaces; evolution equations Keywords:hydrodynamics; Frobenius algebra; Lie algebra; Poisson brackets PDFBibTeX XMLCite \textit{A. A. Balinskii} and \textit{S. P. Novikov}, Sov. Math., Dokl. 32, 228--231 (1985; Zbl 0606.58018); translation from Dokl. Akad. Nauk SSSR 283, 1036--1039 (1985)