73,168 research outputs found
Hydrodynamic chains and a classification of their Poisson brackets
Necessary and sufficient conditions for an existence of the Poisson brackets
significantly simplify in the Liouville coordinates. The corresponding
equations can be integrated. Thus, a description of local Hamiltonian
structures is a first step in a description of integrable hydrodynamic chains.
The concept of Poisson bracket is introduced. Several new Poisson brackets
are presented
Divergence operators and odd Poisson brackets
We define the divergence operators on a graded algebra, and we show that,
given an odd Poisson bracket on the algebra, the operator that maps an element
to the divergence of the hamiltonian derivation that it defines is a generator
of the bracket. This is the "odd laplacian", , of Batalin-Vilkovisky
quantization. We then study the generators of odd Poisson brackets on
supermanifolds, where divergences of graded vector fields can be defined either
in terms of berezinian volumes or of graded connections. Examples include
generators of the Schouten bracket of multivectors on a manifold (the
supermanifold being the cotangent bundle where the coordinates in the fibres
are odd) and generators of the Koszul-Schouten bracket of forms on a Poisson
manifold (the supermanifold being the tangent bundle, with odd coordinates on
the fibres).Comment: 27 pages; new Section 1, introduction and conclusion re-written,
typos correcte
A note on symplectic and Poisson linearization of semisimple Lie algebra actions
In this note we prove that an analytic symplectic action of a semisimple Lie
algebra can be locally linearized in Darboux coordinates. This result yields
simultaneous analytic linearization for Hamiltonian vector fields in a
neighbourhood of a common zero. We also provide an example of smooth
non-linearizable Hamiltonian action with semisimple linear part. The smooth
analogue only holds if the semisimple Lie algebra is of compact type. An
analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic
equivariant Weinstein splitting theorem for general Poisson manifolds are also
obtained in the Poisson setting.Comment: 13 page
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