Quadratic Poisson brackets compatible with an algebra structure
A. A. Balinsky, Yu. Burman
TL;DR
This paper investigates quadratic Poisson brackets on algebraic structures, introduces a compatibility notion, and provides criteria and classifications for such brackets, including coboundary cases, with various examples.
Contribution
It defines a new compatibility concept for quadratic Poisson brackets with algebra structures and offers an effective criterion and classification methods for these brackets.
Findings
A new notion of compatibility for quadratic Poisson brackets is introduced.
An effective criterion for compatibility is established.
Coboundary brackets within compatible brackets are classified and enumerated.
Abstract
Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among compatible brackets, a subclass of coboundary brackets is described, and such brackets are enumerated in a number of examples.
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