Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Poisson Algebras and 3D Superintegrable Hamiltonian Systems   Using a Poisson bracket representation, in 3D, of the Lie algebra
$\mathfrak{sl}(2)$, we first use highest weight representations to embed this
into larger Lie algebras. These are then interpreted as symmetry and conformal
symmetry algebras of the "kinetic energy", related to the quadratic Casimir
function. We then consider the potentials which can be added, whilst remaining
integrable, leading to families of separable systems, depending upon arbitrary
functions of a single variable. Adding further integrals, in the
superintegrable case, restricts these functions to specific forms, depending
upon a finite number of arbitrary parameters. The Poisson algebras of these
superintegrable systems are studied. The automorphisms of the symmetry algebra
of the kinetic energy are extended to the full Poisson algebra, enabling us to
build the full set of Poisson relations.

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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex