Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Lie bracket of undirected curves on a surface   A Lie bracket defined on the linear span of the free homotopy classes of
undirected closed curves was discovered in stages passing through Thurston's
earthquake deformations, Wolpert's corresponding calculations with Hamiltonian
vector fields and Goldman's algebraic treatment of the latter leading to a Lie
bracket on the span of directed closed curves. The purpose of this work is to
deepen the understanding of the former Lie bracket which will be referred to as
the Thurston-Wolpert-Goldman Lie bracket or, briefly, the TWG bracket.
  We give a local direct geometric definition of the TWG bracket and use this
geometric point of view to prove three results: firstly, the center of the
TWG-bracket is the Lie sub algebra generated by the class of the trivial loop
and the classes of loops parallel to boundary components or punctures; secondly
the analogous result hold for the centers of the universal enveloping algebra
and of the symmetric algebra determined by the TWG Lie algebra; and thirdly, in
terms of the natural basis, the TWG bracket of two non-central curves is always
a linear combination of non-central curves. We also give a brief and more
illuminating proof of a known result, namely, the TWG bracket counts
intersection.
  We conclude by discussing substantial computer evidence suggesting an
unexpected and strong conjectural statement relating the intersection structure
of curves and the TWG bracket, namely, if the TWG bracket of two distinct
undirected curves is zero then these curve classes have disjoint
representatives.
  The main tools are basic hyperbolic geometry and Thurston's earthquake
theory.

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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: cs
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys