Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hopf algebras of planar binary trees: an operated algebra approach   Parallel to operated algebras built on top of planar rooted trees via the
grafting operator $B^+$, we introduce and study $\vee$-algebras and more
generally $\vee_\Omega$-algebras based on planar binary trees. Involving an
analogy of the Hochschild 1-cocycle condition, cocycle $\vee_\Omega$-bialgebras
(resp.~$\vee_\Omega$-Hopf algebras) are also introduced and their free objects
are constructed via decorated planar binary trees. As a special case, the
well-known Loday-Ronco Hopf algebra $H_{LR}$ is a free cocycle $\vee$-Hopf
algebra. By means of admissible cuts, a combinatorial description of the
coproduct $\Delta_{LR(\Omega)}$ on decorated planar binary trees is given, as
in the Connes-Kreimer Hopf algebra by admissible cuts.

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The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: solv-int