Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalized bialgebras and triples of operads   We introduce the notion of generalized bialgebra, which includes the
classical notion of bialgebra (Hopf algebra) and many others. We prove that,
under some mild conditions, a connected generalized bialgebra is completely
determined by its primitive part. This structure theorem extends the classical
Poincar\'e-Birkhoff-Witt theorem and the Cartier-Milnor-Moore theorem, valid
for cocommutative bialgebras, to a large class of generalized bialgebras.
Technically we work in the theory of operads which permits us to give a
conceptual proof of our main theorem. It unifies several results, generalizing
PBW and CMM, scattered in the literature. We treat many explicit examples and
suggest a few conjectures.

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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int